Optimal Generalized Quasi-Polycyclic Codes over ${\mathbb{F}}_{\mathit{q}}+\mathit{u}{\mathbb{F}}_{\mathit{q}}$

Abstract

This paper develops a structural and constructive theory of right generalized quasi-polycyclic (GQPC) codes over the finite chain ring $R={\mathbb{F}}_q+u{\mathbb{F}}_q$ with $u^2=0$, extending the existing field-based GQPC framework to a ring-theoretic setting. Right GQPC codes over R are modeled as $R\left[x\right]$-submodules of direct products of polycyclic ambient algebras $R\left[x\right]/⟨x^{e_i}-{\alpha }_i\left(x\right)⟩$, induced by vectors ${\alpha }_i\in R^{e_i}$, thereby unifying right quasi-polycyclic and generalized quasi-cyclic codes over $R.$ Under explicit and verifiable factorization conditions on the defining polynomials, we establish a Chinese Remainder Theorem decomposition that reduces right GQPC codes to collections of shorter codes over finite chain-ring extensions of R. This decomposition yields a characterization of ρ-generator right GQPC codes and leads to a canonical normalized generating set with an upper-triangular structure. As a consequence, we obtain an explicit rank formula in terms of the diagonal generator polynomials, together with an effective normalization algorithm. To demonstrate the coding-theoretic impact of the framework, we combine these structural results with a distance-compatible Gray map $\Phi :R\to {\mathbb{F}}_q^2$ and construct new q-ary linear codes from 2-generator right GQPC codes of index 2 over R. For $q=9$ and $q=3$, the resulting Gray images attain optimal or near-optimal parameters with respect to the best-known bounds, confirming that right GQPC codes over ${\mathbb{F}}_q+u{\mathbb{F}}_q$ constitute a robust and effective ring-based source of high-quality linear codes.

1. Introduction

Classical cyclic, constacyclic, and quasi-cyclic (QC) codes over a finite field ${\mathbb{F}}_q$ admit a well-established algebraic description in terms of ideals and submodules of polynomial quotient rings such as ${\mathbb{F}}_q\left[x\right]/⟨x^n-1⟩$ and their constacyclic analog; see, for example, ref. [1] and the quasi-twisted framework in [2]. Polycyclic and quasi-polycyclic (QPC) codes extend these classical families by replacing the cyclic shift with a more general polycyclic shift induced by an arbitrary vector or, equivalently, by a polynomial constraint [3,4,5]. This viewpoint provides a unified framework encompassing cyclic, constacyclic, and several other shift-invariant code families.

Building on this idea, generalized quasi-cyclic (GQC) and generalized quasi-polycyclic (GQPC) codes have been introduced as flexible algebraic constructions that unify many classical codes while offering additional degrees of freedom for code design. Over finite fields, these families have proved particularly effective for the systematic construction of good and optimal linear codes [6,7,8,9]. In particular, right GQPC codes over ${\mathbb{F}}_q$ admit a rich module-theoretic structure, normalized generating sets, and explicit dimension formulas, which have been successfully combined with computer search techniques to produce new record-breaking codes [9].

In parallel with these developments over fields, linear codes over finite rings have attracted considerable attention, motivated in part by the observation that many high-quality nonbinary linear codes arise naturally as Gray images of structured ring-linear codes. Among the most studied ambient rings in this context are finite chain and semi-local rings that admit both a tractable ideal structure and distance-compatible Gray maps. A particularly important example is the finite chain ring

$$R={\mathbb{F}}_q+u{\mathbb{F}}_q,u^2=0,$$

(1)

which is a commutative local ring of cardinality $\left|R\right|=q^2$, with maximal ideal $⟨u⟩$ and residue field $R/⟨u⟩\cong {\mathbb{F}}_q$. Codes over R exhibit a rich algebraic structure and, via suitable ${\mathbb{F}}_q$-linear Gray maps, give rise to families of ${\mathbb{F}}_q$-linear codes with excellent parameters. This has led to extensive work on cyclic, constacyclic, polycyclic, and generalized quasi-cyclic type codes over R and related rings; see, for instance, refs. [10,11,12] and the references therein.

What distinguishes right GQPC codes from these classical families is the nature of their defining shift structure. Cyclic, constacyclic, and quasi-cyclic codes over rings are each characterized by invariance under a single shift operator. Right GQPC codes, by contrast, are defined by multiple independent polycyclic constraints encoded in distinct polynomials $x^{e_i}-{\alpha }_i\left(x\right)$. This structural difference has two immediate consequences. First, the ambient space becomes a product ring $R_{{\alpha }_1,e_1}\times \cdots \times R_{{\alpha }_{\mathsf{\ell }},e_{\mathsf{\ell }}}$, which admits a finer CRT decomposition than the single-block settings typical of classical constructions. Second, the existence of numerous constraints provides enhanced design flexibility: each block can be customized separately while maintaining the overall algebraic consistency of the code, potentially resulting in parameter combinations unattainable with single-shift families.

Despite these advances, a clear gap remains between the field-based theory of right GQPC codes and the ring-based literature. While quasi-polycyclic and skew quasi-polycyclic codes over finite chain rings have been studied in [8,13], and generalized quasi-cyclic codes over $R={\mathbb{F}}_q+u{\mathbb{F}}_q$ have been analyzed in [10], a systematic treatment of right generalized quasi-polycyclic codes directly over the chain ring R, parallel to the field-based theory of [9], does not appear to be available. To the best of our knowledge, right GQPC codes have previously been investigated only over finite fields, and no comprehensive ring-theoretic framework has been developed for this class of codes over ${\mathbb{F}}_q+u{\mathbb{F}}_q.$

Beyond their theoretical significance, families of quasi-polycyclic codes have practical relevance in communication systems, data storage, and quantum error-correcting code constructions [3,14]. Such structured algebraic frameworks enable systematic parameter adjustment and efficient encoding, while Gray map constructions produce q-ary linear codes with competitive parameters suitable for these applications.

The purpose of this paper is to address this gap by initiating a systematic study of right generalized quasi-polycyclic codes over the chain ring R and by developing a ring-theoretic framework that parallels and extends the existing field-based theory. We model right GQPC codes of block length (e₁, …, e_ℓ) and index ℓ as $R\left[x\right]$-submodules of the ambient module

$$\mathcal{S}=R_{{\alpha }_1,e_1}\times \cdots \times R_{{\alpha }_{\mathsf{\ell }},e_{\mathsf{\ell }}},R_{{\alpha }_i,e_i}=R\left[x\right]/⟨x^{e_i}-{\alpha }_i\left(x\right)⟩,$$

(2)

where the polynomials $x^{e_i}-{\alpha }_i\left(x\right)$ encode right polycyclic shifts induced by vectors ${\alpha }_i\in R^{e_i}$. This ambient-module approach extends to the chain-ring setting, the methods used for polycyclic and GQPC codes over finite fields [4,5,9].

Our first main contribution is a structural characterization of right generalized quasi-polycyclic (GQPC) codes over R as $R\left[x\right]$-submodules of the ambient module $\mathcal{S}$. Under mild and verifiable factorization hypotheses on the defining polynomials $x^{e_i}-{\alpha }_i\left(x\right)$, we establish a Chinese Remainder Theorem (CRT) decomposition of $\mathcal{S}$ and, consequently, of any right GQPC code $C\subseteq \mathcal{S}$ into a direct sum of constituent codes over finite chain-ring extensions of R. This result extends to the chain-ring setting, the CRT-based decompositions known for polycyclic and GQC/GQPC codes over finite fields and certain classes of rings [4,6,9,10]. Within this framework, we characterize $\rho$-generator right GQPC codes in terms of their constituent modules and show that the minimal number of polynomial generators coincides with the maximum rank among the constituents.

A second contribution is the introduction of normalized generating sets for right GQPC codes over R, given by upper-triangular families of generator polynomials satisfying natural divisibility and degree conditions. We provide an explicit normalization procedure that transforms an arbitrary generating set into a normalized one, thereby extending the normalized form theory for GQC codes developed in [7,9] to the present ring-theoretic setting. This yields a transparent and explicit rank formula for right GQPC codes in terms of the degrees of the diagonal generator polynomials. When combined with a distance-compatible Gray map $\Phi :R^N\to {\mathbb{F}}_q^{2N}$, these results lead to precise dimension formulas for the associated ${\mathbb{F}}_q$-linear codes.

Finally, we illustrate the theory by constructing explicit families of 2-generator right GQPC codes of index 2 over R. By combining the structural results with systematic computer searches using Magma [15], we obtain Gray images whose parameters are optimal or near-optimal with respect to the best-known bounds reported in [16]. These examples demonstrate that right GQPC codes over ${\mathbb{F}}_q+u{\mathbb{F}}_q$ constitute a genuinely new and effective ring-based source of high-quality linear codes, extending constructions that were previously available only in the field setting.

The paper is organized as follows. Section 2 recalls basic facts on the chain ring R, right polycyclic codes over R, and Gray maps. Section 3 introduces right GQPC codes over R and establishes the CRT decomposition into constituent codes. Section 4 develops normalized generating sets and derives rank and dimension formulas. Section 5 presents explicit constructions and optimal or near-optimal Gray-image codes.

2. Preliminaries

Throughout the paper, let p be a prime, let $q=p^m$ with $m\ge 1$, and denote by ${\mathbb{F}}_q$ the finite field with q elements. We work over the commutative ring

$$R={\mathbb{F}}_q+u{\mathbb{F}}_q=\{a+ub\mid a,b\in {\mathbb{F}}_q\},u^2=0.$$

(3)

The ring R is a finite commutative local chain ring with maximal ideal $\mathfrak{m}=⟨u⟩$, nilpotency index 2, and residue field $R/\mathfrak{m}\cong {\mathbb{F}}_q$. Its group of units is given by

$$R^{\times }=\{a+ub\in R\mid a\in {\mathbb{F}}_q^{\times }\}.$$

(4)

The lattice of ideals of R consists precisely of

$$\left\{0\right\}\subset ⟨u⟩\subset R,$$

(5)

and hence R is a principal ideal ring. Moreover, R is a finite Frobenius ring and therefore admits a nondegenerate associative bilinear form. As a consequence, duality theory and MacWilliams-type identities for linear codes over R closely parallel those over finite fields.

As an ${\mathbb{F}}_q$-vector space, R has dimension 2 with basis $\{1,u\}$. Thus $R^n$ is a free ${\mathbb{F}}_q$-module of rank $2n$ and a free R-module of rank n. This double module structure underlies the construction of Gray maps and allows one to systematically relate R-linear codes to ${\mathbb{F}}_q$-linear codes, a mechanism widely exploited for cyclic, polycyclic, and GQC-type codes over R; see, for instance, refs. [10,11].

A linear code of length n over R is an R-submodule $C\subseteq R^n$. Such a code admits a generator matrix $G\in R^{k\times n}$, where $k={rk}_R\left(C\right)$ denotes the R-rank of C. Every element $c=(c_0,\dots ,c_{n-1})\in R^n$ can be written uniquely as $c_i=a_i+u{b}_i$ with $a_i,b_i\in {\mathbb{F}}_q$. We employ the standard Gray map

$$\Phi :R⟶{\mathbb{F}}_q^2,\Phi (a+ub)=(b,a+b),$$

(6)

extended coordinatewise to an ${\mathbb{F}}_q$-linear bijection $\Phi :R^n\to {\mathbb{F}}_q^{2n}$. For a code $C\subseteq R^n$, we define its Gray image by

$$\Phi \left(C\right)=\{\Phi \left(c\right)\mid c\in C\}\subseteq {\mathbb{F}}_q^{2n}.$$

(7)

Distances on $R^n$ are measured via the Gray map. For $c\in R^n$, define its weight by

$$w\left(c\right)=w_H(\Phi \left(c\right)),$$

(8)

and for a nonzero code C,

$$d\left(C\right)=min\left\{w\right(c)\mid c\in C\setminus \{0\left\}\right\}.$$

(9)

With this convention, $\Phi$ is an isometry from $(R^n,w)$ onto $({\mathbb{F}}_q^{2n},w_H)$. Consequently, the minimum distance of C coincides with that of its Gray image $\Phi \left(C\right)$, allowing one to transfer distance bounds directly between the ring and field settings [17,18,19].

Let $n\in {\mathbb{Z}}^{+}$ and let $\alpha =({\alpha }_0,\dots ,{\alpha }_{n-1})\in R^n$ with ${\alpha }_0\in R^{\times }$. Define the right polycyclic shift induced by $\alpha$ as the map

$$P_r^{\alpha }:R^n⟶R^n,(y_0,\dots ,y_{n-1})⟼y_{n-1}\alpha +(0,y_0,\dots ,y_{n-2}),$$

(10)

where $y_{n-1}\alpha$ denotes componentwise multiplication of the scalar $y_{n-1}$ with the vector $\alpha$.

An R-linear code $C\subseteq R^n$ is called a right polycyclic (RPC) code induced by α if $P_r^{\alpha }\left(c\right)\in C$ for all $c\in C$.

As in the field case, to the vector $\alpha$ we associate the polynomial

$$\alpha \left(x\right)={\alpha }_0+{\alpha }_{1}x+\dots +{\alpha }_{n-1}{x}^{n-1}\in R\left[x\right],P_{n,\alpha }\left(x\right)=x^n-\alpha \left(x\right).$$

(11)

We then consider the quotient ring

$$R_{\alpha ,n}=R\left[x\right]/⟨P_{n,\alpha }\left(x\right)⟩.$$

(12)

The map

$$\phi :R^n⟶R_{\alpha ,n},(c_0,\dots ,c_{n-1})⟼c_0+c_{1}x+\cdots +c_{n-1}{x}^{n-1},$$

(13)

is an R-module isomorphism. Under this identification, multiplication by x in $R_{\alpha ,n}$ corresponds exactly to the right polycyclic shift $P_r^{\alpha }$, that is,

$$x·\phi \left(c\right)⟷P_r^{\alpha }\left(c\right).$$

Hence, an R-linear code $C\subseteq R^n$ is an RPC code induced by $\alpha$ if and only if $\phi \left(C\right)$ is an $R\left[x\right]$-submodule, equivalently a left ideal, of $R_{\alpha ,n}$.

Throughout the paper, we assume that the polynomial $P_{n,\alpha }\left(x\right)$ is monic and regular in $R\left[x\right]$, that is, it is not a zero divisor. Under this hypothesis, $R_{\alpha ,n}$ is a finite Frobenius ring. We further restrict to the case where $R_{\alpha ,n}$ is a principal ideal ring. Under these assumptions, every RPC code in $R^n$ is generated by a single polynomial $g\left(x\right)\in R\left[x\right]$ satisfying

$$⟨g\left(x\right)⟩=\phi \left(C\right)\mathrm{and}g\left(x\right)\mid P_{n,\alpha }\left(x\right)\mathrm{in}R\left[x\right],$$

exactly as in the polycyclic theory over finite fields and over more general finite chain rings; see, for example, refs. [4,8,11].

3. Algebraic Structures of GQPC Codes over $\mathit{R}={\mathbb{F}}_{\mathit{q}}+\mathit{u}{\mathbb{F}}_{\mathit{q}}$

In this section, we extend generalized quasi-polycyclic (GQPC) codes from the field setting to the finite chain ring $R={\mathbb{F}}_q+u{\mathbb{F}}_q$ with $u^2=0$. We adopt an ambient-module viewpoint: right GQPC codes are precisely $R\left[x\right]$-submodules of a direct product of polycyclic ambient algebras $R\left[x\right]/⟨x^{e_i}-{\alpha }_i\left(x\right)⟩$. Under suitable factorization hypotheses on the defining polynomials, we establish a Chinese remainder theorem (CRT) decomposition into constituent codes over finite chain-ring extensions of R. This decomposition is the key tool underlying generator bounds, rank formulas, and the constructive examples presented in later sections.

Let $e_1,\dots ,e_{\mathsf{\ell }}\in {\mathbb{Z}}^{+}$ be block lengths and set

$$N=\sum _{i=1}^{\mathsf{\ell }}{e}_i.$$

(14)

For each $i\in \{1,\dots ,\mathsf{\ell }\}$, fix a vector ${\alpha }_i=({\alpha }_{i,0},\dots ,{\alpha }_{i,e_i-1})\in R^{e_i}$ with ${\alpha }_{i,0}\in R^{\times }$. Associate the polynomial

$${\alpha }_i\left(x\right)=\sum _{k=0}^{e_i-1}{\alpha }_{i,k}{x}^k\in R\left[x\right],P_i\left(x\right)=P_{e_i,{\alpha }_i}\left(x\right)=x^{e_i}-{\alpha }_i\left(x\right)\in R\left[x\right].$$

(15)

The i-th ambient algebra is

$$R_{{\alpha }_i,e_i}=R\left[x\right]/⟨P_i\left(x\right)⟩.$$

(16)

We consider the direct product

$$\mathcal{S}=\prod _{i=1}^{\mathsf{\ell }}{R}_{{\alpha }_i,e_i}=R_{{\alpha }_1,e_1}\times \cdots \times R_{{\alpha }_{\mathsf{\ell }},e_{\mathsf{\ell }}},$$

(17)

which is naturally an $R\left[x\right]$-module under componentwise multiplication:

$$f\left(x\right)·(s_1,\dots ,s_{\mathsf{\ell }})=(f\left(x\right)s_1,\dots ,f\left(x\right)s_{\mathsf{\ell }}),f\left(x\right)\in R\left[x\right],s_i\in R_{{\alpha }_i,e_i}.$$

(18)

We now formalize the polynomial model for block vectors. For each i, define the R-linear map

$${\phi }_i:R^{e_i}⟶R_{{\alpha }_i,e_i},(c_0,\dots ,c_{e_i-1})⟼\sum _{k=0}^{e_i-1}{c}_k{x}^{k}mod⟨P_i\left(x\right)⟩.$$

(19)

Then define

$$M={\phi }_1\times \cdots \times {\phi }_{\mathsf{\ell }}:R^{e_1}\times \cdots \times R^{e_{\mathsf{\ell }}}⟶\mathcal{S}.$$

(20)

Lemma 1.

The map M is an isomorphism of R-modules. Moreover, for each i, every element of $R_{{\alpha }_i,e_i}$ has a unique representative in $R\left[x\right]$ of degree less than $e_i$, and thus ${\phi }_i$ is bijective.

Proof. 

Since $P_i\left(x\right)$ is monic of degree $e_i$, every coset in $R\left[x\right]/⟨P_i\left(x\right)⟩$ has a unique representative $a_0+\cdots +a_{e_i-1}{x}^{e_i-1}$. Thus ${\phi }_i$ is surjective and injective. The map is R-linear by construction. Hence ${\phi }_i$ is an R-module isomorphism. Taking the product over i yields the claim for M. □

For each i, define the right polycyclic shift induced by ${\alpha }_i$ by

$$P_r^{{\alpha }_i}:R^{e_i}\to R^{e_i},(y_0,\dots ,y_{e_i-1})\mapsto y_{e_i-1}{\alpha }_i+(0,y_0,\dots ,y_{e_i-2}),$$

(21)

where $y_{e_i-1}{\alpha }_i=(y_{e_i-1}{\alpha }_{i,0},\dots ,y_{e_i-1}{\alpha }_{i,e_i-1})$.

Lemma 2.

For each i, multiplication by x in the quotient ring $R_{{\alpha }_i,e_i}$ corresponds to the right polycyclic shift $P_r^{{\alpha }_i}$ on $R^{e_i}$. More precisely, for any $c=(c_0,\dots ,c_{e_i-1})\in R^{e_i}$,

$$x·{\phi }_i\left(c\right)={\phi }_i\left(P_r^{{\alpha }_i}\left(c\right)\right).$$

Proof. 

Write ${\phi }_i\left(c\right)=c\left(x\right)={\sum }_{k=0}^{e_i-1}{c}_k{x}^k$. In $R_{{\alpha }_i,e_i}$ we have the relation $x^{e_i}\equiv {\alpha }_i\left(x\right)$ modulo $P_i\left(x\right)$. Hence

$$xc\left(x\right)=\sum _{k=0}^{e_i-2}{c}_k{x}^{k+1}+c_{e_i-1}{x}^{e_i}\equiv \sum _{k=0}^{e_i-2}{c}_k{x}^{k+1}+c_{e_i-1}{\alpha }_i\left(x\right).$$

Collecting coefficients of $1,x,\dots ,x^{e_i-1}$ gives exactly the coordinate vector $P_r^{{\alpha }_i}\left(c\right)$, proving the identity. □

Definition 1.

A linear code $C\subseteq R^{e_1}\times \cdots \times R^{e_{\mathsf{\ell }}}$ is called a right generalized quasi-polycyclic (GQPC) code over R with block lengths $(e_1,\dots ,e_{\mathsf{\ell }})$ (index ℓ) if it is invariant under the blockwise right polycyclic shift, i.e., for every $c=(c^{\left(1\right)},\dots ,c^{\left(\mathsf{\ell }\right)})\in C$,

$$(P_r^{{\alpha }_1}\left(c^{\left(1\right)}\right),\dots ,P_r^{{\alpha }_{\mathsf{\ell }}}\left(c^{\left(\mathsf{\ell }\right)}\right))\in C.$$

The next result is the fundamental equivalence between the shift definition and the polynomial $R\left[x\right]$-module model.

Proposition 1.

A linear code $C\subseteq R^{e_1}\times \cdots \times R^{e_{\mathsf{\ell }}}$ is a right GQPC code if and only if $M\left(C\right)\subseteq \mathcal{S}$ is an $R\left[x\right]$-submodule. Consequently, right GQPC codes over R are in bijection with $R\left[x\right]$-submodules of $\mathcal{S}$.

Proof. 

Assume C is right GQPC. For $c\in C$, invariance implies that the shifted block vector $\tilde{c}\in C$. By Lemma 2, $M\left(\tilde{c}\right)=x·M\left(c\right)$. Hence $M\left(C\right)$ is closed under multiplication by x. Since $M\left(C\right)$ is an R-submodule and $R\left[x\right]$ is generated over R by x, it follows that $M\left(C\right)$ is an $R\left[x\right]$-submodule of $\mathcal{S}$.

Conversely, if $M\left(C\right)$ is an $R\left[x\right]$-submodule, then $x·M\left(c\right)\in M\left(C\right)$ for all $c\in C$. Applying $M^{-1}$ and Lemma 2 shows the blockwise shift of c lies in C. Thus C is right GQPC. □

Remark 1.

The definition contains several familiar families: (i) if $\mathsf{\ell }=1$ then right GQPC codes coincide with right polycyclic (RPC) codes of length $e_1$; (ii) if ${\alpha }_i=(1,0,\dots ,0)$ for all i, then $P_i\left(x\right)=x^{e_i}-1$ and we recover generalized quasi-cyclic (GQC) codes over R; (iii) if $e_1=\cdots =e_{\mathsf{\ell }}$, then one obtains right quasi-polycyclic (QPC) codes of index ℓ.

We now turn to the CRT decomposition. Since R is a chain ring with maximal ideal $⟨u⟩$, we write $\overline{(·)}:R\left[x\right]\to (R/⟨u⟩)\left[x\right]\cong {\mathbb{F}}_q\left[x\right]$ for coefficientwise reduction modulo u. A typical sufficient condition enabling CRT methods over R is that the reductions $\overline{P_i}\left(x\right)$ are squarefree, in which case coprime factorizations lift from ${\mathbb{F}}_q\left[x\right]$ to $R\left[x\right]$.

Lemma 3.

Let $f\left(x\right)\in R\left[x\right]$ be monic and suppose $\overline{f}\left(x\right)=\overline{a}\left(x\right)\overline{b}\left(x\right)$ in ${\mathbb{F}}_q\left[x\right]$ with $gcd(\overline{a},\overline{b})=1$. Then there exist monic polynomials $a\left(x\right), b\left(x\right)\in R\left[x\right]$ such that $f\left(x\right)=a\left(x\right)b\left(x\right)$ and $\overline{a}=amodu$, $\overline{b}=bmodu$. Moreover, a and b are coprime in $R\left[x\right]$.

Proof. 

Since $u^2=0$, this is the standard Hensel lifting step for the extension $0\to u{\mathbb{F}}_q\to R\to {\mathbb{F}}_q\to 0$. Coprimeness of $\overline{a}$ and $\overline{b}$ yields $\overline{s}\overline{a}+\overline{t}\overline{b}=1$ in ${\mathbb{F}}_q\left[x\right]$. Lifting $\overline{s},\overline{t}$ to $s,t\in R\left[x\right]$ gives $sa+tb\equiv 1(modu)$. A correction term in $uR\left[x\right]$ adjusts the product to match f, and the uniqueness of the correction modulo u follows from $u^2=0$. Coprimeness in $R\left[x\right]$ follows because any common divisor would reduce to a common divisor of $\overline{a}$ and $\overline{b}$. □

To state the decomposition uniformly across blocks, we impose the following factorization hypothesis, which is the natural analog of the “common basic factor set” assumption used for GQC/GQPC codes over fields.

From now on, we work under the following assumptions.

H1. 

Each defining polynomial $P_i\left(x\right)\in R\left[x\right]$ is monic and regular, so that $R_{{\alpha }_i,e_i}=R\left[x\right]/⟨P_i\left(x\right)⟩$ is a finite Frobenius ring.

H2. 

Each ambient ring $R_{{\alpha }_i,e_i}$ is a principal ideal ring. Consequently, every right polycyclic (RPC) code over $R_{{\alpha }_i,e_i}$ is generated by a single polynomial divisor of $P_i\left(x\right)$.

H3. 

The reductions $\overline{P_i}\left(x\right)\in {\mathbb{F}}_q\left[x\right]$ are squarefree and admit a common basic factor set $\{g_1\left(x\right),\dots ,g_s\left(x\right)\}$, where the $g_j\left(x\right)$ are pairwise coprime monic polynomials in $R\left[x\right]$ lifting irreducible factors over ${\mathbb{F}}_q\left[x\right]$.

These hypotheses ensure that the Chinese remainder decomposition and normalized generating set theory developed below apply uniformly across all blocks.

Remark 2.

By (H3), there exist monic polynomials $g_1\left(x\right),\dots ,g_s\left(x\right)\in R\left[x\right]$ which are pairwise coprime in $R\left[x\right]$ such that for each $i\in \{1,\dots ,\mathsf{\ell }\}$, we assume that

$$P_i\left(x\right)=\prod _{j=1}^s{g}_j{\left(x\right)}^{m_{i,j}},m_{i,j}\in \{0,1\}.$$

(22)

Equivalently, each $g_j$ either divides $P_i$ (then $m_{i,j}=1$) or does not occur (then $m_{i,j}=0$), and the family $\{g_1,\dots ,g_s\}$ records all pairwise-coprime basic factors appearing among the $P_i$’s.

For each j, define the finite R-algebra

$${\Gamma }_j=R\left[x\right]/⟨g_j\left(x\right)⟩.$$

(23)

For each pair $(i,j)$, define

$${\Gamma }_{i,j}=\left\{\begin{array}{cc}{\Gamma }_j,\hfill & \mathrm{if}{m}_{i,j}=1,\hfill \\ \left\{0\right\},\hfill & \mathrm{if}{m}_{i,j}=0.\hfill \end{array}\right.$$

(24)

Under Remark 2, CRT yields an explicit decomposition of each block ambient ring $R_{{\alpha }_i,e_i}$, and hence of $\mathcal{S}$.

Proposition 2.

Under Remark 2, for each i there is an $R\left[x\right]$-module isomorphism

$${\theta }_i:R_{{\alpha }_i,e_i}⟶\underset{j=1}{\overset{s}{⨁}}{\Gamma }_{i,j},$$

(25)

and consequently an $R\left[x\right]$-module isomorphism

$$\Theta ={\theta }_1\times \cdots \times {\theta }_{\mathsf{\ell }}:\mathcal{S}⟶\underset{j=1}{\overset{s}{⨁}}\left({\Gamma }_{1,j}\times \cdots \times {\Gamma }_{\mathsf{\ell },j}\right).$$

(26)

Proof. 

Fix i. Since the $g_j$ are pairwise coprime, the product ${\prod }_{j:m_{i,j}=1}{g}_j\left(x\right)$ is a coprime factorization of $P_i\left(x\right)$. The Chinese remainder theorem gives

$$R\left[x\right]/⟨P_i\left(x\right)⟩\cong \underset{j:m_{i,j}=1}{⨁}R\left[x\right]/⟨g_j\left(x\right)⟩,$$

which is (25) upon inserting the zero components for $m_{i,j}=0$. The global isomorphism (26) is obtained by taking the product over i. □

Proposition 3.

Let $g\left(x\right)\in R\left[x\right]$ be monic and set $\Gamma =R\left[x\right]/⟨g\left(x\right)⟩$. Let $\overline{g}\left(x\right)\in {\mathbb{F}}_q\left[x\right]$ be the reduction of $g\left(x\right)$ modulo $⟨u⟩$. Assume that $g\left(x\right)$ is regular in $R\left[x\right]$ (i.e., not a zero divisor). Then the following are equivalent:

(i) 

Γ is a finite chain ring with maximal ideal $⟨u⟩\Gamma$ and residue field $\Gamma /⟨u⟩\Gamma \cong {\mathbb{F}}_{q^{deg\left(\overline{g}\right)}}$.

(ii) 

$\overline{g}\left(x\right)$ is irreducible in ${\mathbb{F}}_q\left[x\right]$.

(iii) 

$g\left(x\right)$ is basic irreducible, i.e., $\overline{g}\left(x\right)$ is irreducible, and $g\left(x\right)$ is a Hensel lift of $\overline{g}\left(x\right)$ from ${\mathbb{F}}_q\left[x\right]$ to $R\left[x\right]$.

In this case, Γ is a free R-module of rank $deg\left(g\right)=deg\left(\overline{g}\right)$, and

$$|\Gamma |={\left|R\right|}^{deg\left(g\right)}=q^{2deg\left(g\right)}.$$

Proof. 

Since g is monic, $\Gamma$ is a finite R-algebra and $\Gamma /⟨u⟩\Gamma \cong {\mathbb{F}}_q\left[x\right]/⟨\overline{g}\left(x\right)⟩$. If $\overline{g}$ is irreducible, then ${\mathbb{F}}_q\left[x\right]/⟨\overline{g}⟩$ is a field, hence $\Gamma /⟨u⟩\Gamma$ is a field. Therefore $\Gamma$ is local with maximal ideal $⟨u⟩\Gamma$, because the only nonunits are precisely the elements whose residue modulo u is zero. Since ${⟨u⟩}^2=0$ in R, we have ${(⟨u⟩\Gamma )}^2=0$, so the ideals form the chain

$$\left\{0\right\}\subset ⟨u⟩\Gamma \subset \Gamma ,$$

hence $\Gamma$ is a finite chain ring. This proves $\left(ii\right)\Rightarrow \left(i\right)$, and also gives the residue field $\Gamma /⟨u⟩\Gamma \cong {\mathbb{F}}_{q^{deg\left(\overline{g}\right)}}$.

Conversely, if $\Gamma$ is a chain ring with maximal ideal $⟨u⟩\Gamma$, then the residue ring $\Gamma /⟨u⟩\Gamma$ must be a field, hence ${\mathbb{F}}_q\left[x\right]/⟨\overline{g}⟩$ is a field, so $\overline{g}$ is irreducible. Thus $\left(i\right)\Rightarrow \left(ii\right)$. The equivalence $\left(ii\right)\iff \left(iii\right)$ is by definition of basic irreducibility. Finally, the R-basis $\{1,x,\dots ,x^{deg\left(g\right)-1}\}$ of $\Gamma$ yields the claimed rank and size formulas. □

Theorem  1.

Assume Remark 2 and let

$$\Theta :\mathcal{S}⟶\underset{j=1}{\overset{s}{⨁}}\left({\Gamma }_{1,j}\times \cdots \times {\Gamma }_{\mathsf{\ell },j}\right)$$

(27)

be the CRT isomorphism in (26). Equip $R^N\cong R^{e_1}\times \cdots \times R^{e_{\mathsf{\ell }}}$ with the Euclidean inner product $⟨x,y⟩={\sum }_{t=1}^N{x}_t{y}_t,$ and transport it to $\mathcal{S}$ via the identification M. For a code $C\subseteq \mathcal{S}$, let $C^{\perp }$ denote its Euclidean dual. If $C\subseteq \mathcal{S}$ is a right GQPC code and

$$\Theta \left(C\right)=\underset{j=1}{\overset{s}{⨁}}{C}_j$$

(28)

is its constituent decomposition as in Proposition 4, then

$$\Theta \left(C^{\perp }\right)=\underset{j=1}{\overset{s}{⨁}}{C}_j^{\perp },$$

(29)

where $C_j^{\perp }$ denotes the Euclidean dual of $C_j$ inside ${\Gamma }_{1,j}\times \cdots \times {\Gamma }_{\mathsf{\ell },j}$ with respect to the induced componentwise inner product. In particular,

$$C^{\perp }\cong \underset{j=1}{\overset{s}{⨁}}{C}_j^{\perp }$$

(30)

as R-modules, and $C^{\perp }$ is again a right GQPC code.

Proof. 

Via the CRT idempotents ${\epsilon }_1,\dots ,{\epsilon }_s$, the ambient module $\mathcal{S}$ decomposes as a direct sum of R-modules

$$\mathcal{S}=\underset{j=1}{\overset{s}{⨁}}{\epsilon }_j\mathcal{S},$$

and under the isomorphism $\Theta$, each summand ${\epsilon }_j\mathcal{S}$ is identified with ${\Gamma }_{1,j}\times \cdots \times {\Gamma }_{\mathsf{\ell },j}$.

The Euclidean inner product on $R^N$, transported to $\mathcal{S}$, is the standard coordinatewise dot product across all blocks. Consequently, for $x=(x_1,\dots ,x_s)$ and $y=(y_1,\dots ,y_s)$ with $x_j,y_j\in {\epsilon }_j\mathcal{S}$, we have

$$⟨x,y⟩=\sum _{j=1}^s⟨x_j,y_j⟩,$$

where each $⟨x_j,y_j⟩$ is the induced Euclidean inner product on ${\epsilon }_j\mathcal{S}\cong {\Gamma }_{1,j}\times \cdots \times {\Gamma }_{\mathsf{\ell },j}$.

Now write $\Theta \left(C\right)={⨁}_{j=1}^s{C}_j$ with $C_j\subseteq {\epsilon }_j\mathcal{S}$. Let $z=(z_1,\dots ,z_s)\in \mathcal{S}$. Then $z\in C^{\perp }$ if and only if

$$0=⟨z,c⟩=\sum _{j=1}^s⟨z_j,c_j⟩\mathrm{for}\mathrm{all}c=(c_1,\dots ,c_s)\in C,$$

with $c_j\in C_j$. This holds if and only if $⟨z_j,c_j⟩=0$ for all $c_j\in C_j$ and all j, that is, $z_j\in C_j^{\perp }$ for each j.

Therefore,

$$C^{\perp }=\underset{j=1}{\overset{s}{⨁}}{C}_j^{\perp },$$

and applying $\Theta$ yields (29).

Finally, since each $C_j^{\perp }$ is stable under the induced $R\left[x\right]$-action, $\Theta \left(C^{\perp }\right)$ is an $R\left[x\right]$-submodule of the CRT ambient module. By Proposition 1, this implies that $C^{\perp }$ is again a right GQPC code. □

Corollary 1.

Under the hypotheses of Theorem 1, we have

$$\left|C\right|·|C^{\perp }{|=|R|}^N,\mathit{and}|C_j|·|C_j^{\perp }|=|{\Gamma }_{1,j}\times \cdots \times {\Gamma }_{\mathsf{\ell },j}|$$

for each j. In particular, if $C_j\cong {\Gamma }_j^{r_j}$ is free over ${\Gamma }_j$, then

$$|C_j|=|{\Gamma }_j{|}^{r_j},|C_j^{\perp }|=|{\Gamma }_j{|}^{\mathsf{\ell }-r_j}(\mathit{when}\mathit{all}{m}_{i,j}=1),$$

and analogous formulas hold when some components ${\Gamma }_{i,j}$ vanish.

It is useful (and later essential) to record the CRT idempotents explicitly. For fixed i, set

$$G_i\left(x\right)=\prod _{j:m_{i,j}=1}{g}_j\left(x\right)=P_i\left(x\right),G_{i,j}\left(x\right)={\displaystyle \frac{G_i\left(x\right)}{g_j\left(x\right)}}(m_{i,j}=1).$$

Since $gcd(G_{i,j},g_j)=1$, there exist polynomials $a_{i,j}\left(x\right),b_{i,j}\left(x\right)\in R\left[x\right]$ satisfying

$$a_{i,j}\left(x\right)G_{i,j}\left(x\right)+b_{i,j}\left(x\right)g_j\left(x\right)=1.$$

(31)

Define the CRT idempotent in $R_{{\alpha }_i,e_i}$ by

$${\epsilon }_{i,j}\left(x\right)=a_{i,j}\left(x\right)G_{i,j}\left(x\right)mod⟨P_i\left(x\right)⟩\in R_{{\alpha }_i,e_i}.$$

(32)

Lemma 4.

For each fixed i, the elements ${\epsilon }_{i,j}$ (with $m_{i,j}=1$) satisfy

$${\epsilon }_{i,j}^2={\epsilon }_{i,j},{\epsilon }_{i,j}{\epsilon }_{i,k}=0(j\ne k),\sum _{j:m_{i,j}=1}{\epsilon }_{i,j}=1\mathit{in}{R}_{{\alpha }_i,e_i}.$$

Moreover, multiplication by ${\epsilon }_{i,j}$ is the projection onto the CRT component $R\left[x\right]/⟨g_j⟩$ in (25).

Proof. 

From (31), we have ${\epsilon }_{i,j}\equiv 1(mod⟨g_j⟩)$ and ${\epsilon }_{i,j}\equiv 0(mod⟨g_k⟩)$ for $k\ne j$ (since then $g_k\mid G_{i,j}$). Thus, under the CRT isomorphism ${\theta }_i$, ${\epsilon }_{i,j}$ maps to the standard basis vector with a 1 in the j-th component and 0 elsewhere. The stated idempotent identities follow immediately from this description. □

Define global idempotents in $\mathcal{S}$ by

$${\epsilon }_j=({\epsilon }_{1,j},\dots ,{\epsilon }_{\mathsf{\ell },j})\in \mathcal{S},(1\le j\le s),$$

(33)

with the convention ${\epsilon }_{i,j}=0$ when $m_{i,j}=0$. Then ${\epsilon }_j^2={\epsilon }_j$, ${\epsilon }_j{\epsilon }_k=0$ for $j\ne k$, and ${\sum }_{j=1}^s{\epsilon }_j=1$ in $\mathcal{S}$. In particular,

$$\mathcal{S}=\underset{j=1}{\overset{s}{⨁}}{\epsilon }_j\mathcal{S},{\epsilon }_j\mathcal{S}\cong {\Gamma }_{1,j}\times \cdots \times {\Gamma }_{\mathsf{\ell },j}\mathrm{as}R\left[x\right]\text{-}\mathrm{modules}.$$

(34)

We now decompose GQPC codes into constituents.

Proposition 4.

Let $C\subseteq \mathcal{S}$ be a right GQPC code over R. Under Remark 2, define

$$C_j=\Theta \left(C\right)\cap \left({\Gamma }_{1,j}\times \cdots \times {\Gamma }_{\mathsf{\ell },j}\right)\subseteq {\Gamma }_{1,j}\times \cdots \times {\Gamma }_{\mathsf{\ell },j},1\le j\le s.$$

(35)

Then each $C_j$ is an $R\left[x\right]$-submodule (equivalently, a ${\Gamma }_j$-submodule), and

$$\Theta \left(C\right)=\underset{j=1}{\overset{s}{⨁}}{C}_j,\mathit{hence}C\cong \underset{j=1}{\overset{s}{⨁}}{C}_j$$

(36)

as $R\left[x\right]$-modules. The submodules $C_j$ are called the constituent codes of C.

Proof. 

Since $\Theta$ is an $R\left[x\right]$-module isomorphism, $\Theta \left(C\right)$ is an $R\left[x\right]$-submodule of the ambient direct sum. Each summand ${\Gamma }_{1,j}\times \cdots \times {\Gamma }_{\mathsf{\ell },j}$ is stable under the $R\left[x\right]$-action, so the intersection $C_j$ in (35) is an $R\left[x\right]$-submodule.

To prove (36), note first that $\Theta \left(C\right)\subseteq {⨁}_j{C}_j$ is automatic. Conversely, let $y={\sum }_{j=1}^s{y}_j$ with $y_j\in C_j$. Since $y_j\in \Theta \left(C\right)$, choose $c^{\left(j\right)}\in C$ with $\Theta \left(c^{\left(j\right)}\right)$ having j-component equal to $y_j$. Using the global idempotents ${\epsilon }_j$ (transported through $\Theta$), we have

$$\sum _{j=1}^s{\epsilon }_j·\Theta \left(c^{\left(j\right)}\right)\in \Theta \left(C\right),$$

and its j-th component equals $y_j$. Therefore ${\sum }_j{y}_j\in \Theta \left(C\right)$, proving the reverse inclusion. Directness follows from orthogonality of the ${\epsilon }_j$. □

This decomposition immediately yields cardinality and rank information. Since R is finite and all constituents are finite R-modules, we obtain multiplicativity of sizes.

Corollary 2.

Under Remark 2, for any right GQPC code $C\subseteq \mathcal{S}$ with constituents $C_j$,

$$\left|C\right|=\prod _{j=1}^s\left|C_j\right|.$$

In particular, if each $C_j$ is a free ${\Gamma }_j$-module of rank $r_j$, then

$${\left|C_j\right|=\left|{\Gamma }_j\right|}^{r_j}\mathit{and}\left|C\right|=\prod _{j=1}^s{\left|{\Gamma }_j\right|}^{r_j}.$$

Proof. 

The first statement follows from the direct sum decomposition $C\cong {⨁}_j{C}_j$. If $C_j\cong {\Gamma }_j^{r_j}$ as a ${\Gamma }_j$-module, then $|C_j|=|{\Gamma }_j{|}^{r_j}$, and multiplying over j gives the claimed formula. □

We now discuss generator numbers. Let $B=R\left[x\right]$. By Proposition 1, a right GQPC code $C\subseteq \mathcal{S}$ is a B-submodule. Define ${\mu }_B\left(C\right)$ to be the minimal integer $\rho$ such that

$$C=B{b}_1+\cdots +B{b}_{\rho }\subseteq \mathcal{S}.$$

(37)

Lemma 5.

If $C\cong {⨁}_{j=1}^s{C}_j$ as in Proposition 4, then

$${\mu }_B\left(C\right)=\underset{1\le j\le s}{max}{\mu }_B\left(C_j\right).$$

Proof. 

If C is generated by r elements over B, then each image under the projection onto a constituent is generated by at most r elements, hence ${\mu }_B\left(C_j\right)\le r$ for all j, giving ${max}_j{\mu }_B\left(C_j\right)\le {\mu }_B\left(C\right)$.

For the reverse inequality, let $r={max}_j{\mu }_B\left(C_j\right)$. For each j pick B-generators $c_{j,1},\dots ,c_{j,r}$ of $C_j$ (padding with zeros if needed) and define $d_t=(c_{1,t},\dots ,c_{s,t})\in {⨁}_j{C}_j$. Then any element of ${⨁}_j{C}_j$ is a B-linear combination of the $d_t$, so ${\mu }_B\left({⨁}_j{C}_j\right)\le r$, hence ${\mu }_B\left(C\right)\le r$. □

The crucial point is that the B-action on the j-th constituent factors through the quotient ${\Gamma }_j=R\left[x\right]/⟨g_j⟩$. Thus, generator counts should be measured over ${\Gamma }_j$, not over R.

Lemma 6.

For each j, the B-action on ${\Gamma }_{1,j}\times \cdots \times {\Gamma }_{\mathsf{\ell },j}$ factors through the canonical surjection $B\twoheadrightarrow {\Gamma }_j$. Consequently, for any B-submodule $D\subseteq {\Gamma }_{1,j}\times \cdots \times {\Gamma }_{\mathsf{\ell },j}$,

$${\mu }_B\left(D\right)={\mu }_{{\Gamma }_j}\left(D\right),$$

where ${\mu }_{{\Gamma }_j}\left(D\right)$ denotes the minimal number of generators of D as a ${\Gamma }_j$-module.

Proof. 

In each nonzero component, multiplication by $f\left(x\right)\in B$ is computed modulo $⟨g_j⟩$, hence if $f\in ⟨g_j⟩$ then f acts as zero. Therefore, the action depends only on the class of f in ${\Gamma }_j$. Thus, the categories of B-submodules and ${\Gamma }_j$-submodules coincide, and generator numbers agree. □

Theorem  2.

Let $C\subseteq \mathcal{S}$ be a right GQPC code over R satisfying Remark 2, and let $C_j$ be its constituent codes. Assume each $C_j$ is a free ${\Gamma }_j$-module of rank $r_j$. Then the minimal number $\rho ={\mu }_B\left(C\right)$ of polynomial generators of C satisfies

$$\rho =\underset{1\le j\le s}{max}{r}_j=\underset{1\le j\le s}{max}{\mathrm{rk}}_{{\Gamma }_j}\left(C_j\right).$$

(38)

In particular, C is a 1-generator right GQPC code if and only if $r_j\le 1$ for all j.

Proof. 

By Lemma 5, ${\mu }_B\left(C\right)={max}_j{\mu }_B\left(C_j\right).$ By Lemma 6, ${\mu }_B\left(C_j\right)={\mu }_{{\Gamma }_j}\left(C_j\right)$. If $C_j\cong {\Gamma }_j^{r_j}$, then $C_j$ needs exactly $r_j$ generators as a ${\Gamma }_j$-module, so ${\mu }_{{\Gamma }_j}\left(C_j\right)=r_j$. Substituting yields (38). □

Remark 3.

When $g_j\left(x\right)$ is regular, ${\Gamma }_j$ is a finite chain-ring extension of R and is a free R-module of rank $deg\left(g_j\right)$. Hence if $C_j$ is free of rank $r_j$ over ${\Gamma }_j$, then $C_j$ is free over R of rank $deg\left(g_j\right)r_j$, and

$$\left|C_j\right|={\left|R\right|}^{deg\left(g_j\right)r_j}=q^{2deg\left(g_j\right)r_j}.$$

Thus, the generator count is governed by constituent ranks over the extension rings ${\Gamma }_j$, exactly mirroring the field case where constituents live over extension fields.

Example 1.

Let $q=3$ and $R={\mathbb{F}}_3+u{\mathbb{F}}_3.$ Take $\mathsf{\ell }=1$, $e_1=4$, and choose $\alpha =(-1,0,0,0)\in R^4$, so that

$$P_1\left(x\right)=x^4-\alpha \left(x\right)=x^4+1\in R\left[x\right].$$

Over ${\mathbb{F}}_3\left[x\right]$ we have the factorization

$$\overline{P_1}\left(x\right)=x^4+1=(x^2-x-1)(x^2+x-1),$$

and both quadratic factors are irreducible in ${\mathbb{F}}_3\left[x\right]$. Let

$$g_1\left(x\right)=x^2-x-1,g_2\left(x\right)=x^2+x-1,$$

viewed in $R\left[x\right]$. Then $g_1,g_2$ are basic irreducible (hence regular), and

$$R_{\alpha ,4}=R\left[x\right]/⟨x^4+1⟩\cong {\Gamma }_1\times {\Gamma }_2,{\Gamma }_j=R\left[x\right]/⟨g_j\left(x\right)⟩.$$

(39)

Let $C\subseteq R_{\alpha ,4}$ be the right GQPC code whose CRT decomposition is

$$\Theta \left(C\right)=C_1\oplus C_2,C_1={\Gamma }_1,C_2=0.$$

(40)

Then, $C_1$ is free of ${\Gamma }_1$-rank 1 and $C_2$ has rank 0. Hence by Theorem 2,

$$\rho =max\{1,0\}=1,$$

(41)

so C is a 1-generator right GQPC code. Moreover, by Corollary 2,

$$\left|C\right|=|C_1|=|{\Gamma }_1{|=|R|}^{deg\left(g_1\right)}=3^{2·2}=3^4.$$

(42)

Finally, Theorem 1 gives

$$\Theta \left(C^{\perp }\right)=C_1^{\perp }\oplus C_2^{\perp },$$

(43)

so the dual code decomposes componentwise as predicted.

4. Existence of Normalized Generating Sets

We now show that every right GQPC code with finite R-rank admits a normalized generating set. The construction proceeds block by block using the fact that each blockwise RPC code is principal.

4.1. Normalized Generating Sets

Let $C\subseteq \mathcal{S}$ be a right GQPC code of block length $(e_1,\dots ,e_{\mathsf{\ell }})$ and index ℓ. By Proposition 4, C is an $R\left[x\right]$-submodule of $\mathcal{S}$. In this subsection, we describe a special type of generating set for C with an upper-triangular structure relative to the block decomposition.

We will work with generating families of size at most ℓ; if necessary, one can always discard redundant generators until a minimal generating set is obtained.

The concept of normalized generating sets was introduced for right GQPC codes over finite fields (see, for instance, ref. [9]). We extend this concept to the setting of finite chain rings as follows.

Definition 2.

Let $C\subseteq \mathcal{S}$ be a right GQPC code. A set $\{{\alpha }_1\left(x\right),\dots ,{\alpha }_{\mathsf{\ell }}\left(x\right)\}\subseteq \mathcal{S}$ is called a normalized generating set of C if the following conditions hold:

(i) 

For each $1\le i\le \mathsf{\ell }$, the vector ${\alpha }_i\left(x\right)$ has the form

$${\alpha }_i\left(x\right)=(g_{i1}\left(x\right),g_{i2}\left(x\right),\dots ,g_{ii}\left(x\right),0,\dots ,0),$$

(44)

where $g_{ij}\left(x\right)\in R_{{\alpha }_j,e_j}$ for all $1\le j\le i$.

(ii) 

For each $1\le i\le \mathsf{\ell }$, the element $g_{ii}\left(x\right)$ is represented by a monic polynomial in $R\left[x\right]$ of degree less than $e_i$ and satisfies

$$P_{e_i,{\alpha }_i}\left(x\right)=q_i\left(x\right)g_{ii}\left(x\right)$$

(45)

in $R\left[x\right]$ for some polynomial $q_i\left(x\right)\in R\left[x\right]$, i.e., $g_{ii}\left(x\right)$ divides $P_{e_i,{\alpha }_i}\left(x\right)$ in $R\left[x\right]$.

(iii) 

For each $1\le i<\mathsf{\ell }$, the off-diagonal entry $g_{i+1,i}\left(x\right)$ is represented by a polynomial in $R\left[x\right]$ of degree strictly smaller than $deg{g}_{ii}\left(x\right)$, after choosing representatives of all elements in $R_{{\alpha }_i,e_i}$ of degree less than $e_i$.

If, in addition,

$$C={⟨{\alpha }_1\left(x\right),\dots ,{\alpha }_{\mathsf{\ell }}\left(x\right)⟩}_{R\left[x\right]},$$

(46)

we say that $\{{\alpha }_1\left(x\right),\dots ,{\alpha }_{\mathsf{\ell }}\left(x\right)\}$ is a normalized generating set of C.

In matrix form, if we arrange the entries $g_{ij}\left(x\right)$ into an $\mathsf{\ell }\times \mathsf{\ell }$ matrix ${\left(g_{ij}\left(x\right)\right)}_{1\le j\le i\le \mathsf{\ell }}$, then this matrix is upper-triangular with respect to the block index. The diagonal entries $g_{ii}\left(x\right)$ are monic divisors of the ambient polynomials $P_{e_i,{\alpha }_i}\left(x\right)$, and the essential off-diagonal entries satisfy a degree reduction condition.

We now provide a straightforward way for constructing normalized generating sets for right GQPC codes over finite chain rings. The subsequent theorem, which embodies a major structural result of this study, establishes this construction.

Theorem  3.

Let $C\subseteq \mathcal{S}$ be a right GQPC code of block length $(e_1,\dots ,e_{\mathsf{\ell }})$ and index ℓ over R, and assume that C has finite rank as an R-module. Then there exists a normalized generating set

$$\{{\alpha }_1\left(x\right),\dots ,{\alpha }_{\mathsf{\ell }}\left(x\right)\}$$

(47)

of C in the sense of Definition 2.

Proof. 

We proceed in several steps.

Step 1: Choosing an initial generating set. Since C has finite rank as an R-module and $R\left[x\right]$ is Noetherian, C is finitely generated as an $R\left[x\right]$-module. Choose an arbitrary finite generating set

$$\{{\Delta }_1\left(x\right),\dots ,{\Delta }_r\left(x\right)\}\subseteq \mathcal{S}$$

such that

$$C=R\left[x\right]{\Delta }_1\left(x\right)+\dots +R\left[x\right]{\Delta }_r\left(x\right).$$

If $r>\mathsf{\ell }$, we may discard redundant generators: by standard module theory, one can successively remove generators whose removal does not change the $R\left[x\right]$-span, until a generating set of size at most ℓ is obtained. Thus, without loss of generality, we may assume that $r\le \mathsf{\ell }.$ If $r<\mathsf{\ell }$, we may formally enlarge the family to ℓ elements by adding zeros; the additional zero rows will be harmless in what follows. For notational convenience, we therefore assume that we have a generating family $\{{\Delta }_1\left(x\right),\dots ,{\Delta }_{\mathsf{\ell }}\left(x\right)\}$ of size exactly ℓ. Each ${\Delta }_k\left(x\right)$ can be written in block form as

$${\Delta }_k\left(x\right)=\left(b_{k1}\left(x\right),\dots ,b_{k\mathsf{\ell }}\left(x\right)\right)\in R_{{\alpha }_1,e_1}\times \cdots \times R_{{\alpha }_{\mathsf{\ell }},e_{\mathsf{\ell }}}.$$

Step 2: Construction of the last row ${\alpha }_{\mathsf{\ell }}\left(x\right)$. Consider the projection map

$${\pi }_{\mathsf{\ell }}:\mathcal{S}\to R_{{\alpha }_{\mathsf{\ell }},e_{\mathsf{\ell }}},(s_1\left(x\right),\dots ,s_{\mathsf{\ell }}\left(x\right))\mapsto s_{\mathsf{\ell }}\left(x\right),$$

and its restriction to C. The image ${\pi }_{\mathsf{\ell }}\left(C\right)$ is an RPC code in $R_{{\alpha }_{\mathsf{\ell }},e_{\mathsf{\ell }}}$: it is an $R\left[x\right]$-submodule, hence a left ideal of the principal ideal ring $R_{{\alpha }_{\mathsf{\ell }},e_{\mathsf{\ell }}}$, and therefore generated by a single element. By the discussion in Section 2, there exists a unique monic polynomial $g_{\mathsf{\ell }\mathsf{\ell }}\left(x\right)\in R\left[x\right]$ of degree less than $e_{\mathsf{\ell }}$ that divides $P_{e_{\mathsf{\ell }},{\alpha }_{\mathsf{\ell }}}\left(x\right)$ and generates ${\pi }_{\mathsf{\ell }}\left(C\right)$ as an $R\left[x\right]$-submodule of $R_{{\alpha }_{\mathsf{\ell }},e_{\mathsf{\ell }}}$.

By definition of the image, there exists some $c^{\left(\mathsf{\ell }\right)}\in C$ such that

$${\pi }_{\mathsf{\ell }}\left(M\left(c^{\left(\mathsf{\ell }\right)}\right)\right)=g_{\mathsf{\ell }\mathsf{\ell }}\left(x\right).$$

Write

$${\alpha }_{\mathsf{\ell }}\left(x\right)=M\left(c^{\left(\mathsf{\ell }\right)}\right)=\left(g_{\mathsf{\ell }1}\left(x\right),\dots ,g_{\mathsf{\ell },\mathsf{\ell }-1}\left(x\right),g_{\mathsf{\ell }\mathsf{\ell }}\left(x\right)\right)\in \mathcal{S}.$$

Replace the last generator ${\Delta }_{\mathsf{\ell }}\left(x\right)$ in our generating set by ${\alpha }_{\mathsf{\ell }}\left(x\right)$. Since ${\alpha }_{\mathsf{\ell }}\left(x\right)$ is a linear combination of elements of C (indeed, it is one of them), the $R\left[x\right]$-submodule generated by the new family still coincides with C. Next we eliminate the ℓ-th block of the other generators. For each $1\le k\le \mathsf{\ell }-1$, the element $b_{k\mathsf{\ell }}\left(x\right)$ lies in ${\pi }_{\mathsf{\ell }}\left(C\right)$, which is generated by $g_{\mathsf{\ell }\mathsf{\ell }}\left(x\right)$. Hence there is a polynomial $h_k\left(x\right)\in R\left[x\right]$ such that $b_{k\mathsf{\ell }}\left(x\right)=h_k\left(x\right)g_{\mathsf{\ell }\mathsf{\ell }}\left(x\right)\mathrm{in}{R}_{{\alpha }_{\mathsf{\ell }},e_{\mathsf{\ell }}}.$ Replace ${\Delta }_k\left(x\right)\mathrm{by}{\Delta }_k^{\prime }\left(x\right)={\Delta }_k\left(x\right)-h_k\left(x\right){\alpha }_{\mathsf{\ell }}\left(x\right).$ Then,

$${\pi }_{\mathsf{\ell }}\left({\Delta }_k^{\prime }\left(x\right)\right)=b_{k\mathsf{\ell }}\left(x\right)-h_k\left(x\right)g_{\mathsf{\ell }\mathsf{\ell }}\left(x\right)=0$$

in $R_{{\alpha }_{\mathsf{\ell }},e_{\mathsf{\ell }}}$, so the last component of ${\Delta }_k^{\prime }\left(x\right)$ is zero. On the other hand, the $R\left[x\right]$-span of $\{{\Delta }_1\left(x\right),\dots ,{\Delta }_{\mathsf{\ell }-1}\left(x\right),{\alpha }_{\mathsf{\ell }}\left(x\right)\}$ coincides with that of $\{{\Delta }_1^{\prime }\left(x\right),\dots ,{\Delta }_{\mathsf{\ell }-1}^{\prime }\left(x\right),{\alpha }_{\mathsf{\ell }}\left(x\right)\}$, since we have only performed elementary operations of the form ${\Delta }_k\mapsto {\Delta }_k-h_k{\alpha }_{\mathsf{\ell }}$, which do not change the generated submodule. After this step, we have a generating set $\{{\Delta }_1^{\prime }\left(x\right),\dots ,{\Delta }_{\mathsf{\ell }-1}^{\prime }\left(x\right),{\alpha }_{\mathsf{\ell }}\left(x\right)\}$ for C, with the property that the last component of every generator except ${\alpha }_{\mathsf{\ell }}\left(x\right)$ is zero.

Step 3: Inductive construction of the remaining rows. We now proceed inductively on the block index. Assume that for some integer i with $1<i\le \mathsf{\ell }$ we have already constructed generators ${\alpha }_i\left(x\right),{\alpha }_{i+1}\left(x\right),\dots ,{\alpha }_{\mathsf{\ell }}\left(x\right)$ such that:

(i)

For each k with $i\le k\le \mathsf{\ell }$, the vector ${\alpha }_k\left(x\right)$ has the form

$${\alpha }_k\left(x\right)=(g_{k1}\left(x\right),\dots ,g_{kk}\left(x\right),0,\dots ,0),$$

so its components beyond the k-th block are zero.

(ii)

The last $\mathsf{\ell }-i+1$ generators $\{{\alpha }_i\left(x\right),\dots ,{\alpha }_{\mathsf{\ell }}\left(x\right)\}$ together with some modified versions of the remaining generators form a generating set of C.

(iii)

For each k with $i\le k\le \mathsf{\ell }$, the entry $g_{kk}\left(x\right)$ is monic, of degree $<e_k$, and divides $P_{e_k,{\alpha }_k}\left(x\right)$.

We show how to construct ${\alpha }_{i-1}\left(x\right)$ with the desired properties and simultaneously ensure that all generators except ${\alpha }_{i-1}\left(x\right)$ have zero in block $i-1$.

Consider the projection

$${\pi }_{i-1}:\mathcal{S}\to R_{{\alpha }_{i-1},e_{i-1}},(s_1\left(x\right),\dots ,s_{\mathsf{\ell }}\left(x\right))\mapsto s_{i-1}\left(x\right),$$

and its restriction to C. The image ${\pi }_{i-1}\left(C\right)$ is an RPC code in $R_{{\alpha }_{i-1},e_{i-1}}$, hence a principal ideal generated by a unique monic divisor $g_{i-1,i-1}\left(x\right)$ of $P_{e_{i-1},{\alpha }_{i-1}}\left(x\right)$. Choose an element $c^{(i-1)}\in C$ such that ${\pi }_{i-1}\left(M\left(c^{(i-1)}\right)\right)=g_{i-1,i-1}\left(x\right),$ and write

$${\alpha }_{i-1}\left(x\right)=M\left(c^{(i-1)}\right)=\left(g_{i-1,1}\left(x\right),\dots ,g_{i-1,i-2}\left(x\right),g_{i-1,i-1}\left(x\right),0,\dots ,0\right).$$

Notice that we can always subtract suitable $R\left[x\right]$-multiples of the already constructed generators ${\alpha }_i\left(x\right),\dots$$,{\alpha }_{\mathsf{\ell }}\left(x\right)$ to annihilate the components of ${\alpha }_{i-1}\left(x\right)$ in blocks $j>i-1$: these blocks are already controlled by ${\alpha }_i\left(x\right),\dots ,{\alpha }_{\mathsf{\ell }}\left(x\right)$, and the set of generators $\{{\alpha }_{i-1}\left(x\right),{\alpha }_i\left(x\right),\dots ,{\alpha }_{\mathsf{\ell }}\left(x\right)\}$ obtained in this way still generates the same submodule of $\mathcal{S}$.

Now modify the remaining generators (those not yet fixed as some ${\alpha }_k\left(x\right)$) to eliminate their components in block $i-1$. If $G\left(x\right)=(g_1\left(x\right),\dots ,g_{\mathsf{\ell }}\left(x\right))$ is such a generator, then its $(i-1)$-st component $g_{i-1}\left(x\right)$ lies in the RPC code ${\pi }_{i-1}\left(C\right)=R\left[x\right]g_{i-1,i-1}\left(x\right)$, so there exists $h\left(x\right)\in R\left[x\right]$ such that $g_{i-1}\left(x\right)=h\left(x\right)g_{i-1,i-1}\left(x\right).$ Replacing $G\left(x\right)\mathrm{by}{G}^{\prime }\left(x\right)=G\left(x\right)-h\left(x\right){\alpha }_{i-1}\left(x\right),$ we obtain a new generator with the same $R\left[x\right]$-span but with zero component in block $i-1$. Repeating this modification for each remaining generator, we end up with a generating family of C in which the only generator having a possibly nonzero component in block $i-1$ is ${\alpha }_{i-1}\left(x\right)$, and its entries beyond the $(i-1)$-st block are zero. By descending induction on i from ℓ down to 1, this procedure yields generators ${\alpha }_1\left(x\right),\dots ,{\alpha }_{\mathsf{\ell }}\left(x\right)$ such that each ${\alpha }_i\left(x\right)$ has the required triangular shape and each diagonal entry $g_{ii}\left(x\right)$ is a monic divisor of $P_{e_i,{\alpha }_i}\left(x\right)$.

Step 4: Degree reduction on off-diagonal entries. We now enforce the degree condition on the off-diagonal entries $g_{i+1,i}\left(x\right)$. Fix i with $1\le i<\mathsf{\ell }$. Consider the i-th and $(i+1)$-st generators:

$${\alpha }_i\left(x\right)=(g_{i1}\left(x\right),\dots ,g_{ii}\left(x\right),0,\dots ,0),$$

$${\alpha }_{i+1}\left(x\right)=(g_{i+1,1}\left(x\right),\dots ,g_{i+1,i}\left(x\right),g_{i+1,i+1}\left(x\right),0,\dots ,0).$$

All components $g_{ij}\left(x\right)$ with $j\le i$ belong to $R_{{\alpha }_j,e_j}$. In particular, $g_{i+1,i}\left(x\right)\in R_{{\alpha }_i,e_i}$, so it can be represented by a polynomial in $R\left[x\right]$ with degree less than $e_i$. Let ${\tilde{g}}_{ii}\left(x\right),{\tilde{g}}_{i+1,i}\left(x\right)\in R\left[x\right]$ be fixed representatives of $g_{ii}\left(x\right)$, respectively $g_{i+1,i}\left(x\right)$, of degree does not exceed or equal $e_i$. Since ${\tilde{g}}_{ii}\left(x\right)$ divides $P_{e_i,{\alpha }_i}\left(x\right)$ in $R\left[x\right]$, it divides every element of the ideal corresponding to the RPC code generated by $g_{ii}\left(x\right)$. Perform polynomial division of ${\tilde{g}}_{i+1,i}\left(x\right)$ by ${\tilde{g}}_{ii}\left(x\right)$ in $R\left[x\right]$: there exist polynomials $q\left(x\right),r\left(x\right)\in R\left[x\right]$ such that ${\tilde{g}}_{i+1,i}\left(x\right)=q\left(x\right){\tilde{g}}_{ii}\left(x\right)+r\left(x\right),$ with either $r\left(x\right)=0$ or $degr\left(x\right)<deg{\tilde{g}}_{ii}\left(x\right)$. Now replace the generator ${\alpha }_{i+1}\left(x\right)$ by ${\alpha }_{i+1}^{\prime }\left(x\right)={\alpha }_{i+1}\left(x\right)-q\left(x\right){\alpha }_i\left(x\right).$ This operation leaves the $R\left[x\right]$-span of the generators unchanged, since it is an elementary row operation in the $R\left[x\right]$-module. By construction, the i-th component of ${\alpha }_{i+1}^{\prime }\left(x\right)$ is exactly $r\left(x\right)$, viewed as an element of $R_{{\alpha }_i,e_i}$. Moreover, the $(i+1)$-st component remains $g_{i+1,i+1}\left(x\right)$, because the $(i+1)$-st component of ${\alpha }_i\left(x\right)$ is zero. Thus, we have replaced $g_{i+1,i}\left(x\right)$ by an element of strictly smaller degree than $g_{ii}\left(x\right)$ (or by zero), without affecting any other diagonal entry and without changing the generated code.

Repeating this procedure for each pair $(i,i+1)$ with $1\le i<\mathsf{\ell }$ gives a generating family $\{{\alpha }_1\left(x\right),\dots ,{\alpha }_{\mathsf{\ell }}\left(x\right)\}$ satisfying all the conditions in Definition 2.

Altogether, we have constructed a normalized generating set of C, which completes the proof. □

Remark 4.

The principal ideal ring assumption in (H2) guarantees that each projected RPC code admits a single polynomial generator. Without this hypothesis, block ideals over $R_{{\alpha }_i,e_i}$ may require multiple generators, and the existence of a normalized generating set would require a more general module-theoretic framework, which is beyond the scope of the present work.

4.2. Rank Formula

We now derive a formula for the R-rank of a right GQPC code in terms of the diagonal polynomials in a normalized generating set. Throughout this subsection, we assume that the ambient rings $R_{{\alpha }_i,e_i}$ are principal ideal rings and that the diagonal generators $g_{ii}\left(x\right)$ are chosen so that multiplication by $g_{ii}\left(x\right)$ is injective on $R\left[x\right]/⟨q_i\left(x\right)⟩$ (equivalently, $g_{ii}\left(x\right)$ is regular modulo $⟨q_i\left(x\right)⟩$).

Let $C\subseteq \mathcal{S}$ be a right GQPC code over R, and assume that $\{{\alpha }_1\left(x\right),\dots ,{\alpha }_{\mathsf{\ell }}\left(x\right)\}$ is a normalized generating set of C. By Definition 2, for each $1\le i\le \mathsf{\ell }$ we have

$${\alpha }_i\left(x\right)=(g_{i1}\left(x\right),\dots ,g_{ii}\left(x\right),0,\dots ,0),$$

and there exists a monic polynomial $q_i\left(x\right)\in R\left[x\right]$ such that

$$P_{e_i,{\alpha }_i}\left(x\right)=q_i\left(x\right)g_{ii}\left(x\right)$$

with $deg{g}_{ii}\left(x\right)<e_i$.

The key point is that each diagonal element $g_{ii}\left(x\right)$ determines an RPC code in block i of rank $e_i-deg{g}_{ii}\left(x\right)$, and the triangular shape ensures that these contributions to the rank are additive.

Lemma 7.

Let $1\le i\le \mathsf{\ell }$, and let $g\left(x\right)\in R\left[x\right]$ be a monic polynomial of degree $d<e_i$ such that $g\left(x\right)\mid P_{e_i,{\alpha }_i}\left(x\right)$ in $R\left[x\right]$. Write

$$P_{e_i,{\alpha }_i}\left(x\right)=q\left(x\right)g\left(x\right)$$

with $q\left(x\right)\in R\left[x\right]$ monic. Assume moreover that $g\left(x\right)$ is regular modulo $⟨q\left(x\right)⟩$, i.e., the class of $g\left(x\right)$ is not a zero divisor in the quotient ring $R\left[x\right]/⟨q\left(x\right)⟩$. Let

$$D=R\left[x\right]g\left(x\right)\subseteq R_{{\alpha }_i,e_i}=R\left[x\right]/⟨P_{e_i,{\alpha }_i}\left(x\right)⟩.$$

Then D is a free R-module of rank $e_i-d$.

Proof. 

Let $P\left(x\right)=P_{e_i,{\alpha }_i}\left(x\right)$. Since $P\left(x\right)=q\left(x\right)g\left(x\right)$, multiplication by $g\left(x\right)$ defines an $R\left[x\right]$-module homomorphism

$$\phi :R\left[x\right]/⟨q\left(x\right)⟩⟶D,\left[h\right(x\left)\right]⟼\left[h\right(x\left)g\right(x\left)\right]\mathrm{in}R\left[x\right]/⟨P\left(x\right)⟩.$$

This map is surjective by definition of D.

To determine the kernel, note that $\phi \left(\right[h\left]\right)=0$ if and only if $h\left(x\right)g\left(x\right)\in ⟨P\left(x\right)⟩=⟨q\left(x\right)g\left(x\right)⟩$, i.e., there exists $t\left(x\right)\in R\left[x\right]$ such that

$$h\left(x\right)g\left(x\right)=t\left(x\right)q\left(x\right)g\left(x\right).$$

Equivalently,

$$\left(h\left(x\right)-t\left(x\right)q\left(x\right)\right)g\left(x\right)=0\mathrm{in}R\left[x\right]/⟨q\left(x\right)⟩.$$

By the regularity assumption on $g\left(x\right)$ in $R\left[x\right]/⟨q\left(x\right)⟩$, this implies

$$h\left(x\right)-t\left(x\right)q\left(x\right)\in ⟨q\left(x\right)⟩,$$

hence $h\left(x\right)\in ⟨q\left(x\right)⟩$. Therefore, $ker\left(\phi \right)=⟨q\left(x\right)⟩$, so $\phi$ is injective.

Thus, $\phi$ is an isomorphism of R-modules:

$$D\cong R\left[x\right]/⟨q\left(x\right)⟩.$$

Since $q\left(x\right)$ is monic of degree $e_i-d$, every class in $R\left[x\right]/⟨q\left(x\right)⟩$ has a unique representative of degree $<e_i-d$, and $\{1,x,\dots ,x^{e_i-d-1}\}$ forms an R-basis. Hence, $R\left[x\right]/⟨q\left(x\right)⟩$ is free of rank $e_i-d$, and therefore, so is D. □

We now apply this lemma to the successive blocks, using the triangular structure.

For $1\le i\le \mathsf{\ell }$, let

$${\mathcal{S}}^{\left(i\right)}=R_{{\alpha }_1,e_1}\times \cdots \times R_{{\alpha }_i,e_i}\times 0\times \cdots \times 0\subseteq \mathcal{S}$$

be the submodule consisting of elements supported in the first i blocks. Set

$$C^{\left(i\right)}=C\cap {\mathcal{S}}^{\left(i\right)}.$$

Thus, we obtain an ascending chain of $R\left[x\right]$-submodules

$$0=C^{\left(0\right)}\subseteq C^{\left(1\right)}\subseteq \dots \subseteq C^{\left(\mathsf{\ell }\right)}=C.$$

(48)

The following theorem describes the fundamental filtration structure of a right GQPC code and determines the module structure of its successive quotients. It is one of the main structural results of this work.

Theorem  4.

For each $1\le i\le \mathsf{\ell }$, the submodule $C^{\left(i\right)}=C\cap {\mathcal{S}}^{\left(i\right)}$ is generated, as an $R\left[x\right]$-module, by $\{{\alpha }_1\left(x\right),\dots ,{\alpha }_i\left(x\right)\}$. Moreover, write

$$P_{e_i,{\alpha }_i}\left(x\right)=q_i\left(x\right)g_{ii}\left(x\right)$$

with $q_i\left(x\right)$ monic, and assume that $g_{ii}\left(x\right)$ is regular modulo $⟨q_i\left(x\right)⟩$, i.e., the class of $g_{ii}\left(x\right)$ is not a zero divisor in $R\left[x\right]/⟨q_i\left(x\right)⟩$. Then the quotient $C^{\left(i\right)}/C^{(i-1)}$ is isomorphic, as an $R\left[x\right]$-module, to the RPC code $R\left[x\right]g_{ii}\left(x\right)\subseteq R_{{\alpha }_i,e_i}$. In particular,

$${rk}_R\left(C^{\left(i\right)}/C^{(i-1)}\right)=e_i-deg{g}_{ii}\left(x\right).$$

Proof. 

Fix $i\in \{1,\dots ,\mathsf{\ell }\}$.

Step 1: $C^{\left(i\right)}$ is generated by ${\alpha }_1\left(x\right),\dots ,{\alpha }_i\left(x\right)$. By assumption, $\{{\alpha }_1\left(x\right),\dots ,{\alpha }_{\mathsf{\ell }}\left(x\right)\}$ is a normalized generating set of C, so

$$C={⟨{\alpha }_1\left(x\right),\dots ,{\alpha }_{\mathsf{\ell }}\left(x\right)⟩}_{R\left[x\right]}.$$

Also, by definition,

$${\mathcal{S}}^{\left(i\right)}=R_{{\alpha }_1,e_1}\times \cdots \times R_{{\alpha }_i,e_i}\times 0\times \cdots \times 0\subseteq \mathcal{S},C^{\left(i\right)}=C\cap {\mathcal{S}}^{\left(i\right)}.$$

The triangular shape in Definition 2 means that, for each k, the vector ${\alpha }_k\left(x\right)$ has the form

$${\alpha }_k\left(x\right)=(g_{k1}\left(x\right),\dots ,g_{kk}\left(x\right),0,\dots ,0).$$

In particular, ${\alpha }_k\left(x\right)\in {\mathcal{S}}^{\left(i\right)}$ whenever $k\le i$, while for $k>i$ the first i blocks of ${\alpha }_k\left(x\right)$ are still defined but ${\alpha }_k\left(x\right)$ has zero components in blocks $i+1,\dots ,\mathsf{\ell }$. Hence, its contribution beyond block i is irrelevant when we intersect with ${\mathcal{S}}^{\left(i\right)}$.

We now prove both inclusions.

(i)

Show ${⟨{\alpha }_1,\dots ,{\alpha }_i⟩}_{R\left[x\right]}\subseteq C^{\left(i\right)}$. Each ${\alpha }_k\left(x\right)$ for $k\le i$ belongs to C and lies in ${\mathcal{S}}^{\left(i\right)}$, so any $R\left[x\right]$-linear combination of ${\alpha }_1\left(x\right),\dots ,{\alpha }_i\left(x\right)$ lies in $C\cap {\mathcal{S}}^{\left(i\right)}=C^{\left(i\right)}$. Thus,

$${⟨{\alpha }_1\left(x\right),\dots ,{\alpha }_i\left(x\right)⟩}_{R\left[x\right]}\subseteq C^{\left(i\right)}.$$

(ii)

Show $C^{\left(i\right)}\subseteq {⟨{\alpha }_1,\dots ,{\alpha }_i⟩}_{R\left[x\right]}$. Take any $c\left(x\right)\in C^{\left(i\right)}$. Since $c\left(x\right)\in C$ and C is generated by all ${\alpha }_k\left(x\right)$, there exist polynomials $h_1\left(x\right),\dots ,h_{\mathsf{\ell }}\left(x\right)\in R\left[x\right]$ such that

$$c\left(x\right)=\sum _{k=1}^{\mathsf{\ell }}{h}_k\left(x\right){\alpha }_k\left(x\right).$$

Because $c\left(x\right)\in {\mathcal{S}}^{\left(i\right)}$, its blocks $i+1,\dots ,\mathsf{\ell }$ are all zero. We first examine block ℓ. The only generator ${\alpha }_k\left(x\right)$ that can contribute a nonzero ℓ-th block is ${\alpha }_{\mathsf{\ell }}\left(x\right)$ itself, since ${\alpha }_k\left(x\right)$ has zero entries beyond block k. Hence, the ℓ-th block of $c\left(x\right)$ equals $h_{\mathsf{\ell }}\left(x\right)g_{\mathsf{\ell }\mathsf{\ell }}\left(x\right)$ and must be zero. Thus,

$$h_{\mathsf{\ell }}\left(x\right)g_{\mathsf{\ell }\mathsf{\ell }}\left(x\right)=0\mathrm{in}{R}_{{\alpha }_{\mathsf{\ell }},e_{\mathsf{\ell }}}.$$

By the construction of a normalized generating set, $g_{\mathsf{\ell }\mathsf{\ell }}\left(x\right)$ generates the image of C in the ℓ-th block. Therefore, the only way to ensure that the ℓ-th block is zero is for the term $h_{\mathsf{\ell }}\left(x\right){\alpha }_{\mathsf{\ell }}\left(x\right)$ to contribute nothing to ${\mathcal{S}}^{\left(i\right)}$. Repeating this argument block by block, descending from ℓ to $i+1$, shows that all contributions from ${\alpha }_{i+1}\left(x\right),\dots ,{\alpha }_{\mathsf{\ell }}\left(x\right)$ must vanish when restricted to ${\mathcal{S}}^{\left(i\right)}$. Consequently, $c\left(x\right)$ can be written using only ${\alpha }_1\left(x\right),\dots ,{\alpha }_i\left(x\right)$, that is,

$$c\left(x\right)\in {⟨{\alpha }_1\left(x\right),\dots ,{\alpha }_i\left(x\right)⟩}_{R\left[x\right]}.$$

Combining (i) and (ii) yields

$$C^{\left(i\right)}={⟨{\alpha }_1\left(x\right),\dots ,{\alpha }_i\left(x\right)⟩}_{R\left[x\right]}.$$

Step 2: Description of ${\pi }_i\left(C^{\left(i\right)}\right)$. Let

$${\pi }_i:{\mathcal{S}}^{\left(i\right)}⟶R_{{\alpha }_i,e_i},(s_1\left(x\right),\dots ,s_i\left(x\right),0,\dots ,0)⟼s_i\left(x\right)$$

be the projection onto the i-th block. Since ${\pi }_i$ is an $R\left[x\right]$-module homomorphism, ${\pi }_i\left(C^{\left(i\right)}\right)$ is an $R\left[x\right]$-submodule of $R_{{\alpha }_i,e_i}$, and hence an RPC code in block i.

Now evaluate ${\pi }_i$ on the generators of $C^{\left(i\right)}$. For $k<i$, the i-th block of ${\alpha }_k\left(x\right)$ is zero, again by the triangular shape. For $k=i$, the i-th block of ${\alpha }_i\left(x\right)$ is $g_{ii}\left(x\right)$. Therefore,

$${\pi }_i\left(C^{\left(i\right)}\right)={⟨{\pi }_i\left({\alpha }_1\left(x\right)\right),\dots ,{\pi }_i\left({\alpha }_i\left(x\right)\right)⟩}_{R\left[x\right]}={⟨0,\dots ,0,g_{ii}\left(x\right)⟩}_{R\left[x\right]}=R\left[x\right]g_{ii}\left(x\right).$$

Step 3: Identification of the kernel $ker\left({\pi }_i{|}_{C^{\left(i\right)}}\right)$. First, observe that every element of $C^{(i-1)}$ has zero i-th block by the definition of ${\mathcal{S}}^{(i-1)}$. Hence,

$$C^{(i-1)}\subseteq ker\left({\pi }_i{|}_{C^{\left(i\right)}}\right).$$

For the reverse inclusion, take $c\left(x\right)\in C^{\left(i\right)}$ such that ${\pi }_i\left(c\left(x\right)\right)=0$. Using Step 1, write

$$c\left(x\right)=\sum _{k=1}^i{h}_k\left(x\right){\alpha }_k\left(x\right),h_k\left(x\right)\in R\left[x\right].$$

Applying ${\pi }_i$ and using the facts that ${\pi }_i\left({\alpha }_k\left(x\right)\right)=0$ for $k<i$ and ${\pi }_i\left({\alpha }_i\left(x\right)\right)=g_{ii}\left(x\right)$, we obtain

$$0={\pi }_i\left(c\left(x\right)\right)=h_i\left(x\right)g_{ii}\left(x\right)\mathrm{in}{R}_{{\alpha }_i,e_i}.$$

Equivalently,

$$h_i\left(x\right)g_{ii}\left(x\right)\in ⟨P_{e_i,{\alpha }_i}\left(x\right)⟩=⟨q_i\left(x\right)g_{ii}\left(x\right)⟩\mathrm{in}R\left[x\right].$$

Thus, there exists $t\left(x\right)\in R\left[x\right]$ such that

$$h_i\left(x\right)g_{ii}\left(x\right)=t\left(x\right)q_i\left(x\right)g_{ii}\left(x\right).$$

Passing to the quotient $R\left[x\right]/⟨q_i\left(x\right)⟩$, this becomes

$$\left(h_i\left(x\right)-t\left(x\right)q_i\left(x\right)\right)g_{ii}\left(x\right)=0\mathrm{in}R\left[x\right]/⟨q_i\left(x\right)⟩.$$

By the hypothesis that $g_{ii}\left(x\right)$ is regular modulo $⟨q_i\left(x\right)⟩$, multiplication by $g_{ii}\left(x\right)$ is injective on $R\left[x\right]/⟨q_i\left(x\right)⟩$. Hence,

$$h_i\left(x\right)-t\left(x\right)q_i\left(x\right)\in ⟨q_i\left(x\right)⟩,\mathrm{and}\mathrm{therefore}{h}_i\left(x\right)\in ⟨q_i\left(x\right)⟩.$$

Consequently, there exists $r\left(x\right)\in R\left[x\right]$ such that $h_i\left(x\right)=r\left(x\right)q_i\left(x\right)$, and thus

$$h_i\left(x\right){\alpha }_i\left(x\right)=r\left(x\right)q_i\left(x\right){\alpha }_i\left(x\right).$$

However, in the i-th block we have

$$q_i\left(x\right)·g_{ii}\left(x\right)=P_{e_i,{\alpha }_i}\left(x\right)=0\mathrm{in}{R}_{{\alpha }_i,e_i},$$

so the i-th block of $q_i\left(x\right){\alpha }_i\left(x\right)$ is zero, and all blocks with index $>i$ are already zero in ${\alpha }_i\left(x\right)$. Hence, $q_i\left(x\right){\alpha }_i\left(x\right)\in {\mathcal{S}}^{(i-1)}$, which implies that $h_i\left(x\right){\alpha }_i\left(x\right)\in {\mathcal{S}}^{(i-1)}$. Therefore,

$$c\left(x\right)=\sum _{k=1}^{i-1}{h}_k\left(x\right){\alpha }_k\left(x\right)+h_i\left(x\right){\alpha }_i\left(x\right)\in {\mathcal{S}}^{(i-1)}.$$

Since $c\left(x\right)\in C$, we conclude that $c\left(x\right)\in C\cap {\mathcal{S}}^{(i-1)}=C^{(i-1)}$. Hence,

$$ker\left({\pi }_i{|}_{C^{\left(i\right)}}\right)\subseteq C^{(i-1)}.$$

Together with the first inclusion, this yields

$$ker\left({\pi }_i{|}_{C^{\left(i\right)}}\right)=C^{(i-1)}.$$

Step 4: The quotient and the rank. From Steps 2 and 3, the restriction ${\pi }_i{|}_{C^{\left(i\right)}}$ yields a short exact sequence of $R\left[x\right]$-modules

$$0⟶C^{(i-1)}⟶C^{\left(i\right)}\stackrel{{\pi }_i}{\to }R\left[x\right]g_{ii}\left(x\right)⟶0,$$

and hence

$$C^{\left(i\right)}/C^{(i-1)}\cong R\left[x\right]g_{ii}\left(x\right)$$

as $R\left[x\right]$-modules.

By Lemma 7, under the same regularity hypothesis, $R\left[x\right]g_{ii}\left(x\right)$ is a free R-module of rank $e_i-deg{g}_{ii}\left(x\right)$. Therefore,

$${rk}_R\left(C^{\left(i\right)}/C^{(i-1)}\right)=e_i-deg{g}_{ii}\left(x\right).$$

This completes the proof. □

Corollary 3.

Assume the hypotheses of Theorem 4 hold for every $i\in \{1,\dots ,\mathsf{\ell }\}$. In particular, for each i write

$$P_{e_i,{\alpha }_i}\left(x\right)=q_i\left(x\right)g_{ii}\left(x\right)$$

with $q_i\left(x\right)$ monic, and assume that the class of $g_{ii}\left(x\right)$ is not a zero divisor in $R\left[x\right]/⟨q_i\left(x\right)⟩$. Then, for each i, the R-module $C^{\left(i\right)}$ is free and

$${rk}_R\left(C^{\left(i\right)}\right)=\sum _{t=1}^i\left(e_t-deg{g}_{tt}\left(x\right)\right).$$

(49)

In particular,

$${rk}_R\left(C\right)=\sum _{t=1}^{\mathsf{\ell }}\left(e_t-deg{g}_{tt}\left(x\right)\right).$$

(50)

Proof. 

Fix $i\in \{1,\dots ,\mathsf{\ell }\}$. By Theorem 4, for each $t\in \{1,\dots ,i\}$ we have an $R\left[x\right]$-module isomorphism

$$C^{\left(t\right)}/C^{(t-1)}\cong R\left[x\right]g_{tt}\left(x\right)\subseteq R_{{\alpha }_t,e_t},$$

and moreover

$${rk}_R\left(C^{\left(t\right)}/C^{(t-1)}\right)=e_t-deg{g}_{tt}\left(x\right).$$

(51)

Step 1. By (51), the quotient $C^{\left(t\right)}/C^{(t-1)}$ has finite R-rank. Furthermore, by Theorem 4 it is isomorphic to the RPC code $R\left[x\right]g_{tt}\left(x\right)$ inside $R_{{\alpha }_t,e_t}$. Under the standing regularity hypothesis (the class of $g_{tt}\left(x\right)$ is not a zero divisor in $R\left[x\right]/⟨q_t\left(x\right)⟩$), Lemma 7 applies and shows that $R\left[x\right]g_{tt}\left(x\right)$ is a freeR-module of rank $e_t-deg{g}_{tt}\left(x\right)$. Hence, each quotient $C^{\left(t\right)}/C^{(t-1)}$ is a free R-module.

Step 2. We proceed by induction on t to compute ${rk}_R\left(C^{\left(t\right)}\right)$. We prove (49) by induction on $t\in \{0,1,\dots ,i\}$.

Base case $t=0$. We have $C^{\left(0\right)}=0$, hence ${rk}_R\left(C^{\left(0\right)}\right)=0$, which agrees with the empty sum.

Inductive step. Assume

$${rk}_R\left(C^{(t-1)}\right)=\sum _{j=1}^{t-1}\left(e_j-deg{g}_{jj}\left(x\right)\right)\mathrm{for}\mathrm{some}1\le t\le i.$$

Consider the short exact sequence of R-modules

$$0⟶C^{(t-1)}⟶C^{\left(t\right)}⟶C^{\left(t\right)}/C^{(t-1)}⟶0.$$

(52)

By Step 1, the quotient $C^{\left(t\right)}/C^{(t-1)}$ is a free R-module. Over a finite chain ring (such as $R={\mathbb{F}}_q+u{\mathbb{F}}_q$), every short exact sequence (52) with a free quotient splits as R-modules, so

$$C^{\left(t\right)}\cong C^{(t-1)}\oplus \left(C^{\left(t\right)}/C^{(t-1)}\right)\mathrm{as}R\text{-}\mathrm{modules}.$$

Taking R-ranks and using (51) gives

$${rk}_R\left(C^{\left(t\right)}\right)={rk}_R\left(C^{(t-1)}\right)+{rk}_R\left(C^{\left(t\right)}/C^{(t-1)}\right)=\sum _{j=1}^{t-1}\left(e_j-deg{g}_{jj}\left(x\right)\right)+\left(e_t-deg{g}_{tt}\left(x\right)\right),$$

which is exactly the desired formula for $C^{\left(t\right)}$. This completes the induction and proves (49).

Finally, taking $i=\mathsf{\ell }$ in (49) yields (50). □

Corollary 4.

Assume the hypotheses of Theorem 4 for $\mathsf{\ell }=2$. Then,

$${rk}_R\left(C\right)=(e_1-deg{g}_{11}\left(x\right))+(e_2-deg{g}_{22}\left(x\right)).$$

(53)

Proof. 

Apply Corollary 3 with $\mathsf{\ell }=2$. □

As an illustration of the practical use of these Equations (50) and (53), consider, for instance, when $(e_1,e_2)=(8,8)$ and $(deg{g}_{11},deg{g}_{22})=(4,3)$ from Table 1. By Corollary 3, we obtain

$${rk}_R\left(C\right)=(8-4)+(8-3)=9,$$

(54)

and hence the Gray image has ${\mathbb{F}}_3$-dimension $2{rk}_R\left(C\right)=18.$ Corollary 4 is the special case $\mathsf{\ell }=2,$ and similarly determines the rank for the remaining codes listed in the table.

Table 1. Codes over ${\mathbb{F}}_9$ obtained as Gray images of 2-generator right GQPC codes of index 2 over $R={\mathbb{F}}_9+u{\mathbb{F}}_9$.

Block $({\mathit{e}}_1,{\mathit{e}}_2)$	${[\mathit{n},\mathit{k},\mathit{d}]}_9$	$(deg{\mathit{g}}_{11},deg{\mathit{g}}_{21},deg{\mathit{g}}_{22})$	${rk}_{\mathit{R}}\left(\mathit{C}\right)$	Status
(6, 6)	[24, 16, 4]	(2, 1, 2)	8	optimal
(7, 6)	[26, 18, 4]	(2, 1, 2)	9	optimal
(7, 7)	[28, 16, 6]	(3, 2, 3)	8	optimal
(8, 7)	[30, 18, 5]	(3, 2, 3)	9	near-optimal
(8, 8)	[32, 18, 6]	(4, 3, 3)	9	optimal
(9, 8)	[34, 22, 6]	(3, 2, 3)	11	optimal
(9, 9)	[36, 24, 6]	(3, 2, 3)	12	optimal
(10, 9)	[38, 26, 5]	(3, 2, 3)	13	near-optimal
(10, 10)	[40, 28, 6]	(3, 2, 3)	14	optimal

5. Examples and Optimal Codes via the Gray Map

In this section, we illustrate the structural results of Section 3 and Section 4 by constructing explicit families of right generalized quasi-polycyclic (GQPC) codes over the chain ring $R={\mathbb{F}}_q+u{\mathbb{F}}_q$ with $u^2=0$, and by analyzing the parameters of their Gray images. We employ the Gray map

$$\Phi :R⟶{\mathbb{F}}_q^2,\Phi (a+ub)=(b,a+b),$$

extended coordinatewise to $\Phi :R^N\to {\mathbb{F}}_q^{2N}$. As recalled in Section 2, $\Phi$ is an ${\mathbb{F}}_q$-linear isometry. Consequently,

$$d\left(C\right)=d\left(\Phi \left(C\right)\right)$$

(55)

for every R-linear code $C\subseteq R^N$. Although we restrict attention here to index 2, this case already captures the essential interaction between blockwise polycyclic constraints and off-diagonal coupling, providing sufficient algebraic flexibility to produce optimal and near-optimal Gray-image codes while keeping the structural analysis and computational verification tractable. All computations were performed in Magma (V2.29-4) [15], and optimality claims are with respect to the best-known parameters reported in [16].

Throughout this section, we restrict to 2-generator right GQPC codes of index 2 with block lengths $(e_1,e_2)$ and normalized generators of the form

$$C={⟨{\alpha }_1\left(x\right),{\alpha }_2\left(x\right)⟩}_{R\left[x\right]}\subseteq R^{e_1}\times R^{e_2},{\alpha }_1\left(x\right)=(g_{11}\left(x\right),0),{\alpha }_2\left(x\right)=(g_{21}\left(x\right),g_{22}\left(x\right)),$$

where $g_{11}\left(x\right)$ and $g_{22}\left(x\right)$ are monic divisors of the defining polynomials $P_{e_1,{\alpha }_1}\left(x\right)$ and $P_{e_2,{\alpha }_2}\left(x\right)$, respectively, and $deg{g}_{21}\left(x\right)<deg{g}_{11}\left(x\right)$. Let $N=e_1+e_2$. By Corollary 4,

$${rk}_R\left(C\right)=(e_1-deg{g}_{11}\left(x\right))+(e_2-deg{g}_{22}\left(x\right)),$$

and the Gray image $\Phi \left(C\right)$ is an ${\mathbb{F}}_q$-linear code of length $2N$ and dimension $2{rk}_R\left(C\right)$.

Example 2.

This example illustrates the CRT verification for the ${\mathbb{F}}_9$ constructions. Let ${\mathbb{F}}_9={\mathbb{F}}_3\left(\omega \right)$ with ${\omega }^2+\omega +2=0$, and let $R={\mathbb{F}}_9+u{\mathbb{F}}_9$. For $(e_1,e_2)=(6,6)$, consider

$$P_1\left(x\right)=x^6+(1+u)x+1,P_2\left(x\right)=x^6+u{x}^2+x+1.$$

Reducing modulo $⟨u⟩$ yields

$$\overline{P_1}\left(x\right)=\overline{P_2}\left(x\right)=x^6+x+1\in {\mathbb{F}}_9\left[x\right],$$

which is squarefree and factors as

$$x^6+x+1=(x^2+x+1)(x^4+x^3+2x^2+1).$$

Hence, the hypotheses of the Chinese Remainder Theorem in Section 3 are satisfied. The same verification applies to $e\in \{7,8,9,10\}$ using $P_1\left(x\right)=x^e+(1+u)x+1$ and $P_2\left(x\right)=x^e+u{x}^2+x+1$.

The parameters ${[n,k,d]}_q$ satisfy $n=2(e_1+e_2)$ and $k=2{rk}_R\left(C\right)$, in accordance with Corollary 4.

Remark 5.

For a given alphabet size q and parameters $(n,k)$, let $d^{*}(n,k,q)$ denote the largest minimum distance among all known q-ary linear codes with parameters ${[n,k,d]}_q$, as reported in Grassl’s tables [16]. We label a code optimal if $d=d^{*}(n,k,q)$, and near-optimal if $d=d^{*}(n,k,q)-1$.

Example 3.

In this example and Table 2, we verify CRT for the ternary constructions. Let $R={\mathbb{F}}_3+u{\mathbb{F}}_3$. For $(e_1,e_2)=(5,5)$, take

$$P_1\left(x\right)=x^5+(1+u)x+2,P_2\left(x\right)=x^5+u{x}^2+x+1.$$

Modulo $⟨u⟩$,

$$\overline{P_1}\left(x\right)=x^5+x+2=(x^2+1)(x^3+2x^2+1),\overline{P_2}\left(x\right)=x^5+x+1=(x^2+x+2)(x^3+2x+1),$$

which are squarefree with coprime factors. The same conclusion holds for $e\in \{6,7,8,9\}$.

Table 2. Ternary codes obtained as Gray images of 2-generator right GQPC codes of index 2 over $R={\mathbb{F}}_3+u{\mathbb{F}}_3$.

Block $({\mathit{e}}_1,{\mathit{e}}_2)$	${[\mathit{n},\mathit{k},\mathit{d}]}_3$	$(deg{\mathit{g}}_{11},deg{\mathit{g}}_{21},deg{\mathit{g}}_{22})$	${rk}_{\mathit{R}}\left(\mathit{C}\right)$	Status
(5, 5)	[20, 12, 5]	(2, 1, 2)	6	optimal
(6, 5)	[22, 14, 5]	(2, 1, 2)	7	optimal
(6, 6)	[24, 16, 5]	(2, 1, 2)	8	optimal
(7, 6)	[26, 18, 5]	(2, 1, 2)	9	optimal
(7, 7)	[28, 20, 5]	(2, 1, 2)	10	near-optimal
(8, 7)	[30, 22, 5]	(2, 1, 2)	11	optimal
(8, 8)	[32, 24, 6]	(2, 1, 2)	12	optimal
(9, 8)	[34, 26, 6]	(2, 1, 2)	13	optimal
(9, 9)	[36, 28, 6]	(2, 1, 2)	14	optimal

Table 2. Ternary codes obtained as Gray images of 2-generator right GQPC codes of index 2 over $R={\mathbb{F}}_3+u{\mathbb{F}}_3$.

Block $({\mathit{e}}_1,{\mathit{e}}_2)$	${[\mathit{n},\mathit{k},\mathit{d}]}_3$	$(deg{\mathit{g}}_{11},deg{\mathit{g}}_{21},deg{\mathit{g}}_{22})$	${rk}_{\mathit{R}}\left(\mathit{C}\right)$	Status
(5, 5)	[20, 12, 5]	(2, 1, 2)	6	optimal
(6, 5)	[22, 14, 5]	(2, 1, 2)	7	optimal
(6, 6)	[24, 16, 5]	(2, 1, 2)	8	optimal
(7, 6)	[26, 18, 5]	(2, 1, 2)	9	optimal
(7, 7)	[28, 20, 5]	(2, 1, 2)	10	near-optimal
(8, 7)	[30, 22, 5]	(2, 1, 2)	11	optimal
(8, 8)	[32, 24, 6]	(2, 1, 2)	12	optimal
(9, 8)	[34, 26, 6]	(2, 1, 2)	13	optimal
(9, 9)	[36, 28, 6]	(2, 1, 2)	14	optimal

Example 4

(Optimality). All examples in Table 3 satisfy

$$k=2\left[(e_1-d_1)+(e_2-d_2)\right],$$

in exact agreement with Corollary 4. The code attains the minimum distance equal to the best-known value for its length and dimension as reported in Grassl’s online tables [16], and is therefore optimal or near-optimal with respect to currently known bounds.

Table 3. Properties of selected right GQPC codes and their Gray images, $(d_1,d_2)=(deg{g}_{11},deg{g}_{22})$.

$[\mathit{n},\mathit{k},\mathit{d}]$	Alphabet	Block $({\mathit{e}}_1,{\mathit{e}}_2)$	$({\mathit{d}}_1,{\mathit{d}}_2)$	${rk}_{\mathit{R}}\left(\mathit{C}\right)$
$[24,16,4]$	${\mathbb{F}}_9$	(6, 6)	(2, 2)	8
$[28,16,6]$	${\mathbb{F}}_9$	(7, 7)	(3, 3)	8
$[32,18,6]$	${\mathbb{F}}_9$	(8, 8)	(4, 3)	9
$[36,24,6]$	${\mathbb{F}}_9$	(9, 9)	(3, 3)	12
$[40,28,6]$	${\mathbb{F}}_9$	(10, 10)	(3, 3)	14
$[20,12,5]$	${\mathbb{F}}_3$	(5, 5)	(2, 2)	6
$[24,16,5]$	${\mathbb{F}}_3$	(6, 6)	(2, 2)	8
$[32,24,6]$	${\mathbb{F}}_3$	(8, 8)	(2, 2)	12
$[36,28,6]$	${\mathbb{F}}_3$	(9, 9)	(2, 2)	14

Table 3. Properties of selected right GQPC codes and their Gray images, $(d_1,d_2)=(deg{g}_{11},deg{g}_{22})$.

$[\mathit{n},\mathit{k},\mathit{d}]$	Alphabet	Block $({\mathit{e}}_1,{\mathit{e}}_2)$	$({\mathit{d}}_1,{\mathit{d}}_2)$	${rk}_{\mathit{R}}\left(\mathit{C}\right)$
$[24,16,4]$	${\mathbb{F}}_9$	(6, 6)	(2, 2)	8
$[28,16,6]$	${\mathbb{F}}_9$	(7, 7)	(3, 3)	8
$[32,18,6]$	${\mathbb{F}}_9$	(8, 8)	(4, 3)	9
$[36,24,6]$	${\mathbb{F}}_9$	(9, 9)	(3, 3)	12
$[40,28,6]$	${\mathbb{F}}_9$	(10, 10)	(3, 3)	14
$[20,12,5]$	${\mathbb{F}}_3$	(5, 5)	(2, 2)	6
$[24,16,5]$	${\mathbb{F}}_3$	(6, 6)	(2, 2)	8
$[32,24,6]$	${\mathbb{F}}_3$	(8, 8)	(2, 2)	12
$[36,28,6]$	${\mathbb{F}}_3$	(9, 9)	(2, 2)	14

Example 5.

Let $q=3$ and $R={\mathbb{F}}_3+u{\mathbb{F}}_3$ with $u^2=0$. Consider an index-2 right GQPC code

$$C={⟨{\alpha }_1\left(x\right),{\alpha }_2\left(x\right)⟩}_{R\left[x\right]}\subseteq R^{e_1}\times R^{e_2},$$

with normalized generators

$${\alpha }_1\left(x\right)=(g_{11}\left(x\right),0),{\alpha }_2\left(x\right)=(g_{21}\left(x\right),g_{22}\left(x\right)),$$

where

$$g_{11}\left(x\right)=x^4+x+2,g_{21}\left(x\right)=u(x^2+1),g_{22}\left(x\right)=x^3+2x+1.$$

Here, $g_{11}\left(x\right)$ and $g_{22}\left(x\right)$ are monic divisors of the corresponding defining polynomials $P_1\left(x\right)$ and $P_2\left(x\right)$ in $R\left[x\right]$, and $deg{g}_{21}\left(x\right)<deg{g}_{11}\left(x\right)$. Hence $\{{\alpha }_1\left(x\right),{\alpha }_2\left(x\right)\}$ satisfies the triangular support condition of Definition 2. Assuming the regularity hypotheses of Theorem 4 for $\mathsf{\ell }=2$, the R-rank of C is given by the index-2 rank formula

$${rk}_R\left(C\right)=(e_1-deg{g}_{11}\left(x\right))+(e_2-deg{g}_{22}\left(x\right)),$$

(56)

in accordance with Corollary 4.

6. Conclusions

This paper developed a structural and constructive theory of right generalized quasi-polycyclic (GQPC) codes over the finite chain ring $R={\mathbb{F}}_q+u{\mathbb{F}}_q$ with $u^2=0$. GQPC codes were characterized as $R\left[x\right]$-submodules of suitable product rings, and, under natural factorization assumptions, a Chinese Remainder Theorem decomposition was obtained, reducing their study to codes over finite chain ring extensions. A key contribution is the introduction of normalized generating sets with an upper-triangular structure, leading to an explicit rank formula in terms of the diagonal generator polynomials. Via the Gray map, this framework yields precise dimension formulas for the associated q-ary linear codes. The construction produces several optimal and near-optimal ternary and ${\mathbb{F}}_9$-ary linear codes, demonstrating that the framework is both structurally transparent and computationally effective.

From an application perspective, the families of right GQPC codes studied here hold practical relevance for communication systems, data storage, and quantum error-correcting code constructions [3,14]. The results show that the right GQPC codes over ${\mathbb{F}}_q+u{\mathbb{F}}_q$ offer a meticulously organized algebraic structure that facilitates systematic parameter adjustment and improve encoding efficiency. These codes produce q-ary linear codes with competitive parameters through distance-compatible Gray maps and provide a suitable basis for quantum code constructions using CSS-type and similar techniques.

The current study is limited to the finite chain ring $R={\mathbb{F}}_q+u{\mathbb{F}}_q$ and to standard factorization criteria that ensure CRT decompositions and normalized generating forms. A natural direction for future work is to broaden the study to encompass larger categories of finite local and non-chain rings. Further examination of duality and LCD features, an in-depth structural analysis of constituent modules, and the formulation of decoding algorithms tailored to the ring-based framework represent significant opportunities for ongoing research. In general, the results show that right GQPC codes over ${\mathbb{F}}_q+u{\mathbb{F}}_q$ make a strong and flexible algebraic foundation for constructing high-quality linear codes, while also raising a number of theoretical and practical questions that warrant further investigation.