Optimal GQPC Codes over the Finite Field F_q

Abstract

This paper presents the algebraic structure of generalized quasi-polycyclic (GQPC) codes, which is a generalization of the right quasi-polycyclic (QPC) and generalized quasi-cyclic (GQC) codes over a finite field $F_q$. Here, we mainly study the multi-generator polynomial of the right GQPC codes of index l. In this regard, we use the Chinese Remainder Theorem to decompose the right GQPC codes into their constituent codes. Further, we determine the dimension of a right GQPC code and provide a method for finding a normalized generating set for a multi-generator right GQPC code. As a by-product, we provide some examples of GQPC codes and obtain several optimal and near-optimal 2-generator right GQPC codes of index 2 over $F_2$.

1. Introduction

The class of polycyclic (Pseudocyclic) codes emerges as the generalized class of linear codes from the study of cyclic, negacyclic, and constacyclic codes. In 1990, Krishna and Sarwate [1] introduced the term pseudocyclic code while studying constacyclic code. Later, Lopez-Permouth et al. [2] renamed pseudocyclic codes as polycyclic codes. Recall that a right polycyclic (RPC) code on a finite field $F_q$ of length n is the ideal of the quotient ring $F_q\left[z\right]/⟨f\left(z\right)⟩$ where $f\left(z\right)$ is a monic polynomial of degree n. Many researchers have studied the algebraic structure of polycyclic codes in various ways over different code alphabets, considering both finite fields and rings [3,4,5,6,7,8]. Further, some of the studies [9,10] have explored quantum MDS codes as applications of RPC codes. Later, Lopez-Permouth et al. [2] established the duality relationship between polycyclic codes and sequential codes.

On the other hand, we have another generalization of the family of cyclic codes, known as quasi-cyclic (QC) codes. A QC code of length n and index l is defined as a nonempty subspace C of ${F_q^n}^l$ with the property that the cyclic shift of any arbitrary codeword of C by l components is again a codeword of C. Note that for $l=1$, the family of QC codes is in fact the cyclic codes. Further, an important subclass of QC codes is the 1-generator QC codes. Similar to the cyclic codes [11,12], these codes employ a 1-generator polynomial to achieve a compact representation and facilitate efficient encoding. For more details, refer to [13,14]. Recently, Hassan et al. [15] extended the QC and quasi-twisted frameworks to the 1-generator right (or left) quasi-polycyclic (QPC) codes by replacing the cyclic shift with the right (or left) polycyclic shift. However, from the structural point of view, these codes can be realized as an $F_q\left[z\right]$-submodule of ${\left(F_q\left[z\right]/⟨f\left(z\right)⟩\right)}^l$. This approach has also facilitated the construction of many best-known linear and quantum codes over finite fields and rings, as reported in [15,16,17]. However, their work primarily considered on 1-generator QPC codes, providing a valuable tabulation of optimal codes for short lengths and dimensions. In addition to this framework, generalized quasi-cyclic (GQC) and generalized quasi-twisted (multi-twisted) codes have further extensions and remain an active area of research. In particular, many researchers extended the concept of QC codes to the GQC setting in both commutative and noncommutative polynomial rings [18,19,20,21,22,23,24,25,26]. These generalizations unify several well-known code families under a single algebraic structure. It has led to the explicit construction of numerous new linear codes, including many that meet or improve the previously best-known parameters.

Ou-Azzou et al. [27] studied GQPC codes in skew polynomial rings with derivations over finite fields. Subsequently, Kabbouch et al. [28] extended the notions of GQC and QPC codes to define right GQPC codes over $F_q$. Both works focused on 1-generator right GQPC codes and reported several good linear codes, with Ou-Azzou’s constructions corresponding to short code lengths and small dimensions. In this research paper, we extend the concept of right QPC to GQPC codes over the finite fields $F_q$ with l induced vectors ${\mathtt{a}}_1^{t_1},{\mathtt{a}}_2^{t_2},\dots ,{\mathtt{a}}_l^{t_l}$ in $F_q^{t_i}$ for all $i=1,2,\dots ,l$, respectively. That is, we do not restrict our work to a 1-generator right GQPC code only. Our main contributions in this paper are as follows:

• Establish the decomposition theorem for the right GQPC codes of index l.
• Discuss a method to obtain a normalized generating set for the $\rho$-generator right GQPC codes of index l.
• Using the MAGMA software [29], constructing many optimal and near-optimal 2-generator right GQPC codes whose parameters achieve the bounds given in the Codes Table [30].

This study proceeds as follows: Section 2 recalls the basic concepts of RPC codes over the finite field $F_q$. In Section 3, we describe the structure of the right GQPC code and demonstrate the code decomposition theorem with examples. Section 4 contains the normalized generating set of the right GQPC code as the $\rho$-generator polynomial of the index l. Additionally, this section presents a table listing optimal and near-optimal GQPC codes of index 2 derived from the 2-generator polynomial set. Section 5 concludes the work.

2. Background

In this section, we recall the basic concepts of right polycyclic (RPC) codes over the finite field $F_q$ that we will use throughout this paper. Let $F_q$ be a finite field with q elements, where $q=p^d$ for some prime number p and an integer $d\ge 1$. A subspace C of $F_q^n$ (n-dimensional vector space) is called a linear code of length n over $F_q$, and elements of C are called codewords. Let $\mathtt{c}=(c_0,c_1,\dots ,c_{n-1})$ be a codeword in C. Then the number of nonzero elements in $\mathtt{c}$ is called its (Hamming) weight. It is denoted by $w_H\left(\mathtt{c}\right)$. The weight $w_H\left(C\right)$ of C is the minimum among the weights of all its nonzero codewords. Let $\mathtt{a}=(a_0,a_1,\dots ,a_{n-1})$, $\mathtt{b}=(b_0,b_1,\dots ,b_{n-1})\in F_q^n$. Then the (Hamming) distance between $\mathtt{a}$ and $\mathtt{b}$ is defined as $d(\mathtt{a},\mathtt{b})=|\{i:a_i\ne b_i\}|$, equivalently $d(a,b)=w_H(a-b)$. The distance of code C, denoted by $d_H\left(C\right)$, is defined as $d_H\left(C\right)=min\{d(\mathtt{a},\mathtt{b}):\mathtt{a},\mathtt{b}\in C,\mathtt{a}\ne \mathtt{b}\}$. For a linear code $C\subseteq F_q^n$, $d_H\left(C\right)=w_H\left(C\right)$. Note that a k-dimensional linear code C of length n over the finite field $F_q$ is represented by $[n,k,d_H]$, where $d_H$ is the distance of C.

Definition 1. 

Let $(y_0,y_1,\dots ,y_{n-1})$ and $\mathtt{a}=(a_0,a_1,\dots ,a_{n-1})$ be two vectors in $F_q^n$ with $a_0\ne 0$. The right polyshift operator $P_r^{\mathtt{a}}:F_q^n\mapsto F_q^n$ is given by

$$\begin{array}{cc}\hfill & (y_0,y_1,\dots ,y_{n-1})\mapsto y_{n-1}\mathtt{a}+(0,y_0,\dots ,y_{n-2}).\hfill \end{array}$$

Definition 2. 

Let $C\subseteq F_q^n$ be a linear code over $F_q$ and $\mathtt{a}=(a_0,a_1,\dots ,a_{n-1})\in F_q^n$. Then C is called an RPC code induced by a vector $\mathtt{a}$ if for every codeword $\mathtt{c}=(c_0,c_1,\dots ,c_{n-1})\in C$, the right polyshift of $\mathtt{c}$ is $P_r^{\mathtt{a}}\left(\mathtt{c}\right)=c_{n-1}\mathtt{a}+(0,c_0,\dots ,c_{n-2})\in C$. For a linear code C, $P_r^{\mathtt{a}}$ is called the RPC shift induced by $\mathtt{a}$.

In light of the above definition, we can express $P_r^{\mathtt{a}}\left(\mathtt{c}\right)$ in terms of the following matrix:

$$\begin{array}{c}\hfill P_r^{\mathtt{a}}\left(\mathtt{c}\right)=\mathtt{c}{A}_r,\mathrm{where}{A}_r=\left(\begin{array}{cccccc}0& 1& 0& \dots & 0& 0\\ 0& 0& 1& \dots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & 0& 1\\ a_0& a_1& a_2& \dots & 0& a_{n-1}\end{array}\right).\end{array}$$

Hence, a linear code C is a right polycyclic code if it is an invariant subspace under right multiplication by $A_r$.

Remark 1. 

In Definition 2 and throughout the paper, we use the term right polycyclic codes corresponding to the right polyshift, and it has no connection with the concept of right modules.

RPC codes are the generalization of cyclic and constacyclic codes over the field $F_q$. Recall that cyclic and constacyclic codes of length n over $F_q$ can be described as ideals of rings $F_q\left[z\right]/⟨z^n-\lambda ⟩$ and $F_q\left[z\right]/⟨z^n-1⟩$, respectively, where $\lambda$ is a nonzero element of $F_q$. Extending this concept, an RPC code of length n is also an ideal of the ring $F_q\left[z\right]/⟨z^n-\mathtt{a}\left(z\right)⟩$, where $\mathtt{a}\left(z\right)$ is a polynomial over $F_q$ of degree strictly less than n (see [2,16]). For convenience, we recall their structural properties and restate them in our notation.

Let $R=F_q\left[z\right]$ be a polynomial ring over the field $F_q$. Consider a vector $\mathtt{a}=(a_0,a_1,\dots ,a_{n-1})$ in $F_q^n$, and associate this with the polynomial $\mathtt{a}\left(z\right)=a_0+a_{1}z+\cdots +a_{n-1}{z}^{n-1}$ in the polynomial ring R. Let $P_{n,\mathtt{a}}\left(z\right)=z^n-\mathtt{a}\left(z\right)$ be a polynomial of degree n with $P_{n,\mathtt{a}}\left(0\right)\ne 0$. From [31] (Page No. 84), there always exists a positive integer m less than or equal to $q^n-1$ such that $P_{n,\mathtt{a}}\left(z\right)$ divides $z^m-1$. The lowest positive integer m for which this property holds is called the order of the polynomial $P_{n,\mathtt{a}}\left(z\right)$. In the case of $P_{n,\mathtt{a}}\left(0\right)=0$, we can write $P_{n,\mathtt{a}}\left(z\right)=z^{k}Q\left(z\right)$ for some polynomial $Q\left(z\right)\in R$ with $Q\left(0\right)\ne 0$ and a positive integer $k\ge 1$. In addition, the order of $P_{n,\mathtt{a}}\left(z\right)$ is defined as the order of $Q\left(z\right)$. Consider a principal ideal $⟨P_{n,\mathtt{a}}\left(z\right)⟩$ in the polynomial ring R and denote the quotient ring $R/⟨P_{n,\mathtt{a}}\left(z\right)⟩$ by $R_n^{\mathtt{a}}$. Define a $F_q$-linear map

$$\begin{array}{cc}\hfill \mathrm{\Phi }:F_q^n& \mapsto R_n^{\mathtt{a}}\mathrm{given}\mathrm{by}\hfill \\ \hfill (r_0,r_1,\dots ,r_{n-1})& \mapsto r_0+r_{1}z+\cdots +r_{n-1}{z}^{n-1},\hfill \end{array}$$

in the polynomial ring R. Let $\mathtt{c}=(c_0,c_1,\dots ,c_{n-1})$ be an element of the linear code C. Then $\varphi \left(\mathtt{c}\right)=c_0+c_{1}z+\cdots +c_{n-1}{z}^{n-1}$ is the polynomial representation of the codeword $\mathtt{c}$. Observe that $z\varphi \left(\mathtt{c}\right)=c_{0}z+c_1{z}^2+\cdots +c_{n-1}{z}^n$. Since $z^n=\mathtt{a}\left(z\right)$ in $R_n^{\mathtt{a}}$, it follows that $z\varphi \left(\mathtt{c}\right)=c_{0}z+c_1{z}^2+\cdots +c_{n-1}\mathtt{a}\left(z\right)$ which corresponds to the RPC shift $P_r^{\mathtt{a}}\left(\mathtt{c}\right)$ of a codeword $\mathtt{c}$, i.e., $z\varphi \left(\mathtt{c}\right)\in C$. Based on the above discussion, the following proposition presents the ideal structure of an RPC code over the field $F_q$.

Proposition 1 

([16] Section 2). Let C be a linear code over $F_q$ of length n. Then C is an RPC code induced by a vector $\mathtt{a}=(a_0,a_1,\dots ,a_{n-1})$ if and only if C is an ideal of the quotient ring $R_n^{\mathtt{a}}$.

Let C be an RPC code of length n over the field $F_q$. Then the following results [5] hold:

• C is principally generated by a least degree monic polynomial $f\left(z\right)$ such that $f\left(z\right)\mid P_{n,\mathtt{a}}\left(z\right)$.
• C has a basis set $\{f\left(z\right),zf\left(z\right),\dots ,z^{k-1}f\left(z\right)\}$ and the corresponding generating matrix for C is

  $$\begin{array}{c}\hfill M=\left(\begin{array}{c}{\mathrm{\Phi }}^{-1}\left(f\left(z\right)\right)\\ {\mathrm{\Phi }}^{-1}\left(zf\left(z\right)\right)\\ ⋮\\ {\mathrm{\Phi }}^{-1}\left(z^{k-1}f\left(z\right)\right)\end{array}\right),\mathrm{where}k=n-deg\left(f\left(z\right)\right).\end{array}$$
• $dimC$, the dimension of C is $k=n-deg\left(f\right(z\left)\right)$.

Lemma 1. 

Let $P_{n,\mathtt{a}}\left(z\right)=z^n-\mathtt{a}\left(z\right)\in R$, and it factors into monic polynomials as $P_{n,\mathtt{a}}\left(z\right)=b\left(z\right)d\left(z\right)$. Suppose C is an RPC code generated by $d\left(z\right)$ with $b\left(z\right)$ as its corresponding parity-check polynomial. Then a polynomial $r\left(z\right)\in R_n^{\mathtt{a}}$ is a generator of the code C if and only if there exists a polynomial $s\left(z\right)$ in R such that $r\left(z\right)=s\left(z\right)d\left(z\right)$ and $gcd\left(b\right(z),s(z\left)\right)=1$.

Proof. 

Consider $r\left(z\right)=s\left(z\right)d\left(z\right)$, where $gcd\left(b\right(z),s(z\left)\right)=1$. Then, there exist some polynomials $\mu \left(z\right)$ and $\nu \left(z\right)$ in R such that

$$\begin{array}{c}\hfill \mu \left(z\right)s\left(z\right)+\nu \left(z\right)b\left(z\right)=1.\end{array}$$

(1)

If both sides of the Equation (1) are multiplied by $d\left(z\right)$, then we get $\mu \left(z\right)s\left(z\right)d\left(z\right)+\nu \left(z\right)b\left(z\right)d\left(z\right)=d\left(z\right)$. This expression leads to $\mu \left(z\right)r\left(z\right)=d\left(z\right)mod(z^n-\mathtt{a}\left(z\right))$. Hence, $d\left(z\right)\in Rr\left(z\right)$ where $Rr\left(z\right)$ is the ideal of R. Consequently, $Rd\left(z\right)\subseteq Rr\left(z\right)$. Otherwise, since $r\left(z\right)=s\left(z\right)d\left(z\right)$, it is clear that $Rr\left(z\right)\subseteq Rd\left(z\right)$, and hence $Rr\left(z\right)=Rd\left(z\right)=C$. Thus, it follows that $r\left(z\right)=s\left(z\right)d\left(z\right)$ is a generator of C. □

3. Right Generalized Quasi-Polycyclic (GQPC) Codes

Here, we study the structural properties of right GQPC codes over the finite field $F_q$, mainly focusing on the algebraic structure of $\rho$-generator polynomial right GQPC codes. The definition of right QPC codes of length $lm$ and index l given in [16] (Definition 3.2) extends naturally to right GQPC codes over $F_q$ by considering l different values of m. Similarly, a GQC code of block length $(t_1,t_2,\dots ,t_l)$ and index l extends to a right GQPC code by replacing the cyclic shift with the RPC shift induced by l distinct vectors. We begin with the definition of a GQC code.

Definition 3. 

Let $t_1,t_2,\dots ,t_l\in Z^{+}$ (set of positive integers). Suppose $C\subseteq F_q^{t_1}\times F_q^{t_2}\times \cdots \times F_q^{t_l}$ is a linear code of length $N=t_1+t_2+\cdots +t_l$ and index l over the field $F_q$. Consider a codeword $\mathtt{c}=(c_{t_1},c_{t_2},\dots ,c_{t_l})$ in C where $c_{t_i}=(c_{i,0},c_{i,1},\dots ,c_{i,t_i-1})\in F_q^{t_i}$ for all $i=1,2,\dots ,l$. Then C is said to be a GQC code, if for any $\mathtt{c}\in C$,

$$\begin{array}{c}\hfill \mathit{T}\left(\mathtt{c}\right)=(\mathit{T}\left(c_{t_1}\right),\mathit{T}\left(c_{t_2}\right),\dots ,\mathit{T}\left(c_{t_l}\right))\in C,\end{array}$$

where T is the cyclic shift operator.

By replacing the cyclic shift T in each component of the above definition of GQC codes with the RPC shifts $P_r^{{\mathtt{a}}_i}$ induced by the vector ${\mathtt{a}}_i\in F_q^{t_i}$ for $i=1,2,\dots ,l$, we obtain the class of GQPC codes, which naturally includes the right QPC codes as a special case.

Definition 4. 

A linear code $C\subseteq F_q^{t_1}\times F_q^{t_2}\times \cdots \times F_q^{t_l}$ is a right GQPC code of length $N=t_1+t_2+\cdots +t_l$ and index l if for any codeword $\mathtt{c}\in C$, $(P_r^{{\mathtt{a}}_1}\left(c_{t_1}\right),P_r^{{\mathtt{a}}_2}\left(c_{t_2}\right),\dots ,P_r^{{\mathtt{a}}_l}\left(c_{t_l}\right))\in C$, where

$$\begin{array}{cc}\hfill P_r^{{\mathtt{a}}_i}\left(c_{t_i}\right)=& (0,c_{i,0},c_{i,1},\dots ,c_{i,t_i-2})+c_{i,t_i-1}(a_{i,0},a_{i,1},\dots ,a_{i,t_i-1})\hfill \\ \hfill =& {\left(\begin{array}{c}0+c_{i,t_i-1}{a}_{i,0}\\ c_{i,1}+c_{i,t_i-1}{a}_{i,1}\\ ⋮\\ c_{i,t_i-2}+c_{i,t_i-1}{a}_{i,t_i-1}\end{array}\right)}^{T^{\prime }},\hfill \end{array}$$

with induced vector ${\mathtt{a}}_i=(a_{i,0},a_{i,1},\dots ,a_{i,t_i-1})\in F_q^{t_i},\mathit{for}\mathit{all}i=1,2,\dots ,l$. Here, $T^{\prime }$ denotes the transpose of a matrix.

If each ${\mathtt{a}}_i=(1,0,\dots ,0)$, then the right GQPC code is just the GQC code, and if all $t_1=t_2=\cdots =t_l$, then the right GQPC code is the right QPC code.

Consider the quotient ring $R_{t_i}^{{\mathtt{a}}_i}=$$R/⟨P_{t_i,{\mathtt{a}}_i}\left(z\right)⟩$, where $P_{t_i,{\mathtt{a}}_i}\left(z\right)=z^{t_i}-{\mathtt{a}}_i\left(z\right)$ and ${\mathtt{a}}_i\left(z\right)$ is the polynomial representation of the corresponding induced vector ${\mathtt{a}}_i$ for all $i=1,2,\dots ,l$. Denote $\mathbf{S}=R_{t_1}^{{\mathtt{a}}_1}\times R_{t_2}^{{\mathtt{a}}_2}\times \cdots \times R_{t_l}^{{\mathtt{a}}_l}$ (throughout the paper). Let $r\left(z\right)=r_0+r_{1}z+\cdots +r_m{z}^m\in R$, and $s\left(z\right)=(s_1\left(z\right),s_2\left(z\right),\dots ,s_l\left(z\right))$ be an element in the set $\mathbf{S}$. We introduce the multiplication rule over $\mathbf{S}$ by

$$\begin{array}{c}\hfill \nu \left(z\right)·s\left(z\right)=(\nu \left(z\right)·s_1\left(z\right),\nu \left(z\right)·s_2\left(z\right),\dots ,\nu \left(z\right)·s_l\left(z\right)),\end{array}$$

(2)

where $\nu \left(z\right)·s_i\left(z\right)$ belonging to $R_{t_i}^{{\mathtt{a}}_i}$ for all $i=1,2,\dots ,l$. From Equation (2), it can easily be checked that the ring $\mathbf{S}$ is an R-module. We define a map

$$\begin{array}{c}\hfill M:F_q^{t_1}\times F_q^{t_2}\times \cdots \times F_q^{t_l}\mapsto \mathbf{S}\mathrm{given}\mathrm{by}(c_{t_1},c_{t_2},\dots ,c_{t_l})\mapsto (c_{t_1}\left(z\right),c_{t_2}\left(z\right),\dots ,c_{t_l}\left(z\right)),\end{array}$$

(3)

where $c_{t_i}\left(z\right)=c_{i,0}+c_{i,1}\left(z\right)+c_{i,2}\left(z^2\right)+\cdots +c_{i,t_i-1}{z}^{t_i-1}\in R_{t_i}^{{\mathtt{a}}_i}$. Then, we get the following result, which defines the correspondence between the linear and the right GQPC codes.

Proposition 2 

([28]). The map M defined in Equation (3) gives rise to a bijective map between the set of all right GQPC codes of block length $(t_1,t_2,\dots ,t_l)$ and index l over the field $F_q$, and the set of all R-submodules of the R-module $\mathit{S}$.

Let C be a right GQPC code of block length $(t_1,t_2,\dots ,t_l)$ and index l given by $C=(C_{t_1},C_{t_2},\dots ,C_{t_l})$, where each $C_{t_i}$ is an RPC code of length $t_i$ and generated by $g_{t_i}\left(z\right)\in R_{t_i}$ for $i=1,2,\dots ,l$. Then a 1-generator polynomial of C is an l-tuple $(e_1\left(z\right),e_2\left(z\right),\dots ,e_l\left(z\right))$, where each $e_i\left(z\right)=h_i\left(z\right)g_{t_i}\left(z\right)$, $h_i\left(z\right)$ is relatively prime to the check polynomial of $C_{t_i}$ and $g_{t_i}\left(z\right)\mid (z^{t_i}-{\mathtt{a}}_i\left(z\right))$ for all $i=1,2,\dots ,l$. Consider a 1-generator right GQPC code C of block length $(t_1,t_2,\dots ,t_l)$ and index l over R generated by $e\left(z\right)=(e_1\left(z\right),e_2\left(z\right),\dots ,e_l\left(z\right))\in \mathbf{S}$. Then C can be written as

$$\begin{array}{cc}\hfill C=& \{r\left(z\right)(e_1\left(z\right),e_2\left(z\right),\dots ,e_l\left(z\right)):r\left(z\right)\in R\}\hfill \\ \hfill =& \{(r\left(z\right)e_1\left(z\right),r\left(z\right)e_2\left(z\right),\dots ,r\left(z\right)e_l\left(z\right)):r\left(z\right)\in R\}.\hfill \end{array}$$

A right GQPC code C of block length $(t_1,t_2,\dots ,t_l)$ and index l is called a $\rho$-generator (multi-generator) polynomial over $F_q$ if $\rho$ is the smallest positive integer greater than 1 for which there are codewords $b_i\left(z\right)=(b_{i,1}\left(z\right),b_{i,2}\left(z\right),\dots ,b_{i,l}\left(z\right))\in \mathbf{S}$ for $1\le i\le \rho$ in C such that $C=R{b}_1\left(z\right)+R{b}_2\left(z\right)+\cdots +R{b}_{\rho }\left(z\right)$. To further investigate the algebraic structure of $\rho$-generator right GQPC codes, particularly through the factorization of their defining polynomials, we recall the following fundamental result, the Chinese Remainder Theorem (CRT).

Definition 5 

([32]). Consider a set of polynomials $\{p_1\left(z\right),p_2\left(z\right),\dots ,p_L\left(z\right)\}$ in R, which are pairwise coprime over $F_q$. Let $P\left(z\right)$$=p_1\left(z\right)$$p_2\left(z\right)$⋯$p_L\left(z\right)$. If $p_1^{\prime }\left(z\right),p_2^{\prime }\left(z\right),\dots ,p_L^{\prime }\left(z\right)$ are any set of polynomials in R, then there is exactly one polynomial $e\left(z\right)$ having $deg\left(e\right(z\left)\right)<deg\left(P\right(z\left)\right)$ such that

$$\begin{array}{c}\hfill e\left(z\right)\equiv p_i^{\prime }\left(z\right)mod\left(p_i\left(z\right)\right),\mathit{for}\mathit{every}0\le i\le L.\end{array}$$

(4)

Also, let $r_i\left(z\right)\in R$ be such that

$$\begin{array}{c}\hfill {\displaystyle \frac{P\left(z\right)}{p_i\left(z\right)}}{r}_i\left(z\right)\equiv 1mod{p}_i\left(z\right),0\le i\le L.\end{array}$$

Then the solution for Equation (4) is $e\left(z\right)={\sum }_{i=1}^L{\displaystyle \frac{P\left(z\right)}{p_i\left(z\right)}}{r}_i\left(z\right)p_i^{\prime }\left(z\right)$ reduced $(modP(z\left)\right)$.

Code Decomposition

Esmaeili and Yari [23] presented the decomposition theorem of GQC codes that decompose into their shorter constituent codes using the CRT. According to the CRT, the ring $R/⟨z^n-1⟩$ decomposes into a direct product of finite fields if n is coprime with the prime p. Here, we introduce the decomposition theorem for the right GQPC codes over the field $F_q$.

Let $P_{t_i,{\mathtt{a}}_i}\left(z\right)\in R$, such that $P_{t_i,{\mathtt{a}}_i}\left(0\right)\ne 0$ be a monic polynomial of degree $t_i$ and order $n_i$ with the condition $gcd(n_i,q)=1$ for $i=1,2,\dots ,l$. As $gcd(n_i,q)=1$, $z^{n_i}-1$ reduces into irreducible polynomials in R and since $P_{t_i,{\mathtt{a}}_i}\left(z\right)$ divides $z^{n_i}-1$, then $P_{t_i,{\mathtt{a}}_i}\left(z\right)$ factors into completely distinct irreducible polynomials for $i=1,2,\dots ,l$. Assume that the number of irreducible factors of all the $P_{t_i,{\mathtt{a}}_i}\left(z\right)$ decompositions is t for all $i=1,2,\dots ,l$. Hence, we have

$$\begin{array}{cc}\hfill P_{t_i,{\mathtt{a}}_i}\left(z\right)& =z^{t_i}-{\mathtt{a}}_i\left(z\right)\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & \hfill =p_{t,1}{\left(z\right)}^{u_{i,1}}{p}_{t,2}{\left(z\right)}^{u_{i,2}}\dots p_{t,t}{\left(z\right)}^{u_{i,t}},\end{array}$$

(6)

where $p_{t_i,j}\left(z\right)$ are distinct irreducible polynomials for all $1\le j\le t$ and $u_{i,j}\in \{0,1\}$. Now, $u_{i,j}=1$ for some $1\le j\le t$, then $U_{t_j}=R/⟨p_{t,j}\left(z\right)⟩$ is the finite field extension of $F_q$. For $1\le j\le t$ and $i=1,2,\dots ,l$, we define

$$\begin{array}{c}\hfill U_{t_i,j}=\left\{\begin{array}{cc}{U}_{t_j},\hfill & \mathrm{if}{u}_{i,j}=1,\hfill \\ \left\{0\right\},\hfill & \mathrm{if}{u}_{i,j}=0.\hfill \end{array}\right.\end{array}$$

(7)

Consider a root ${\beta }_j$ for each $P_{t_i,j}\left(z\right)$. Now, for $d_{t_i}\left(z\right)\in R_{t_i}^{{\mathtt{a}}_i}=R/⟨z^{t_i}-a_i\left(z\right)⟩$ and $1\le j\le t$, define

$$\begin{array}{c}\hfill d_{t_i,j}=\left\{\begin{array}{cc}{d}_{t_i}\left({\beta }_j\right),\hfill & \mathrm{if}{U}_{t_i,j}=U_{t_i},\hfill \\ 0,\hfill & \mathrm{if}{U}_{t_i,j}=\left\{0\right\}.\hfill \end{array}\right.\end{array}$$

(8)

Using the above Equations (7) and (8) and CRT, for each $i=1,2,\dots ,l$, we get the following ring isomorphism

$$\begin{array}{c}\hfill R_{t_i}^{{\mathtt{a}}_i}\cong {\oplus }_{j=1}^t{U}_{t_i,j},\end{array}$$

(9)

where the isomorphism is given by $d_{t_i}\left(z\right)$ to $(d_{t_i,1}+d_{t_i,2}+\cdots +d_{t_i,t})$. Then

$$\begin{array}{cc}\hfill \mathbf{S}=R_{t_1}^{{\mathtt{a}}_1}\times R_{t_2}^{{\mathtt{a}}_2}\times \cdots \times R_{t_l}^{{\mathtt{a}}_l}& \cong \left({\oplus }_{j=1}^t{U}_{t_1,j}\right)\times \cdots \times \left({\oplus }_{j=1}^t{U}_{t_l,j}\right)\hfill \end{array}$$

(10)

$$\begin{array}{cc}& \cong {\oplus }_{j=1}^t(U_{t_1,j}\times \cdots \times U_{t_l,j}),\hfill \end{array}$$

(11)

where $(d_{t_1}\left(z\right),d_{t_2}\left(z\right),\dots ,d_{t_l}\left(z\right))\in \mathbf{S}$ maps to ${\sum }_{j=1}^t(d_{t_1,j},d_{t_2,j},\dots ,d_{t_l,j})$. Since some $U_{t_{i}j}$ will be $\left\{0\right\}$, the corresponding coordinates of the codewords in that particular constituent will be 0. Hence, from Proposition 2, a right GQPC code can be observed as a submodule of ${\oplus }_{j=1}^t{U}_{t_j}^l$ as for each i, $U_{t_i,j}$ is either $U_{t_j}$ or $\left\{0\right\}\subset U_{t_j}$.

Proposition 3. 

Let the right GQPC code $C\subset \mathit{S}$ of block length $(t_1,t_2,\dots ,t_l)$ and index l be generated as an R-submodule by

$$\begin{array}{c}\hfill \{(d_{t_1}^1\left(z\right),\dots ,d_{t_l}^1\left(z\right)),\left(d_{t_1}^2\left(z\right)\right),\dots ,d_{t_l}^2\left(z\right)),\dots ,(d_{t_1}^s\left(z\right)\dots ,d_{t_l}^s\left(z\right))\}\subset \mathit{S}.\end{array}$$

Then as a submodule of ${\oplus }_{j=1}^t{U}_{t_j}^l$, C takes the form inside ${\oplus }_{j=1}^t{U}_{t_j}^l$, as $C={\oplus }_{j=1}^t{C}_j$, where every $C_j$ is a l length $U_{t_j}$-linear code and written as

$$\begin{array}{c}\hfill C_j={\mathit{Span}}_{U_{t_j}}\{(d_{j,1}^{b,1},\dots ,d_{j,l}^{b,l}):1\le b\le s\},\mathrm{for}1\le j\le t.\end{array}$$

(12)

Proof. 

Since C is a submodule of $\mathbf{S}$, we have

$$\begin{array}{cc}\hfill C=& \{{\nu }_1\left(z\right)(d_{t_1}^1\left(z\right),\dots ,d_{t_l}^1\left(z\right))+{\nu }_2\left(z\right)\left(d_{t_1}^2\left(z\right)\right),\dots ,d_{t_l}^2\left(z\right))+\cdots +{\nu }_s\left(z\right)(d_{t_1}^s\left(z\right)\dots ,d_{t_l}^s\left(z\right)):\hfill \\ & {\nu }_1,{\nu }_2,\dots ,{\nu }_s\in R\}\hfill \end{array}$$

Now, from Equation (10), $C_j\subset (U_{t_1,j}\times U_{t_2,j}\times \cdots \times U_{t_l,j})$ and can be written as

$$\begin{array}{c}\hfill C_j=\{{\nu }_1\left({\beta }_j\right)(d_{t_1,j}^1,d_{t_2,j}^1,\dots ,d_{t_l,j}^1)+\cdots +{\nu }_s\left({\beta }_j\right)(d_{t_1,j}^s,d_{t_2,j}^s,\dots ,d_{t_l,j}^s):{\nu }_1,{\nu }_2,\dots ,{\nu }_s\in R\}.\end{array}$$

As ${\beta }_j$ is a root of $P_{t,j}\left(z\right)$, we have $U_{t_j}=F_q\left({\beta }_j\right)$. Also, each $U_{t_1,j},U_{t_2,j},\dots ,U_{t_l,j}$ is either $U_{t_j}$ or $\left\{0\right\}\subset U_{t_j}$, therefore, ${\nu }_1\left({\beta }_j\right)$,${\nu }_2\left({\beta }_j\right)$,…,${\nu }_s\left({\beta }_j\right)$ take all possible values in $U_{t_j}$ as the polynomials ${\nu }_1\left(z\right),{\nu }_2\left(z\right),\dots ,{\nu }_s\left(z\right)$ vary over R. Thus, we get

$$\begin{array}{c}\hfill C_j={\mathrm{Span}}_{U_{t_j}}\{(d_{j,1}^{b,1},\dots ,d_{j,l}^{b,l}):1\le b\le s\},\mathrm{for}1\le j\le t.\end{array}$$

□

Theorem 1. 

Let C be a ρ-generator right GQPC code of block length $(t_1,t_2,\dots ,t_l)$ and index l over $F_q$. If $C={\oplus }_{i=1}^t{C}_i$, where $C_i$ is of dimension $k_i$ for $i=1,2,\dots ,t$, and $K=max\{k_i:1\le i\le t\}$. Then $\rho =K$. In addition, any right GQPC code C with constituent codes $C_i$ of dimension $k_i$, satisfying $\rho =max{k}_i$, is a ρ-generator right GQPC code.

Proof. 

Let C be a $\rho$-generator right GQPC code generated by the elements $e^j\left(z\right)=(e_1^j\left(z\right),\dots ,e_l^j\left(z\right))$$\in \mathbf{S}$ for $j=1,2,\dots ,\rho$. Then for each i, $C_i$ is spanned as R-submodule by ${\overline{e}}^j\left(z\right)=({\overline{e}}_1^j\left(z\right),\dots ,{\overline{e}}_l^j\left(z\right))$, where ${\overline{e}}_r^j\left(z\right)=e_r^j\left(z\right)mod\left(p_{t,i}\left(z\right)\right)$, if $p_{t,i}\left(z\right)$ is a factor of $P_{t_i,{\mathtt{a}}_i}\left(z\right)$ and ${\overline{e}}_r^j\left(z\right)=0$ otherwise, for all $r=1,2,\dots ,l$. Further, let ${\beta }_i$ be a root of irreducible factor $P_{t,i}\left(z\right)$ of $P_{t_i,{\mathtt{a}}_i}\left(z\right)$. Then, for each $i=,1,2,\dots ,t$, the generator matrix of $C_i$ contains the $j^{\mathrm{th}}$ row as ${\overline{e}}^j=(e_1^j\left({\beta }_i\right),e_2^j\left({\beta }_i\right),\dots ,e_l^j\left({\beta }_i\right)),j=1,2,\dots ,\rho$. Hence, $k_i\le \rho$ for every i, and thus $K=max\{k_i:1\le i\le t\}\le \rho$.

On the other hand, as $K=max{k}_i$, there exists $e_i^j\left(z\right)\in R^l$, $1\le j\le K$, such that $e_i^j\left(z\right)$ span $C_i$, for $1\le i\le t$, as a R-submodule. Then by Proposition 3, for every $1\le j\le K$ there exists $e^j\left(z\right)\in C$ such that $e_i^j\left(z\right)=e^j\left(z\right)mod\left(p_{t,j}\left(z\right)\right)$ and C is generated by $e_i^j\left(z\right)$ for $1\le j\le K$. Thus, $\rho \le K$ which implies that $\rho =K$. □

If C is a 1-generator right GQPC code of block length $(t_1,t_2,\dots ,t_l)$ and index l over $F_q$, then by Theorem 1, every $C_i$, $i=1,2,\dots ,t$, is either an $[l,1]$ linear code or the trivial code over $R/⟨p_{t,j}\left(z\right)⟩$. Conversely, any right GQPC code whose constituent codes $C_i$ are of dimension at most 1 is a 1-generator right GQPC code.

Example 1. 

Let $q=2,t_1=4,t_2=6,$ and $t_3=8$. Consider the monic polynomials

$$\begin{array}{cc}\hfill P_{t_1,{\mathtt{a}}_1}\left(z\right)& =z^4+z^2+z+1\mathit{of}\mathit{order}7,\mathit{where}{\mathtt{a}}_1=(1,1,1,0)\in F_2^4\hfill \\ \hfill & =(z+1)(z^3+z^2+1),\hfill \\ \hfill P_{t_2,{\mathtt{a}}_2}\left(z\right)& =z^6+z^4+z+1\mathit{of}\mathit{order}21,\mathit{where}{\mathtt{a}}_2=(1,1,0,0,1,0,0)\in F_2^6\hfill \\ \hfill & =(z+1)(z^2+z+1)(z^3+z+1),\mathit{and}\hfill \\ \hfill P_{t_3,{\mathtt{a}}_3}\left(z\right)& =z^8+z^6+z^4+z+1\mathit{of}\mathit{order}217,\mathit{where}{\mathtt{a}}_3=(1,1,0,0,1,0,1,0,0)\in F_2^8\hfill \\ \hfill & =(z^3+z^2+1)(z^5+z^4+z^2+z+1).\hfill \end{array}$$

In this case, we have

$$\begin{array}{c}\hfill R_{t_1}^{{\mathtt{a}}_1}\times R_{t_2}^{{\mathtt{a}}_2}\times R_{t_3}^{{\mathtt{a}}_3}=F_2\left[z\right]/⟨P_{t_1,a_1}\left(z\right)⟩\times F_2\left[z\right]/⟨P_{t_2,a_2}\left(z\right)⟩\times F_2\left[z\right]/⟨P_{t_3,a_3}\left(z\right)⟩.\end{array}$$

Suppose $P_{5,1}\left(z\right)=(z+1)$, $P_{5,2}\left(z\right)=(z^2+z+1)$, $P_{5,3}\left(z\right)=(z^3+z^2+1)$, $P_{5,4}\left(z\right)=(z^3+z+1)$ and $P_{5,5}\left(z\right)=(z^5+z^4+z^2+z+1)$. Then, we have $U_{t_1}\cong F_2\cong {\displaystyle \frac{F_2\left[z\right]}{⟨z+1⟩}},U_{t_2}\cong F_4\cong {\displaystyle \frac{F_2\left[z\right]}{⟨z^2+z+1⟩}},U_{t_3}\cong F_8\cong {\displaystyle \frac{F_2\left[z\right]}{⟨z^3+z^2+1⟩}},U_{t_4}\cong F_8\cong {\displaystyle \frac{F_2\left[z\right]}{⟨z^3+z+1⟩}}\mathit{and}{U}_{t_5}\cong F_{32}\cong {\displaystyle \frac{F_2\left[z\right]}{⟨z^5+z^4+z^2+z+1⟩}}$. Now, following the same notations as in Equation (7), we have

$$\begin{array}{c}\hfill \begin{array}{cccccc}{U}_{t_1,1}=F_2,\hfill & U_{t_1,2}=\left\{0\right\},\hfill & U_{t_1,3}=F_8,\hfill & U_{t_1,4}=\left\{0\right\},\hfill & U_{t_1,5}=\left\{0\right\}\hfill & \\ U_{t_2,1}=F_2,\hfill & U_{t_2,2}=F_4,\hfill & U_{t_2,3}=\left\{0\right\},\hfill & U_{t_2,4}=F_8,\hfill & U_{t_2,5}=\left\{0\right\}\hfill & \\ U_{t_3,1}=\left\{0\right\},\hfill & U_{t_3,2}=\left\{0\right\},\hfill & U_{t_3,3}=F_8,\hfill & U_{t_3,4}=\left\{0\right\},\hfill & U_{t_3,5}=F_{32}.\hfill \end{array}\end{array}$$

Thus, we get

$$\begin{array}{cc}\hfill R_{t_1}^{{\mathtt{a}}_1}\times R_{t_2}^{{\mathtt{a}}_2}\times R_{t_3}^{{\mathtt{a}}_3}\cong (U_{t_1}\times U_{t_1}\times \left\{0\right\})& \oplus (\left\{0\right\}\times U_{t_2}\times \left\{0\right\})\oplus (U_{t_3}\times \left\{0\right\}\times U_{t_3})\hfill \\ & \oplus (\left\{0\right\}\times U_{t_4}\times \left\{0\right\})\oplus (\left\{0\right\}\times \left\{0\right\}\times U_{t_5}).\hfill \end{array}$$

We take C as a right GQPC code of block length $(4,6,8)$ generated by $C=⟨z+z^2,z^3+z+1,z^5+z^4+z^2+z+1⟩$. Then the right GQPC code C can be decomposed into $C={\oplus }_{i=1}^{t=5}{C}_i$. Let ${\beta }_i$ be a root of $P_{5,i}\left(z\right)$ for $i=1,2,\dots ,5$, with ${\beta }_1=1$. Then, C has the following components:

$$\begin{array}{cc}\hfill & C_1:\mathit{an}[l=3,1]\mathit{linear}\mathit{code}\mathit{over}{\displaystyle \frac{F_2\left[z\right]}{⟨z+1⟩}}\cong F_2\mathit{with}\mathit{the}\mathit{generator}\mathit{matrix}[0,1,1],\hfill \\ \hfill & C_2:\mathit{an}[l=3,1]\mathit{linear}\mathit{code}\mathit{over}{\displaystyle \frac{F_2\left[z\right]}{⟨z^2+z+1⟩}}\cong F_4\mathit{with}\mathit{generator}\mathit{matrix}[0,{\beta }_2^3+{\beta }_2^2,0],\hfill \\ \hfill & C_4:\mathit{an}[l=3,0]\mathit{trivial}\mathit{linear}\mathit{code}\mathit{over}{\displaystyle \frac{F_2\left[z\right]}{⟨z^3+z+1⟩}}\cong F_8,\hfill \\ \hfill & C_5:\mathit{an}[l=3,0]\mathit{trivial}\mathit{linear}\mathit{code}\mathit{over}{\displaystyle \frac{F_2\left[z\right]}{⟨z^5+z^4+z^2+z+1⟩}}\cong F_{32},\hfill \\ \hfill & C_3:\mathit{an}[l=3,1]\mathit{linear}\mathit{code}\mathit{over}{\displaystyle \frac{F_2\left[z\right]}{⟨z^3+z^2+1⟩}}\cong F_8\mathit{with}\mathit{generator}\mathit{matrix}\hfill \\ & [{\beta }_3+{\beta }_3^2,0,{\beta }_3^3+{\beta }_3^4+{\beta }_3^5].\hfill \end{array}$$

Let $k_i$ be the dimension of $C_i$, $i=1,2,3,4,5$. Then $max\left\{k_i\right\}=1$ which is the number of generator of GQPC code C.

Example 2. 

Let $q=3$, $t_1=4$ and $t_2=6$. Consider the monic polynomials

$$\begin{array}{cc}\hfill P_{t_1,{\mathtt{a}}_1}\left(z\right)& =z^4+2z^3+z^2+z+1\mathit{of}\mathit{order}8,\mathit{where}{\mathtt{a}}_1=(1,1,1,2)\in F_3^4\hfill \\ \hfill & =(z+1)(z+2)(z^2+2z+2);\hfill \\ \hfill P_{t_2,{\mathtt{a}}_2}\left(z\right)& =z^6+2z^5+2z^4+2z^2+z+1\mathit{of}\mathit{order}8,\mathit{where}{\mathtt{a}}_2=(1,1,2,0,2,2)\in F_3^6\hfill \\ \hfill & =(z+1)(z+2)(z^2+1)(z^2+2z+2).\hfill \end{array}$$

In this case, we have

$$\begin{array}{c}\hfill \mathbf{S}=R_{t_1}^{{\mathtt{a}}_1}\times R_{t_2}^{{\mathtt{a}}_2}=F_3\left[z\right]/⟨P_{t_1,{\mathtt{a}}_1}\left(z\right)⟩\times F_3\left[z\right]/⟨P_{t_2,{\mathtt{a}}_2}\left(z\right)⟩.\end{array}$$

Continuing with the same notations as used in Equation (5), let $P_{4,1}\left(z\right)=z+1$, $P_{4,2}\left(z\right)=z+2,P_{4,3}\left(z\right)=z^2+1$ and $P_{4,4}\left(z\right)=z^2+2z+2$. Then, from Equation (8), we have $U_{t_1}\cong F_3\cong {\displaystyle \frac{F_3\left[z\right]}{⟨z+1⟩}},U_{t_2}\cong F_3\cong {\displaystyle \frac{F_3\left[z\right]}{⟨z+2⟩}},U_{t_3}\cong F_{3^2}\cong {\displaystyle \frac{F_3\left[z\right]}{⟨z^2+1⟩}}$ and $U_{t_4}\cong F_{3^2}\cong {\displaystyle \frac{F_3\left[z\right]}{⟨z^2+2z+2⟩}}$. Now, following the notation and simple calculation as given in Equation (7), we have

$$\begin{array}{c}\hfill \begin{array}{ccccc}{U}_{t_1,1}=U_{t_1}=F_3,\hfill & U_{t_1,2}=U_{t_2}=F_3,\hfill & U_{t_1,3}=\left\{0\right\},\hfill & U_{t_1,4}=U_{t_4}=F_{3^2}\hfill & \\ U_{t_2,1}=U_{t_1}=F_3,\hfill & U_{t_2,2}=U_{t_2}=F_3,\hfill & U_{t_2,3}=U_{t_3}=F_{3^2},\hfill & U_{t_2,4}=U_{t_4}=F_{3^2}.\hfill \end{array}\end{array}$$

Using the CRT and following Equations (9) and (10), we get

$$\begin{array}{cc}\hfill \mathbf{S}=R_{t_1}^{{\mathtt{a}}_1}\times R_{t_2}^{{\mathtt{a}}_2}\cong (U_{t_1}\times U_{t_1})& \oplus (U_{t_2}\times U_{t_2})\oplus (\left\{0\right\}\times U_{t_3})\oplus (U_{t_4}\times U_{t_4}).\hfill \end{array}$$

Let $C\subset \mathit{S}$ be a 2-generator right GQPC code of block length $(4,6)$ generated by $e_1\left(z\right)=(z^2+z,z^2+2z)$ and $e_2\left(z\right)=(z^2+2,z^3+z^2+1)$. Then, right GQPC code can be decompose into $C={\oplus }_{j=1}^{t=4}{C}_j$. Furthermore, for $t=4$, let ${\beta }_i$ be a root of $P_{t,i}\left(z\right)$ for $i=1,2,\dots ,t$, where ${\beta }_1=2$ and ${\beta }_2=1$. Then, constituents of code C are as follows:

$$\begin{array}{cc}\hfill & C_1:\mathrm{a}[2,1]\mathit{linear}\mathit{code}\mathit{with}\mathit{the}\mathit{basis}\left\{(0,1)\right\}\mathit{over}{U}_{t_1},\hfill \\ \hfill & C_2:\mathrm{a}[2,1]\mathit{linear}\mathit{code}\mathit{with}\mathit{the}\mathit{basis}\left\{(2,0)\right\}\mathit{over}{U}_{t_2},\hfill \\ \hfill & C_3:\mathrm{a}[2,1]\mathit{linear}\mathit{code}\mathit{with}\mathit{the}\mathit{basis}\{0,{{\beta }_3}^2+2{\beta }_3\}\mathit{over}{U}_{t_3},\hfill \\ \hfill & C_4:\mathrm{a}[2,2]\mathit{linear}\mathit{code}\mathit{with}\mathit{the}\mathit{basis}\{({\beta }_4({\beta }_4+1),2),(({\beta }_4+2)({\beta }_4+1),0)\mathit{over}{U}_{t_4}.\hfill \end{array}$$

Let $k_j$ be the dimension of $C_j$, $j=1,2,3,4$. Then $max\left\{k_j\right\}=2$ which is the number of generator of right GQPC code C.

4. Normalized Generating Set for a GQPC Code

This section provides a normalized generating set of the $\rho$-generator right GQPC code of block length $(t_1,t_2,\dots ,t_l)$ and index l over the field $F_q$, and a method to obtain a normalized generating set for a $\rho$-generator polynomial set that satisfies certain conditions by following the construction given in [20].

Definition 6. 

Let $\{{\mathit{A}}_1,{\mathit{A}}_2,\dots ,{\mathit{A}}_k\}\subset \mathit{S}$ be a formal generating set of a right GQPC code C. If it can be transformed into the following standard form

$$\begin{array}{c}\hfill \{(f_{11}\left(z\right),0,\cdots ,0),(f_{21}\left(z\right),f_{22}\left(z\right),\dots ,0),\dots ,(f_{l1}\left(z\right),f_{l2}\left(z\right),\dots ,f_{ll}\left(z\right))\},\end{array}$$

where $f_{ij}\left(z\right)\in R_{t_j}^{{\mathtt{a}}_j},f_{ii}\left(z\right)\mid (z^{t_i}-{\mathtt{a}}_i\left(z\right))$ for all $i=1,2,\dots ,l$ and $j=1,2,\dots ,i$. Then this standard form of a formal generating set is termed a normalized generating set.

In the following, we provide a method to construct a normalized generating set from any generating set for a right GQPC code C. Assume that

$$\begin{array}{cc}\hfill {\mathbf{A}}_1& =(A_{11}\left(z\right),A_{12}\left(z\right),\dots ,A_{1l}\left(z\right)),\hfill \\ \hfill {\mathbf{A}}_2& =(A_{21}\left(z\right),A_{22}\left(z\right),\dots ,A_{2l}\left(z\right)),\hfill \\ \hfill & ⋮\hfill \\ \hfill {\mathbf{A}}_k& =(A_{k1}\left(z\right),A_{k2}\left(z\right),\dots ,A_{kl}\left(z\right)),\hfill \end{array}$$

form a generating set for the right GQPC code C of block length $(t_1,t_2,\dots ,t_l)$ and index l.

• Step (i): Construct $f_{ll}\left(z\right)=gcd(A_{1l}\left(z\right),A_{2l}\left(z\right),\dots ,A_{kl}\left(z\right),z^{t_l}-a_l\left(z\right))$. Suppose that

  $$\begin{array}{c}\hfill {\displaystyle \frac{A_{1l}\left(z\right)}{f_{ll}\left(z\right)}},{\displaystyle \frac{A_{2l}\left(z\right)}{f_{ll}\left(z\right)}},\dots ,{\displaystyle \frac{A_{kl}\left(z\right)}{f_{ll}\left(z\right)}}\end{array}$$

  are pairwise coprime over $F_q$. Then from the map defined in the proof of Theorem 2, $M_l\left(C\right)=⟨A_{1l}\left(z\right),A_{2l}\left(z\right),$$A_{3l}\left(z\right),\dots ,A_{kl}\left(z\right)⟩$. Since $M_l\left(C\right)$ is an RPC code of length $t_l$, we have $M_l\left(C\right)=⟨f_{ll}\left(z\right)⟩$. Now, using the Euclidean algorithm over $F_q$, there exist $r_1\left(z\right),r_2\left(z\right),\dots ,r_k\left(z\right),u\left(z\right)\in R$ such that

  $$\begin{array}{c}\hfill r_1\left(z\right)A_{1l}\left(z\right)+r_2\left(z\right)A_{2l}\left(z\right)+\cdots +r_k\left(z\right)A_{kl}\left(z\right)+u\left(z\right)(z^{t_l}-{\mathtt{a}}_l\left(z\right))=f_{ll}\left(z\right).\end{array}$$
  If $f_{ll}\left(z\right)\equiv 0mod(z^{t_l}-{\mathtt{a}}_l\left(z\right))$, then take ${\mathbf{a}}_l\left(z\right)=(0,0,\dots ,z^{t_l}-{\mathtt{a}}_l\left(z\right))$. Otherwise, consider

  $$\begin{array}{c}\hfill r_1\left(z\right)A_{1i}\left(z\right)+r_2\left(z\right)A_{2i}\left(z\right)+\cdots +r_k\left(z\right)A_{ki}\left(z\right)=f_{li}\left(z\right)mod(z^{t_i}-{\mathtt{a}}_i\left(z\right))\mathrm{for}i<l.\end{array}$$
  Hence, ${\mathbf{a}}_l\left(z\right)=(f_{l1}\left(z\right),f_{l2}\left(z\right),\dots ,f_{ll}\left(z\right))$.
• Step (ii): Let ${\alpha }_i={\displaystyle \frac{A_{il}\left(z\right)}{f_{ll}\left(z\right)}}$ for $i=1,2,\dots ,l$. Then

  $$\begin{array}{c}\hfill {\mathbf{A}}_i^1={\mathbf{A}}_i-{\alpha }_i{\mathbf{a}}_l\left(z\right)=(A_{i,1}^1\left(z\right),A_{i,2}^1\left(z\right),\dots ,A_{i,l-1}^1\left(z\right),0).\end{array}$$
  Now, consider $f_{l-1,l-1}\left(z\right)={gcd}_q(A_{1,l-1}^1\left(z\right),A_{2,l-1}^1\left(z\right),\dots ,A_{k,l-1}^1\left(z\right),z^{t_{l-1}}-{\mathtt{a}}_{l-1}\left(z\right))$ and as in step (i), assume that ${\displaystyle \frac{A_{1,l-1}^1\left(z\right)}{f_{l-1,l-1}\left(z\right)}},{\displaystyle \frac{A_{2,l-1}^1\left(z\right)}{f_{l-1,l-1}\left(z\right)}},\dots ,{\displaystyle \frac{A_{k,l-1}^1\left(z\right)}{f_{l-1,l-1}\left(z\right)}}$ are pairwise coprime in R. Then using the same procedure as in step (i), we can get

  $$\begin{array}{c}\hfill {\mathbf{a}}_{l-1}\left(z\right)=(f_{l-1,1}\left(z\right),f_{l-1,2}\left(z\right),\dots ,f_{l-1,l-1}\left(z\right),0).\end{array}$$
• Step (iii): Let ${\alpha }_i^1={\displaystyle \frac{A_{i,l-1}^1\left(z\right)}{f_{l-1,l-1}\left(z\right)}}$ for $i=1,2,\dots ,l$. Then

  $$\begin{array}{c}\hfill {\mathbf{A}}_i^2={\mathbf{A}}_i^1-{\alpha }_i^1{\mathbf{a}}_{l-1}\left(z\right)=(A_{i,1}^2\left(z\right),A_{i,2}^2\left(z\right),\dots ,A_{i,l-2}^2\left(z\right),0,0).\end{array}$$
  Employing the same approach as in step (i), we have

  $$\begin{array}{c}\hfill f_{l-2,l-2}\left(z\right)=gcd(A_{1,l-2}^2\left(z\right),A_{2,l-2}^2\left(z\right),\dots ,A_{l,l-2}^2\left(z\right),z^{t_{l-2}}-{\mathtt{a}}_{l-2}\left(z\right)).\end{array}$$
  Then by step (i) and ${\mathbf{A}}_1^2,{\mathbf{A}}_2^2,\dots ,{\mathbf{A}}_l^2$, we can construct

  $$\begin{array}{c}\hfill {\mathbf{a}}_{l-2}\left(z\right)=(f_{l-2,1}\left(z\right),f_{l-2,2}\left(z\right),\dots ,f_{l-2,l-2}\left(z\right),0,0).\end{array}$$

Repeating the above procedure, we finally obtain ${\mathbf{a}}_1\left(z\right)=(f_{1,1}\left(z\right),0,0,\dots ,0)$. Hence, above method can yield a normalized generating set $\{{\mathbf{a}}_1\left(z\right),{\mathbf{a}}_2\left(z\right),\dots ,$${\mathbf{a}}_l\left(z\right)\}$.

Example 3. 

Let $q=2$, $t_1=6$ and $t_2=4$. Consider the monic polynomials

$$\begin{array}{cc}\hfill P_{t_1,{\mathtt{a}}_1}\left(z\right)& =z^6-z^2+1,\mathit{where}{\mathtt{a}}_1=(1,0,1,0,0,0)\in F_2^6\hfill \\ \hfill & ={(z^3+z+1)}^2,\hfill \\ \hfill P_{t_2,{\mathtt{a}}_2}\left(z\right)& =z^4+z^2+z+1,\mathit{where}{\mathtt{a}}_2=(1,1,1,0)\in F_2^4\hfill \\ \hfill & =(z+1)(z^3+z^2+1).\hfill \end{array}$$

Let ${\mathit{A}}_1=\{z^3+z+1,z+1\}$ and ${\mathit{A}}_2=\{0,z(z+1)\}$ be a formal 2-generating set of right GQPC code of index 2 over $F_2$. Here, $\mathit{S}=R_{t_1}^{{\mathtt{a}}_1}\times R_{t_2}^{{\mathtt{a}}_2}=F_2\left[z\right]/⟨P_{t_1,{\mathtt{a}}_1}\left(z\right)⟩\times F_2\left[z\right]/⟨P_{t_2,{\mathtt{a}}_2}\left(z\right)⟩$. As in step (i) of the above method, by a simple calculation

$$\begin{array}{c}\hfill f_{22}\left(z\right)=gcd(z+1,z(z+1),z^4+z^2+z+1)=z+1.\end{array}$$

Now, by the Euclidean algorithm, there exist two polynomials $r_1\left(z\right)=1$ and $r_2\left(z\right)=0$ such that $z+1=r_1\left(z\right)(z+1)+r_2\left(z\right)z(z+1)$. Since $f_{22}\left(z\right)\ne 0(mod{P}_{t_2,{\mathtt{a}}_2}\left(z\right))$,

$$\begin{array}{cc}\hfill f_{21}\left(z\right)& =r_1\left(z\right)·(z^3+z+1)+r_2\left(z\right)·0(mod{z}^6-z^2+1)\hfill \\ \hfill & =z^3+z+1.\hfill \end{array}$$

Hence, ${\mathit{a}}_2\left(z\right)=(z^3+z+1,z+1)$. Following step (ii) of the above method, ${\alpha }_1=1$ and ${\alpha }_2=z$. Then

$$\begin{array}{cc}\hfill {\mathit{A}}_1^1& ={\mathit{A}}_1-{\alpha }_1·{\mathit{a}}_2\left(z\right)\hfill \\ \hfill & =\{z^3+z+1,z+1\}-\{z^3+z+1,z+1\}=\{0,0\},\hfill \\ \hfill {\mathit{A}}_2^1& ={\mathit{A}}_2-{\alpha }_2·{\mathit{a}}_2\left(z\right)\hfill \\ \hfill & =\{0,z(z+1)\}-\{z(z^3+z+1),z(z+1)\}\hfill \\ \hfill & =\{z(z^3+z+1),0\}.\hfill \end{array}$$

Again, following step (i), we obtain $f_{11}\left(z\right)=gcd(z^6-z^2+1,z(z^3+z+1))=z^3+z+1,f_{12}\left(z\right)=0$. Hence, ${\mathit{a}}_1\left(z\right)=\{f_{11}\left(z\right),f_{12}\left(z\right)\}$. Thus, a Normalized generating set for a right GQPC code is $\{{\mathit{a}}_1\left(z\right),{\mathit{a}}_2\left(z\right)\}$.

Example 4. 

Let $q=5$, $t_1=4,t_2=t_3=6$. Consider monic polynomials

$$\begin{array}{cc}\hfill P_{t_1,{\mathtt{a}}_1}\left(z\right)& =z^4+z^2+3z+1,\mathit{where}{\mathtt{a}}_1=(1,3,1,0)\in F_5^4\hfill \\ \hfill & =(z+1)(z+2)(z^2+2z+3),\hfill \\ \hfill P_{t_2,{\mathtt{a}}_2}\left(z\right)& =z^6+2z^4+z^2+z+2,\mathit{where}{\mathtt{a}}_2=(2,1,1,0,2,0)\in F_5^6\hfill \\ \hfill & =(z+2){(z+1)}^2(z^3+z^2+3z+1),\hfill \\ \hfill P_{t_3,{\mathtt{a}}_3}\left(z\right)& =z^6+2z^4+z^2+z+2,\mathit{where}{\mathtt{a}}_3=(2,1,1,0,2,0)\in F_5^6\hfill \\ \hfill & =(z+2){(z+1)}^2(z^3+z^2+3z+1).\hfill \end{array}$$

Let ${\mathit{A}}_1=\{z^3+4z^2+2z+1,z^4+z^2+3z+1,0\}$, ${\mathit{A}}_2=\{0,z^2+3z+2,z^4+z^3+3z^2+z\}$ and ${\mathit{A}}_3=\{z^3+4z+4,0,z^4+4z^3+z^2+3\}$ be a formal 3-generating set of right GQPC code of index 3 over $F_5$. Here, $\mathit{S}=R_{t_1}^{{\mathtt{a}}_1}\times R_{t_2}^{{\mathtt{a}}_2}\times R_{t_3}^{{\mathtt{a}}_3}=F_5\left[z\right]/⟨P_{t_1,{\mathtt{a}}_1}\left(z\right)⟩\times F_5\left[z\right]/⟨P_{t_2,{\mathtt{a}}_2}\left(z\right)⟩\times F_5\left[z\right]/⟨P_{t_3,{\mathtt{a}}_3}\left(z\right)⟩$, and $\{{\mathit{A}}_1,{\mathit{A}}_2,{\mathit{A}}_3\}\subset \mathit{S}$. By following the above method and simple calculation,

$$\begin{array}{c}\hfill f_{33}\left(z\right)=gcd(z^4+4z^3+z^2+3,z^4+z^3+3z^2+z,P_{t_3,a_3}\left(z\right))=z^3+z^2+3z+1.\end{array}$$

Now, by the Euclidean algorithm, there exist polynomials $r_1\left(z\right)=0$, $r_2\left(z\right)=3$, $r_3\left(z\right)=2$ and $u\left(z\right)=0$ such that

$$\begin{array}{c}\hfill f_{33}\left(z\right)=r_1\left(z\right)·0+r_2\left(z\right)·(z^4+z^3+3z^2+z)+r_3\left(z\right)·(z^4+4z^3+z^2+3)+u\left(z\right)P_{t_3,{\mathtt{a}}_3}\left(z\right).\end{array}$$

Since $f_{33}\left(z\right)\ne 0mod{P}_{t_3,{\mathtt{a}}_3}\left(z\right)$,

$$\begin{array}{cc}\hfill f_{32}\left(z\right)& =r_1\left(z\right)(z^4+3z^3+2z+2)+r_2\left(z\right)(z^2+3z+2)+r_3\left(z\right)·0\hfill \\ \hfill & =3z^2+z+1,\mathit{and}\hfill \\ \hfill f_{31}\left(z\right)& =r_1\left(z\right)(z^3+4z^2+2z+1)+r_2\left(z\right)·0+r_3\left(z\right)(z^3+4z+4)\hfill \\ \hfill & =2z^3+3z+3.\hfill \end{array}$$

Hence, ${\mathit{a}}_3\left(z\right)=\{f_{31}\left(z\right),f_{32}\left(z\right),f_{33}\left(z\right)\}$. Following step (ii), ${\alpha }_1=0$, ${\alpha }_2=z$ and ${\alpha }_3=z+3$. Then

$$\begin{array}{cc}\hfill {\mathit{A}}_1^1& ={\mathit{A}}_1-{\alpha }_1·{\mathit{a}}_3\left(z\right)={\mathit{A}}_1,(\mathit{as}{\alpha }_1=0),\hfill \\ \hfill {\mathit{A}}_2^1& ={\mathit{A}}_2-{\alpha }_2·{\mathit{a}}_3\left(z\right)=\{3z^4+2z^2+2z,3z^2+2z+1,0\},\hfill \\ \hfill {\mathit{A}}_3^1& ={\mathit{A}}_3-{\alpha }_3·{\mathit{a}}_3\left(z\right)=\{3z^4+2z^2+2z,2z^3+z+2,0\}.\hfill \end{array}$$

By following step (i) and simple calculation, we get $f_{22}\left(z\right)=gcd(z^4+3z^3+2z+2,3z^2+2z+1,2z^3+z+2)=1$. Now, using Euclidean algorithm, there exist polynomials $r_1^{\prime }\left(z\right)=0$, $r_2^{\prime }\left(z\right)=2z^2+z+3$ and $r_3^{\prime }\left(z\right)=2z+4$ such that

$$\begin{array}{c}\hfill 1=0·(z^4+3z^3+2z+2)+(2z^2+z+3)·(3z^2+2z+1)+(2z+4)·(2z^3+z+2).\end{array}$$

Since $f_{22}\left(z\right)\ne 0(mod{P}_{t_2,{\mathtt{a}}_2}\left(z\right))$, we have

$$\begin{array}{cc}\hfill f_{21}\left(z\right)& =r_1^{\prime }\left(z\right)(z^3+4z^2+2z+1)+r_2^{\prime }\left(z\right)(3z^4+2z^2+2z)+r_3^{\prime }\left(z\right)(3z^4+2z^2+2z)\hfill \\ \hfill & =z^6+4z^5+4zmod{P}_{t_1,{\mathtt{a}}_1}\left(z\right)\hfill \\ \hfill & =3z^3+3z^2+3z+1.\hfill \end{array}$$

Hence, we have ${\mathit{a}}_2\left(z\right)=(f_{21}\left(z\right),f_{22}\left(z\right),0)$. Further, following step (ii), we have

$$\begin{array}{c}\hfill {\alpha }_1^1={\displaystyle \frac{z^4+3z^3+2z+2}{f_{22}\left(z\right)=1}},{\alpha }_2^1={\displaystyle \frac{3z^2+2z+1}{f_{22}\left(z\right)=1}},{\alpha }_3^1={\displaystyle \frac{2z^3+z+2}{f_{22}\left(z\right)=1}}.\end{array}$$

Then

$$\begin{array}{c}\hfill {\mathit{A}}_1^2={\mathit{A}}_1^1-{\alpha }_1^1{\mathit{a}}_2\left(z\right)=\{4z^3+z^2+3z+4,0,0\},\\ \hfill {\mathit{A}}_2^2={\mathit{A}}_2^1-{\alpha }_2^1{\mathit{a}}_2\left(z\right)=\{z^3+4z^2+2z+1,0,0\},\\ \hfill {\mathit{A}}_3^2={\mathit{A}}_3^1-{\alpha }_3^1{\mathit{a}}_2\left(z\right)=\{3z^3+2z^2+z+3,0,0\}.\end{array}$$

Again, following step (i), we get $f_{11}\left(z\right)=gcd(4z^3+z^2+3z+4,z^3+4z^2+2z+1,3z^3+2z^2+z+3)=z^3+4z^2+2z+1$. Further, $f_{12}\left(z\right)=0$ and $f_{13}\left(z\right)=0$. Hence, ${\mathit{a}}_1\left(z\right)=\{f_{11}\left(z\right),f_{12}\left(z\right),f_{13}\left(z\right)\}$. Thus, a normalized generating set for the right GQPC code is $\{{\mathit{a}}_1\left(z\right),{\mathit{a}}_2\left(z\right),{\mathit{a}}_3\left(z\right)\}$.

Theorem 2. 

Let C be a right GQPC code of block length $(t_1,t_2,\dots ,t_l)$ and index l over the field $F_q$, where $N={\sum }_{i=1}^l{t}_i$ is the length of C. Then there is a generating set $\{{\mathit{a}}_1\left(z\right),{\mathit{a}}_2\left(z\right),\dots ,{\mathit{a}}_l\left(z\right)\}$ such that $C=⟨{\mathit{a}}_1\left(z\right),{\mathit{a}}_2\left(z\right),\dots ,{\mathit{a}}_l\left(z\right)⟩$, where

$$\begin{array}{cc}\hfill {\mathit{a}}_1\left(z\right)& =(f_{11}\left(z\right),0,\dots ,0),\hfill \\ \hfill {\mathit{a}}_2\left(z\right)& =(f_{21}\left(z\right),f_{22}\left(z\right),\dots ,0),\hfill \\ \hfill ⋮\\ \hfill {\mathit{a}}_l\left(z\right)& =(f_{l1}\left(z\right),f_{l2}\left(z\right),\dots ,f_{ll}\left(z\right)),\hfill \end{array}$$

and $f_{ij}\left(z\right)\in R_{t_j}^{{\mathtt{a}}_j},f_{ii}\left(z\right)\mid (z^{t_i}-{\mathtt{a}}_i\left(z\right))$ for all $i=1,2,\dots ,l$ and $j=1,2,\dots ,i$.

Proof. 

For index $l=1$, C is an RPC code of length $t_1$. Hence, from Proposition 1, C is an ideal of $R_{t_1}^{{\mathtt{a}}_1}$. Then, there exists a monic polynomial of the least degree $f_{11}\left(z\right)\in R$ such that $f_{11}\left(z\right)\mid z^{t_1}-a_1\left(z\right)$ and $C=⟨f_{11}\left(z\right)⟩$ (see [5] (Proposition 1)). Now, assume that the result holds for $k<l$ over $F_q$. Consider a module homomorphism for all $i=1,2,\dots ,l$,

$$\begin{array}{cc}\hfill M_i:& R_{t_1}^{{\mathtt{a}}_1}\times R_{t_2}^{{\mathtt{a}}_2}\times \cdots \times R_{t_l}^{{\mathtt{a}}_l}\mapsto R_{t_i}^{{\mathtt{a}}_i},\mathrm{given}\mathrm{by}\hfill \\ \hfill & (b_1\left(z\right),b_2\left(z\right),\dots ,b_l\left(z\right))\mapsto b_i\left(z\right).\hfill \end{array}$$

For $i=l$, $M_l\left(C\right)$ is an RPC code in $R_{t_l}^{{\mathtt{a}}_l}$ over $F_q$ of length $t_l$. Then, there exists a monic polynomial of the least degree $f_{ll}\left(z\right)\in R$ such that $f_{ll}\left(z\right)$ divides $z^{t_l}-a_l\left(z\right)$ and $M_l\left(C\right)=⟨f_{ll}\left(z\right)⟩$. Now, consider any codeword $(c_1\left(z\right),c_2\left(z\right),\dots ,c_l\left(z\right))\in C$. Then, there exists a polynomial $\alpha \left(z\right)\in R$ such that $c_l\left(z\right)=\alpha \left(z\right)f_{ll}\left(z\right)$. For the above map $M_l$, $ker\left(M_l\left(C\right)\right)$ is the right GQPC code of index $l-1$. Hence, from our assumption, it can be generated by $\{{\mathbf{a}}_1\left(z\right),{\mathbf{a}}_2\left(z\right),\dots ,{\mathbf{a}}_{l-1}\left(z\right)\}$. If ${\mathbf{a}}_l\left(z\right)\in C$, then $c\left(z\right)-\alpha \left(z\right){\mathbf{a}}_l\left(z\right)\in ker\left(M_l\left(C\right)\right)$. Thus, $c\left(z\right)$ belongs to $⟨{\mathbf{a}}_1\left(z\right),{\mathbf{a}}_2\left(z\right),\dots ,{\mathbf{a}}_l\left(z\right)⟩$. This completes the proof. □

For ease of computation, we present some relationships between constituents of the generator polynomial set in the next result.

Theorem 3. 

Assume that $C=⟨{\mathit{a}}_1\left(z\right),{\mathit{a}}_2\left(z\right),\dots ,{\mathit{a}}_l\left(z\right)⟩$ is the right GQPC code of block length $(t_1,t_2,\dots ,t_l)$ and index l, where ${\mathit{a}}_i\left(z\right)$ same as in Theorem 2 and $N={\sum }_{i=1}^l{t}_i$. Then $deg\left(f_{i+1i}\left(z\right)\right)<deg\left(f_{ii}\left(z\right)\right)$ for all $i=1,2,\dots ,l-1$.

Proof. 

For index $l=2$, let $C=⟨{\mathbf{a}}_1\left(z\right),{\mathbf{a}}_2\left(z\right)⟩$ as given in Theorem 2. Assume that $deg(f_{11}\left(z\right)<deg(f_{21}\left(z\right)$. Then, using the division algorithm, there exist $h_{21}\left(z\right)$ and $r_{21}\left(z\right)\in R$ such that $f_{21}\left(z\right)=h_{21}\left(z\right)f_{11}\left(z\right)+r_{21}\left(z\right)$, where $r_{21}\left(z\right)=0$ or $deg\left(r_{21}\left(z\right)\right)<deg\left(f_{11}\left(z\right)\right)$. Then

$$\begin{array}{cc}\hfill ⟨\begin{array}{c}{\mathbf{a}}_1\left(z\right),{\mathbf{a}}_2\left(z\right)\end{array}⟩& =⟨\begin{array}{c}(f_{11}\left(z\right),0),(f_{21}\left(z\right),f_{22}\left(z\right))\end{array}⟩,\hfill \\ \hfill & =⟨\begin{array}{c}(f_{11}\left(z\right),0),(h_{21}\left(z\right)f_{11}\left(z\right)+r_{21}\left(z\right),f_{22}\left(z\right))\end{array}⟩,\hfill \\ \hfill & =⟨\begin{array}{c}(f_{11}\left(z\right),0),(h_{21}\left(z\right)(f_{11}\left(z\right),0)+(r_{21}\left(z\right),f_{22}\left(z\right)))\end{array}⟩,\hfill \\ \hfill & =⟨\begin{array}{c}(f_{11}\left(z\right),0),(r_{21}\left(z\right),f_{22}\left(z\right))\end{array}⟩.\hfill \end{array}$$

Similarly, for index $l=3$, let $C=⟨{\mathbf{a}}_1\left(z\right),{\mathbf{a}}_2\left(z\right),{\mathbf{a}}_3\left(z\right)⟩$. Assume that $deg\left(r_{21}\left(z\right)\right)<deg\left(f_{31}\left(z\right)\right)$ and $deg(f_{22}\left(z\right)<deg(f_{32}\left(z\right)$. Then by using the divison algorithm there exist $h_{31}\left(z\right),h_{22}\left(z\right),r_{31}\left(z\right),r_{32}\left(z\right)$$\in R$ which satisfy $f_{31}\left(z\right)=h_{31}\left(z\right)r_{21}\left(z\right)+r_{31}\left(z\right)$, where $r_{31}\left(z\right)=0$ or $deg(r_{31}\left(z\right)<deg\left(r_{21}\left(z\right)\right)$ and $f_{32}\left(z\right)=h_{32}\left(z\right)f_{22}\left(z\right)+r_{32}\left(z\right)$, where $r_{32}\left(z\right)=0$ or $deg\left(r_{32}\left(z\right)\right)<deg\left(f_{22}\left(z\right)\right)$. Then

$$\begin{array}{cc}\hfill ⟨\begin{array}{c}{\mathbf{a}}_1\left(z\right),{\mathbf{a}}_2\left(z\right),{\mathbf{a}}_3\left(z\right)\end{array}⟩& =⟨\begin{array}{c}(f_{11}\left(z\right),0,0),(f_{21}\left(z\right),f_{22}\left(z\right),0),(f_{31}\left(z\right),f_{32}\left(z\right),f_{33}\left(z\right))\end{array}⟩\hfill \\ \hfill & =⟨\begin{array}{c}(f_{11}\left(z\right),0,0),(r_{21}\left(z\right),f_{22}\left(z\right),0),(h_{31}\left(z\right)r_{21}\left(z\right)+r_{31}\left(z\right),\\ h_{32}\left(z\right)f_{22}\left(z\right)+r_{32}\left(z\right),f_{33}\left(z\right))\end{array}⟩\hfill \\ \hfill & =⟨\begin{array}{c}(f_{11}\left(z\right),0,0),(r_{21}\left(z\right),f_{22}\left(z\right),0),(h_{31}\left(z\right)r_{21}\left(z\right),h_{32}\left(z\right)f_{22}\left(z\right),0)\\ +(r_{31}\left(z\right),r_{32}\left(z\right),f_{33}\left(z\right))\end{array}⟩\hfill \\ \hfill & =⟨\begin{array}{c}(f_{11}\left(z\right),0,0),(r_{21}\left(z\right),f_{22}\left(z\right),0),(r_{31}\left(z\right),r_{32}\left(z\right),f_{33}\left(z\right))\end{array}⟩.\hfill \end{array}$$

Hence, the statement holds for index $l=3$. Now, assume that the statement holds for index $l-1$. Then consider

$$\begin{array}{cc}\hfill & ⟨\begin{array}{c}{\mathbf{a}}_1\left(z\right),{\mathbf{a}}_2\left(z\right),\dots ,{\mathbf{a}}_{l-1}\left(z\right),{\mathbf{a}}_l\left(z\right)\end{array}⟩\hfill \\ & =⟨\begin{array}{c}(f_{11}\left(z\right),0,\dots ,0),(f_{21}\left(z\right),f_{22}\left(z\right),0,\dots ,0),\dots ,(f_{l-11}\left(z\right),f_{l-12}\left(z\right),\dots ,f_{l-1l-1}\left(z\right),0),\\ (f_{l1}\left(z\right),f_{l2}\left(z\right),\dots ,f_{ll-1}\left(z\right),f_{ll}\left(z\right))\end{array}⟩\hfill \\ & \mathrm{as}\mathrm{we}\mathrm{have}\mathrm{assumed}\mathrm{that}\mathrm{the}\mathrm{statement}\mathrm{holds}\mathrm{for}\mathrm{index}\mathrm{less}\mathrm{than}l,\mathrm{we}\mathrm{thus}\mathrm{have}\hfill \\ & ⟨\begin{array}{c}{\mathbf{a}}_1\left(z\right),{\mathbf{a}}_2\left(z\right),\dots ,{\mathbf{a}}_{l-1}\left(z\right),{\mathbf{a}}_l\left(z\right)\end{array}⟩\hfill \\ \hfill & =⟨\begin{array}{c}(f_{11}\left(z\right),0,\dots ,0),(r_{21}\left(z\right),f_{22}\left(z\right),0,\dots ,0),\dots ,(r_{l-11}\left(z\right),r_{l-12}\left(z\right),\dots ,f_{l-1l-1}\left(z\right),0)\\ ,(f_{l1}\left(z\right),f_{l2}\left(z\right),\dots ,f_{ll-1}\left(z\right),f_{ll}\left(z\right))\end{array}⟩.\hfill \end{array}$$

Now, using similar arguments for $f_{l1}\left(z\right),f_{l2}\left(z\right),\dots ,f_{ll-1}\left(z\right)$ and applying the division algorithm, we can write

$$\begin{array}{cc}\hfill & ⟨\begin{array}{c}{\mathbf{a}}_1\left(z\right),{\mathbf{a}}_2\left(z\right),\dots ,{\mathbf{a}}_{l-1}\left(z\right),{\mathbf{a}}_l\left(z\right)\end{array}⟩\hfill \\ & =⟨\begin{array}{c}(f_{11}\left(z\right),0,\dots ,0),(r_{21}\left(z\right),f_{22}\left(z\right),0,\dots ,0),\dots ,(r_{l-11}\left(z\right),r_{l-12}\left(z\right),\dots ,f_{l-1l-1}\left(z\right),0),\\ (r_{l1}\left(z\right),r_{l2}\left(z\right),\dots ,r_{ll-1}\left(z\right),f_{ll}\left(z\right))\end{array}⟩,\hfill \end{array}$$

where $deg(r_{l1}\left(z\right)<deg\left(r_{l-11}\left(z\right)\right),deg\left(r_{l2}\left(z\right)\right)<deg\left(r_{l-12}\left(z\right)\right),\dots ,deg\left(r_{ll-1}\left(z\right)\right)<deg\left(r_{l-1l-1}\left(z\right)\right)$. Hence, we get the desired result. □

Theorems 2 and 3 describe the generator set for the right GQPC codes of block length $(t_1,t_2,\dots ,t_l)$ and index l. Based on the uniqueness and the structural properties of this generator set, we now establish the following result.

Theorem 4. 

Let C be a right GQPC code of block length $(t_1,t_2,\dots ,t_l)$ and index l, with a normalized generating set as given in the Theorem 2. Then the dimension of C is $deg\left(q_{t_1}\left(z\right)\right)+deg\left(q_{t_2}\left(z\right)\right)+\cdots +deg\left(q_{t_l}\left(z\right)\right)$, where

$$\begin{array}{cc}\hfill & z^{t_1}-{\mathtt{a}}_1\left(z\right)=q_{t_1}\left(z\right)f_{11}\left(z\right),\hfill \\ \hfill & z^{t_2}-{\mathtt{a}}_2\left(z\right)=q_{t_2}\left(z\right)f_{22}\left(z\right),\hfill \\ \hfill & ⋮\hfill \\ \hfill & z^{t_l}-{\mathtt{a}}_l\left(z\right)=q_{t_l}\left(z\right)f_{ll}\left(z\right).\hfill \end{array}$$

Proof. 

To prove the result, we implement the principle of mathematical induction on l. For index $l=1$, $C=⟨f_{11}\left(z\right)⟩$. It is just the right cyclic code of length $t_1$ with induced vector $a_1$. From [5] (Proposition 1), the basis of C is denoted by $S_1^1$ and is given by $S_1^1=\{f_{11}\left(z\right),z{f}_{11}\left(z\right),\dots ,z^{t_1-deg\left(f_{11}\left(z\right)\right)-1}{f}_{11}\left(z\right)\}$ and thus the dimension of C is $t_1-deg\left(f_{11}\left(z\right)\right)$. Now, assume that index $l=2$. Then $C=⟨(f_{11}\left(z\right),0),(r_{21}\left(z\right),f_{22}\left(z\right))⟩$. Consider $S_1^2={\cup }_{i=0}^{deg\left(q_{t_1}\left(z\right)\right)-1}\{z^i\ast (f_{11}\left(z\right),0)\}\bigcup S_2^2={\cup }_{j=0}^{deg\left(q_{t_2}\left(z\right)\right)-1}\{z^j\ast (r_{21}\left(z\right),f_{22}\left(z\right))\}$. If a codeword $c\left(z\right)\in C$, then there exist $q_1\left(z\right),q_2\left(z\right)$ in R such that

$$\begin{array}{cc}\hfill c\left(z\right)& =q_1\left(z\right)(f_{11}\left(z\right),0)+q_2\left(z\right)(r_{21}\left(z\right),f_{22}\left(z\right))\hfill \\ \hfill & =(q_1\left(z\right)f_{11}\left(z\right),0)+(q_2\left(z\right)r_{21}\left(z\right),q_2\left(z\right)f_{22}\left(z\right)).\hfill \end{array}$$

If $deg\left(q_1\left(z\right)\right)\ge deg\left(q_{t_1}\left(z\right)\right)$, then, by the divison algorithm, there exist $Q_1\left(z\right)$ and $Q_1^{\prime }\left(z\right)$ in R such that

$$\begin{array}{c}\hfill q_1\left(z\right)=Q_1\left(z\right)q_{t_1}\left(z\right)+Q_1^{\prime }\left(z\right)\mathrm{with}{Q}_1^{\prime }\left(z\right)=0\mathrm{or}deg(Q_1^{\prime }\left(z\right)<deg\left(q_{t_1}\left(z\right)\right).\end{array}$$

Then,

$$\begin{array}{cc}\hfill (q_1\left(z\right)f_{11}\left(z\right),0)& =((Q_1\left(z\right)q_{t_1}\left(z\right)+Q_1^{\prime }\left(z\right))f_{11}\left(z\right),0)\hfill \\ \hfill & =Q_1\left(z\right)(q_{t_1}\left(z\right)f_{11}\left(z\right),0)+Q_1^{\prime }\left(z\right)(f_{11}\left(z\right),0)\hfill \\ \hfill & =Q_1^{\prime }\left(z\right)(f_{11}\left(z\right),0)\in \mathrm{span}\left(S_1^2\right).\hfill \end{array}$$

Also, if $deg\left(q_2\left(z\right)\right)\ge deg\left(q_{t_2}\left(z\right)\right)$, then by the division algorithm there exist $Q_2\left(z\right)$ and $Q_2^{\prime }\left(z\right)$ in R such that

$$\begin{array}{c}\hfill q_2\left(z\right)=Q_2\left(z\right)q_{t_2}\left(z\right)+Q_2^{\prime }\left(z\right)\mathrm{with}{Q}_2^{\prime }\left(z\right)=0\mathrm{or}deg(Q_2^{\prime }\left(z\right)<deg\left(q_{t_2}\left(z\right)\right).\end{array}$$

Then

$$\begin{array}{cc}\hfill (q_2\left(z\right)r_{21}\left(z\right),f_{22}\left(z\right))& =((Q_2\left(z\right)q_{t_2}\left(z\right)+Q_2^{\prime }\left(z\right))r_{21}\left(z\right),f_{22}\left(z\right))\hfill \\ \hfill & =Q_2\left(z\right)(q_{t_2}\left(z\right)r_{21}\left(z\right),f_{22}\left(z\right))+Q_2^{\prime }\left(z\right)(r_{21}\left(z\right),f_{22}\left(z\right))\hfill \\ \hfill & =Q_2\left(z\right)(q_{t_2}\left(z\right)r_{21}\left(z\right),0)+Q_2^{\prime }\left(z\right)(r_{21}\left(z\right),f_{22}\left(z\right)).\hfill \end{array}$$

Assume that $q_{t_2}\left(z\right)r_{21}\left(z\right)\in ⟨f_{11}\left(z\right)⟩$. Then $Q_2\left(z\right)(q_{t_2}\left(z\right)r_{21}\left(z\right),0)\in ⟨(f_{11}\left(z\right),0)⟩$ and $deg\left(Q_2\left(z\right)\right)<deg\left(q_{t_1}\left(z\right)\right)$. Hence,

$$\begin{array}{cc}\hfill c\left(z\right)& =q_1\left(z\right)(f_{11}\left(z\right),0)+q_2\left(z\right)(r_{21}\left(z\right),f_{22}\left(z\right))\hfill \\ \hfill & =Q_1^{\prime }\left(z\right)(f_{11}\left(z\right),0)+Q_2^{\prime \prime }\left(z\right)(f_{11}\left(z\right),0)+Q_2^{\prime }\left(z\right)(r_{21}\left(z\right),f_{22}\left(z\right))\hfill \\ \hfill & =(Q_1^{\prime }\left(z\right)+Q_2^{\prime \prime }\left(z\right))(f_{11}\left(z\right),0)+Q_2^{\prime }\left(z\right)(r_{21}\left(z\right),f_{22}\left(z\right)),\hfill \end{array}$$

where $deg((Q_1^{\prime }\left(z\right)+Q_2^{\prime \prime }\left(z\right))<deg\left(q_{t_1}\left(z\right)\right)$. Therefore, $c\left(z\right)\in \mathrm{Span}(S_1^2\bigcup S_2^2)$, and clearly, $S_1^2\bigcup S_2^2$ is a linearly independent set. Thus, $dim\left(C\right)=deg\left(q_{t_1}\left(z\right)\right)+deg\left(q_{t_2}\left(z\right)\right)$.

Similarly, suppose that the statement holds for the index $l-1$ over $F_q$. We will also show that this statement holds for the index l. Consider any $c\left(z\right)\in C=⟨{\mathbf{a}}_1\left(z\right),{\mathbf{a}}_2\left(z\right),\dots ,{\mathbf{a}}_l\left(z\right)⟩$. Then there exist $q_1\left(z\right),q_2\left(z\right),\dots ,q_l\left(z\right)$ such that $c\left(z\right)=q_1\left(z\right){\mathbf{a}}_1\left(z\right)$$+q_2\left(z\right){\mathbf{a}}_2\left(z\right)+\cdots +q_l\left(z\right){\mathbf{a}}_l\left(z\right)$. Using the proof of Theorem 2, $ker\left(M_l\left(C\right)\right)=⟨{\mathbf{a}}_1\left(z\right),{\mathbf{a}}_2\left(z\right),$$\dots ,{\mathbf{a}}_{l-1}\left(z\right)⟩$ is a right GQPC code of block length $(t_1,t_2,\dots ,t_{l-1})$ of index $l-1$ over $F_q$. Thus, $ker\left(M_l\left(C\right)\right)={\bigcup }_{i=1}^{l-1}{S}_i$, where $S_i={\cup }_{i=0}^{deg\left(q_{t_i}\left(z\right)\right)-1}\{z^i\ast (r_{i,1}\left(z\right),r_{i,2}\left(z\right),\dots ,f_{ii}\left(z\right),\dots ,0)\}$. Now, consider $q_l\left(z\right){\mathbf{a}}_l\left(z\right)=q_l\left(z\right)(r_{l1}\left(z\right),r_{l2}\left(z\right),\dots ,f_{ll}\left(z\right))$ in $c\left(z\right)$. If $deg\left(q_l\left(z\right)\right)$$>deg\left(q_{t_l}\left(z\right)\right)$ then by the division algorithm, there exist $Q_l\left(z\right)$ and $Q_l^{\prime }\left(z\right)\in R$ such that $q_l\left(z\right)=Q_l\left(z\right)q_{t_l}\left(z\right)+Q_l^{\prime }\left(z\right)$, where $Q_l^{\prime }\left(z\right)=0$ or $deg\left(Q_l^{\prime }\left(z\right)\right)<deg\left(q_{t_l}\left(z\right)\right)$. Hence,

$$\begin{array}{cc}\hfill q_l\left(z\right){\mathbf{a}}_l\left(z\right)& =(Q_l\left(z\right)q_{t_l}\left(z\right)+Q_l^{\prime }\left(z\right))(r_{l1}\left(z\right),r_{l2}\left(z\right),\dots ,f_{ll}\left(z\right))\hfill \\ \hfill & =Q_l\left(z\right)(q_{t_l}\left(z\right)r_{l1}\left(z\right),q_{t_l}\left(z\right)r_{l2}\left(z\right),\dots ,0)+Q_l^{\prime }\left(z\right)(r_{l1}\left(z\right),r_{l2}\left(z\right),\dots ,f_{ll}\left(z\right)).\hfill \end{array}$$

Assume that $q_{t_l}\left(z\right)r_{ll-1}\left(z\right)\in ⟨f_{l-1l-1}\left(z\right)⟩$. Then $(q_{t_l}\left(z\right)r_{l1}\left(z\right),q_{t_l}\left(z\right)r_{l2}\left(z\right),\dots ,0)\in ker\left(M_l\left(C\right)\right)$ and $Q_l^{\prime }\left(z\right)=0\mathrm{or}deg\left(Q_l^{\prime }\left(z\right)\right)<deg\left(q_{t_l}\left(z\right)\right)$. Then we have

$$\begin{array}{c}\hfill q_l\left(z\right){\mathbf{a}}_l\left(z\right)\in \mathrm{Span}\left(\bigcup _{i=1}^{l-1}{S}_i\bigcup S_l\right).\end{array}$$

Therefore, $c\left(z\right)\in \mathrm{Span}\left({\bigcup }_{i=1}^{l-1}{S}_i\bigcup S_l\right)$. Clearly, it is a linearly independent set, as none of its elements can be written as a nonzero linear combination of the remaining elements. Thus, $dim\left(C\right)=deg\left(q_{t_1}\left(z\right)\right)+deg\left(q_{t_2}\left(z\right)\right)+\cdots +deg\left(q_{t_l}\left(z\right)\right)$. □

The next step in our discussion is to characterize the performance of these codes in terms of optimality. To this end, we define the notions of optimal and near-optimal GQPC codes.

Definition 7. 

A linear code C of parameters $[n,k,d]$ over $F_q$ is said to be optimal (with fixed length and dimension) if no other $[n,k,d^{\prime }]$ code over $F_q$ exists in codestable [30] with $d^{\prime }>d$.

Definition 8. 

A linear code C of parameters $[n,k,d]$ over $F_q$ is said to be near-optimal (with fixed length and dimension) if d is one less than that given in codestable [30].

Remark 2. 

It is noted that if we do not fix the dimension, then “optimality" will occur in two ways: one regarding the code rate and the other regarding the error-correcting capability. As $[7,2,4]$ is not optimal over $F_2$, because a better code $[7,3,4]$ exists.

Example 5. 

Let $q=2$, $t_1=36$ and $t_2=2$. Consider the monic polynomials $P_{t_1,{\mathtt{a}}_1}\left(z\right)=z^{36}+z^{34}+z^{32}+z^{30}+z^{28}+z^{26}+z^{24}+z^{22}+z^{21}+z^{18}+z^{16}+z^{14}+z^{11}+z^9+z^8+z^6+z^4+z^2+z+1$, where

$$\begin{array}{c}\hfill {\mathtt{a}}_1=(1,1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0)\in F_2^{36}\end{array}$$

, and $P_{t_2,{\mathtt{a}}_2}\left(z\right)=z^2+z+1$ where as ${\mathtt{a}}_2=(1,1)\in F_2^2$. Following the Theorems 2 and 3, we have a normalized generating set $\{{\mathit{a}}_1\left(z\right),{\mathit{a}}_2\left(z\right)\}$, where

$$\begin{array}{c}\hfill {\mathit{a}}_1\left(z\right)=\{f_{11}\left(z\right)=z^{14}+z^{10}+z^8+z^7+z^6+z^4+z^3+z^2+1,f_{21}\left(z\right)=0\}\mathit{and}\\ \hfill {\mathit{a}}_2\left(z\right)=\{r_{21}\left(z\right)=z^{12}+z^{11}+z^9+z^7+z^4+z^3+z^2+z+1,f_{22}\left(z\right)=1\}.\end{array}$$

Let C be a right GQPC code of block length $(36,2)$ and index 2 with $C=⟨{\mathit{a}}_1\left(z\right),{\mathit{a}}_2\left(z\right)⟩$. Then, from Theorem 4, the dimension of C is $22+2=24$. The generator matrix of C of order $(24\times 38)$ is given by

$$\begin{array}{c}\hfill G=\left[\begin{array}{c}1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0|0,0\\ 0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0|0,0\\ 0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0|0,0\\ 0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0|0,0\\ 0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0|0,0\\ 0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0|0,0\\ 0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0|0,0\\ 0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0|0,0\\ 0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0|0,0\\ 0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0|0,0\\ 0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0|0,0\\ 0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0|0,0\\ 0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0|0,0\\ 0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0|0,0\\ 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0|0,0\\ 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0|0,0\\ 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0,0|0,0\\ 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,0|0,0\\ 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0|0,0\\ 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0|0,0\\ 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0|0,0\\ 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,1|0,0\\ 1,1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0|1,0\\ 0,1,1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0|0,1\end{array}\right].\end{array}$$

Hence, we obtain C as a $[38,24,6]$ right GQPC code over $F_2$ which meets the distance for length 38 and dimension 24 in the codestable [30]. Thus, C is an optimal right GQPC code.

By following Theorems 2–4, we construct several optimal and near-optimal 2-generator right GQPC codes of index 2 by fixing the dimension that are tabulated in Table 1. The optimal codes presented in the table meet the distance bound in the code table [30]. In Table 1, we represent polynomials $P_{t_1,{\mathtt{a}}_1}\left(z\right)=z^{t_1}-{\mathtt{a}}_1\left(z\right)$ and $P_{t_2,{\mathtt{a}}_2}\left(z\right)=z^{t_2}-{\mathtt{a}}_2\left(z\right)$ and the nonzero term of the normalized generating set in terms of their coefficient in ascending order of the degree of the indeterminate. For example, if

$$\begin{array}{cc}\hfill P{t}_1,{\mathtt{a}}_1\left(z\right)& =z^{16}+z^3+1\hfill \\ & =(z^3+z^2+1)(z^{13}+z^{12}+z^{11}+z^9+z^6+z^5+z^4+z^2+1),\hfill \end{array}$$

then $P_{t_1,{\mathtt{a}}_1}\left(z\right)$ is represented as $\left(10010000000000001\right)$, and $g_{t_1}\left(z\right)=z^3+z^2+1$ is represented by $\left(1011\right)$. We use MAGMA software (Magma $V2.28-6$) and the Magma handbook [29] for our calculations to calculate the parameters of the codes.

Table 1. 2-Generator right GQPC codes over $F_2$.

${\mathit{t}}_1,{\mathit{t}}_2,\mathit{N}$	Polynomials ${\mathit{z}}^{{\mathit{t}}_{\mathit{i}}}-{\mathit{a}}_{\mathit{i}}\left(\mathit{z}\right)$, ${\mathit{f}}_{11}\left(\mathit{z}\right),{\mathit{r}}_{21}\left(\mathit{z}\right)$ and ${\mathit{f}}_{22}\left(\mathit{z}\right)$	Parameters	Remarks
$20,2,22$	$P_{t_1,a_1}\left(z\right)=110111110111111111111$,
	$f_{11}=101101100111$, $r_{21}=1001001011$,
	$P_{t_2,a_2}\left(z\right)=101$, $f_{22}=1$	$[22,11,6]$	$\mathrm{Near-optimal},\mathrm{LCD}$
$29,2,31$	$P_{t_1,a_1}\left(z\right)=111101101001011010101010101011$,
	$f_{11}=1011101111011$, $r_{21}=10011100111$,
	$P_{t_2,a_2}\left(z\right)=101$, $f_{22}=1$	$[31,19,6]$	Optimal
$34,2,36$	$P_{t_1,a_1}\left(z\right)=11110111010101011010111011001010101$,
	$f_{11}=11110111101010001111$, $r_{21}=110001100010000011$,
	$P_{t_2,a_2}\left(z\right)=101$, $f_{22}=1$	$[36,17,8]$	Optimal
$36,1,37$	$P_{t_1,a_1}\left(z\right)=1110101011010010101001101010101010101$,
	$f_{11}=10100110111100010001$, $r_{21}=1100010010100001111$,
	$P_{t_2,a_2}\left(z\right)=11$, $f_{22}=1$	$[37,18,8]$	Optimal
$36,1,37$	$P_{t_1,a_1}\left(z\right)=1110101011010010101001101010101010101$,
	$f_{11}=10000101111101$, $r_{21}=1111100101011$,
	$P_{t_2,a_2}\left(z\right)=11$, $f_{22}=1$	$[37,24,6]$	Optimal
$36,1,37$	$P_{t_1,a_1}\left(z\right)=1110101011010010101001101010101010101$,
	$f_{11}=1111101011$, $r_{21}=101011001$,
	$P_{t_2,a_2}\left(z\right)=11$, $f_{22}=1$	$[37,28,4]$	Optimal
$36,2,38$	$P_{t_1,a_1}\left(z\right)=1110101011010010101001101010101010101$,
	$f_{11}=101110111010001$, $r_{21}=1111100101011$,
	$P_{t_2,a_2}\left(z\right)=111$, $f_{22}=1$	$[38,24,6]$	Optimal
$38,2,40$	$P_{t_1,a_1}\left(z\right)=111101101001011010101010101010110100101$,
	$f_{11}=111110111$, $r_{21}=10101101$,
	$P_{t_2,a_2}\left(z\right)=101$, $f_{22}=11$	$[40,31,4]$	Optimal
$38,2,40$	$P_{t_1,a_1}\left(z\right)=111101101001011010101010101010110100101$,
	$f_{11}=10101011101111110111$, $r_{21}=111000100010010001$,
	$P_{t_2,a_2}\left(z\right)=101$, $f_{22}=1$	$[40,21,7]$	Near-optimal
$40,2,42$	$P_{t_1,a_1}\left(z\right)=11110110100101101010101010101011010010111$,
	$f_{11}=10010011$, $r_{21}=101111$,
	$P_{t_2,a_2}\left(z\right)=101$, $f_{22}=1$	$[42,35,4]$	Optimal
$40,2,42$	$P_{t_1,a_1}\left(z\right)=11110110100101101010101010101011010010111$,
	$f_{11}=10111100110110110001$, $r_{21}=1101011101101101111$,
	$P_{t_2,a_2}\left(z\right)=101$, $f_{22}=11$	$[42,22,8]$	$\mathrm{Optimal},\mathrm{LCD}$
$41,1,42$	$P_{t_1,a_1}\left(z\right)=111101110101010011101010101010101010101001$,
	$f_{11}=1001000101011011$, $r_{21}=111000011001001$,
	$P_{t_2,a_2}\left(z\right)=11$, $f_{22}=1$	$[42,27,6]$	Optimal
$47,1,48$	$P_{t_1,a_1}\left(z\right)=101011110101010110101110110010101010101010001001$,
	$f_{11}=11111100101$, $r_{21}=1010100011$,
	$P_{t_2,a_2}\left(z\right)=11$, $f_{22}=1$	$[48,38,4]$	Optimal
$50,2,52$	$P_{t_1,a_1}\left(z\right)=111111110101010011101110101010101001101010100110111$,
	$f_{11}=100010001111$, $r_{21}=1101011001$,
	$P_{t_2,a_2}\left(z\right)=111$, $f_{22}=1$	$[52,41,4]$	Optimal
$50,1,51$	$P_{t_1,a_1}\left(z\right)=111111110101010011101110101010101001101010100110111$,
	$f_{11}=100001110001110100111111$, $r_{21}=11111010000101100010101$,
	$P_{t_2,a_2}\left(z\right)=11$, $f_{22}=1$	$[51,28,8]$	Near-optimal
$51,1,52$	$P_{t_1,a_1}\left(z\right)=1111111101010101101011101100101010101010101011101001$,
	$f_{11}=11111110111011101$, $r_{21}=1010101101001011$,
	$P_{t_2,a_2}\left(z\right)=11$, $f_{22}=1$	$[52,36,6]$	Optimal

Remark 3. 

A linear code C of length n over $F_2$ is called a linear complementary dual (LCD) code if $C\cap C^{\perp }=\left\{0\right\}$, where $C^{\perp }$ denotes the dual code of C with respect to the standard inner product. Based on the above definition of LCD codes, Table 1 lists specific examples constructed under this framework.

5. Conclusions

In this work, we have generalized the concept of QPC and GQC codes over the finite field $F_q$. Here, we have decomposed the right GQPC codes into linear codes using the Chinese Remainder Theorem and characterized the $\rho$-generator right GQPC codes (Section 3). Furthermore, we have discussed the normalized generating set of $\rho$-generator polynomials for the right GQPC codes and their corresponding dimensions and presented a method for finding a normalized generating set from any $\rho$-generator polynomial set. This way, we have obtained several optimal right GQPC codes (Table 1).

However, it will be interesting to further work on the concatenated structure of the right GQPC code. Also, finding the annihilator dual of these codes would be a promising direction. Furthermore, the construction of quantum codes from these codes remains an open problem.