Hamming and Symbol-Pair Distances of Constacyclic Codes of Length 2p^s over ${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{⟨u^2,v^2,uv-vu⟩}}$

Abstract

Let $\mathcal{R}={\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{⟨u^2,v^2,uv-vu⟩}}$, where p is an odd prime and m is a positive integer. For a unit $\alpha$ in $\mathcal{R}$, $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$ are ideals of ${\displaystyle \frac{\mathcal{R}\left[x\right]}{⟨x^{2p^s}-\alpha ⟩}}$, where s is a positive integer. The structure of $\alpha$-constacyclic codes are classified on the distinct cases for the unit $\alpha$: when $\alpha$ is a square in $\mathcal{R}$ and when it is not. In this paper, for all such $\alpha$-constacyclic codes, the Hamming distances are determined using this structure. In addition, their symbol-pair distances are obtained. The symmetry property of Hamming and symbol-pair distances makes analysis easier and maintains consistency by guaranteeing that the distance between codewords is the same regardless of their order. As symmetry preserves invariant distance features across transformations, it improves error detection and correction.

1. Introduction

Digital processing was expensive and slow, and digital communication channels were also slow. In 1948, Claude Shannon’s [1] study showed that using a small amount of redundancy could allow decoders to correct bit errors, improving system parameters like power, file size, and transfer speeds. This led to the development of error-correcting codes. Linear codes introduced by Shannon [1] play a crucial role in information theory for recovering corrupted messages sent through noisy communication channels. Initially, linear codes were studied over finite fields. Since the late 1950s, cyclic codes over finite fields have been extensively researched due to their practical applications and rich algebraic structures. Cyclic codes are generalized into constacyclic codes. As they can be efficiently encoded using basic shift registers, constacyclic codes have practical applications. With the transition from cyclic to constacyclic coding, error correction can be handled with more flexibility. Their uses are widespread in contemporary digital communications, storage systems, wireless networks, and cryptography, making them an essential component of reliable data transfer.

In the finite field ${\mathbb{F}}_q$, $\alpha$-constacyclic codes of length n over ${\mathbb{F}}_q$ are categorized as ideals $\langle \mathsf{\ell }\left(x\right)\rangle$ of the ambient ring ${\displaystyle \frac{{\mathbb{F}}_q\left[x\right]}{\langle x^n-\alpha \rangle }}$, where $\mathsf{\ell }\left(x\right)$ is a divisor of $x^n-\alpha$. The codes are referred to as simple root codes when the length of the code n is relatively prime to the characteristic of ${\mathbb{F}}_q$. If not, they are known as repeated-root codes, which were initially studied in [2,3,4].

A landmark study by Hammons et al. [5] demonstrated that by using the Gray map, linear codes over ${\mathbb{Z}}_4$ can be converted into some good binary nonlinear codes, such as Preparata and Kerdock codes. This work initiated a rigorous study on codes over rings.

Codes over chain rings have been examined by several authors [6,7,8,9,10,11,12,13,14,15]. Recently, cyclic and constacyclic codes over finite non-chain rings have drawn the attention of researchers as they can obtain optimal linear codes and quantum codes. In [16], Yildiz and Karadeniz examined cyclic codes of odd length over the non-chain ring ${\displaystyle \frac{{\mathbb{F}}_2[u,v]}{\langle u^2,v^2,uv-vu\rangle }}$. As the Gray images of these cyclic codes, they found a few good binary codes. Later, they examined $(1+v)$-constacyclic codes over the ring ${\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2$ and obtained cyclic codes over ${\mathbb{F}}_2+u{\mathbb{F}}_2$ as the image of $(1+v)$-constacyclic codes over ${\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2$ under natural Gray map [17]. After that, Haifeng et al. [18] examined $(1-uv)$-constacyclic codes over ${\mathbb{F}}_p+u{\mathbb{F}}_p+v{\mathbb{F}}_p+uv{\mathbb{F}}_p$ and showed that under a Gray map, images of these codes are distance invariant quasi-cyclic code of length $p^{3}n$ and index $p^2$ over ${\mathbb{F}}_p$. Negacyclic codes of odd length over ${\displaystyle \frac{{\mathbb{F}}_p[u,v]}{\langle u^2,v^2,uv-vu\rangle }}$ are studied by Ghosh [19]. Moreover, Bag et al. [20] investigated $({\lambda }_1+u{\lambda }_2)$ and $({\lambda }_1+v{\lambda }_3)$-constacyclic codes of length $p^s$ over ${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2,v^2,uv-vu\rangle }}$. The more general ring ${\displaystyle \frac{{\mathbb{F}}_2[u_1,u_2,\dots ,u_k]}{\langle u_i^2,v_j^2,u_i{v}_j-v_j{u}_i\rangle }}$ was taken into consideration by Dougherty et al. [21], who also examined the general characteristics of cyclic codes over such rings and identified the nontrivial one-generator cyclic codes. These investigations were expanded to cyclic codes over the ring ${\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2,v^2,uv-vu\rangle }}$ by Sobhani and Molakarimi in [22]. The structures of all constacyclic codes of length $p^s$ and $2p^s$ over the ring ${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2,v^2,uv-vu\rangle }}$ are investigated in [23,24]. For recent research on non-chain rings, one can refer [25,26,27,28].

For a given linear code $\mathcal{C}$, with a Hamming distance $d_H$, the error correction capacity of $\mathcal{C}$ is $\lfloor {\displaystyle \frac{d_H-1}{2}}\rfloor$ [29]. The symmetry property of the Hamming distance, which states that $d_H(x,y)=d_H(y,x)$, guarantees consistent evaluation of codeword differences and makes error correction easier. This constancy makes decoding predictable and enhances the effectiveness and dependability of error-correcting codes. As a result, the Hamming distance plays an important role in coding theory. Thus, many researchers are interested in the Hamming distance. In 2010, Dinh [8] computed the Hamming distances of all ($\alpha +u\beta$)-constacyclic codes of length $p^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$. Later, for all $\gamma$ -constacyclic codes of length $p^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$, the Hamming distances are studied in [30]. For all constacyclic codes of length $2p^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$, the Hamming distances are completely determined in [31]. In 2020, the Hamming distance of $(\alpha +u\beta )$-constacyclic codes (where $\alpha +u\beta$ is not a cube) of length $3p^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ was determined in [11]. For $\lambda$-constacyclic codes of length $3p^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$, where $\lambda =\alpha +u\beta$ is a cube and $\lambda$ is not a cube in ${\mathbb{F}}_{p^m}$, the Hamming distance was determined in [32]. Further, Hamming distances of $\alpha$-constacyclic codes of length $p^s$ over ${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2,v^2,uv-vu\rangle }}$ for $\alpha ={\alpha }_1+{\alpha }_{2}u+{\alpha }_{4}uv$, ${\alpha }_1+{\alpha }_{3}v+{\alpha }_{4}uv$, ${\alpha }_1+{\alpha }_{2}u+{\alpha }_{3}v+{\alpha }_{4}uv$, where ${\alpha }_1,{\alpha }_2,{\alpha }_3\in {\mathbb{F}}_{p^m}^{*}$, and ${\alpha }_4\in {\mathbb{F}}_{p^m}$ are determined in [33].

As high-density data storage technologies advance, symbol-pair codes are suggested as an effective way to guard against a specific number of pair errors. Cassuto et al. [34] introduced a novel coding method for symbol-pair read channels in 2010. In this method, the output of the read process is consecutive symbol pairs. The symbol-pair distances of the constacyclic codes over ${\mathbb{F}}_{p^m}$ are studied in [35,36,37]. Further, symbol-pair distances of repeated-root constacyclic codes of length $p^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ and ${\mathbb{F}}_{p^m}\left[u\right]/\langle u^3\rangle$ were studied in [30,38], respectively. Also, several authors studied Maximum Distance Separable symbol-pair codes (see, for example, [39,40,41,42,43]).

By examining the literature, the Hamming and symbol-pair distances of constacyclic codes of length $2p^s$ over $\mathcal{R}$ have remained open. Motivated by all these works, in this paper, we compute the Hamming and symbol-pair distances of constacyclic codes of length $2p^s$ over $\mathcal{R}$. The paper is organized as follows. Section 2 provides some preliminary information. Section 3 gives a brief outline of the $\alpha$-constacyclic code of length $2p^s$ over the ring $\mathcal{R}={\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2,v^2,uv-vu\rangle }}$. In Section 4, we obtain the Hamming distances of $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$. In Section 5, we determine the symbol-pair distance of $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$. The paper is concluded in Section 6.

2. Preliminaries

Let ℜ be a finite commutative ring with identity. A code $\mathcal{C}$ of length n over ℜ is a nonempty subset of ${\Re }^n$. An element of $\mathcal{C}$ is called a codeword. If $\mathcal{C}$ is an ℜ-submodule of ${\Re }^n$, then $\mathcal{C}$ is said to be linear denoted by $[n,k,d]$, where k is the dimension of the code and d is the distance of code. Let $\alpha$ be a unit of ℜ. The $\alpha$-constacyclic shift ${\sigma }_{\alpha }$ on ${\Re }^n$ is the shift,

$${\sigma }_{\alpha }({\zeta }_0,{\zeta }_1,\dots ,{\zeta }_{n-1})=(\alpha {\zeta }_{n-1},{\zeta }_0,\dots ,{\zeta }_{n-2})$$

and a code $\mathcal{C}$ is said to be $\alpha$-constacyclic if $\mathcal{C}$ is closed under the $\alpha$-constacyclic shift ${\sigma }_{\alpha }$. If $\alpha$ is equal to 1 (or −1), then the $\alpha$-constacyclic codes are referred to as cyclic (or negacyclic) codes.

Each codeword in $\mathcal{C}$, denoted by $\zeta =({\zeta }_0,{\zeta }_1,\dots ,{\zeta }_{n-1})$, is uniquely identified using its polynomial representation, $\zeta \left(x\right)={\zeta }_0+{\zeta }_{1}x+\cdots +{\zeta }_{n-1}{x}^{n-1}$. From that, the following proposition follows from [29,44].

Proposition 1.

A linear code $\mathcal{C}$ of length n over ℜ is an α-constacyclic if and only if $\mathcal{C}$ is an ideal of ${\displaystyle \frac{\Re \left[x\right]}{\langle x^n-\alpha \rangle }}$.

Let $\zeta =({\zeta }_0,{\zeta }_1,\dots ,{\zeta }_{n-1})$ and ${\zeta }^{\prime }=({\zeta }_0^{\prime },{\zeta }_1^{\prime },\dots ,{\zeta }_{n-1}^{\prime })$ be two vectors in ${\Re }^n$, where ℜ is a code alphabet. The Hamming weight of $\zeta$, represented as $w{t}_H\left(\zeta \right)$, is the nonzero entries of a codeword $\zeta$. The Hamming distance $d_H(\zeta ,{\zeta }^{\prime })$, of two words $\zeta$ and ${\zeta }^{\prime }$, equals the number of components in which they differ. i.e., $d_H(\zeta ,{\zeta }^{\prime })=w{t}_H(\zeta -{\zeta }^{\prime })$. The Hamming distance of code $\mathcal{C}$ contains at least two words, $d_H\left(\mathcal{C}\right)=min\{d(\zeta ,{\zeta }^{\prime }):\zeta ,{\zeta }^{\prime }\in \mathcal{C},\zeta \ne {\zeta }^{\prime }\}$.

The symbol-pair distance is given by using the Hamming distance over the alphabet $(\Re ,\Re )$ as follows:

$$d_{sp}(\zeta ,{\zeta }^{\prime })=\left|\{i:({\zeta }_i,{\zeta }_{i+1})\ne ({\zeta }_i^{\prime },{\zeta }_{i+1}^{\prime })\}\right|.$$

Then, $d_{sp}\left(\mathcal{C}\right)=\underset{\underset{\zeta \ne {\zeta }^{\prime }}{\zeta ,{\zeta }^{\prime }\in \mathcal{C}}}{min}\left\{d_{sp}(\zeta ,{\zeta }^{\prime })\right\}$ is the symbol-pair distance of code $\mathcal{C}$ [34].

Theorem 1

([34]). For $x,y$ in ${\mathcal{R}}^n$, let $0<d_H(x,y)<n$ be the Hamming distance between x and y. Then,

$$d_H(x,y)+1\le d_{sp}(x,y)\le 2d_H(x,y).$$

(1)

In the extreme cases in which $d_H(x,y)$ equals 0 or n, clearly $d_{sp}(x,y)=d_H(x,y)$.

Theorem 2

([45]). Let ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ be two $[n_1,k_1,d_1]$ and $[n_2,k_2,d_2]$ linear codes over ${\mathbb{F}}_{p^m}$. Then, their direct sum is defined as ${\mathcal{C}}_1\oplus {\mathcal{C}}_2=\left\{({\zeta }_1,{\zeta }_2)\right|{\zeta }_1\in {\mathcal{C}}_1,{\zeta }_2\in {\mathcal{C}}_2\}$, representing $[n_1+n_2,k_1+k_2,min\{d_1,d_2\}]$ linear codes over ${\mathbb{F}}_{p^m}$.

3. $\mathit{\alpha }$-Constacyclic Code of Length $\mathbf{2}{\mathit{p}}^{\mathit{s}}$ over the Ring $\mathcal{R}$

Let p be an odd prime number and m be a positive integer. Let $\mathcal{R}={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}+uv{\mathbb{F}}_{p^m}$ ($u^2=0$, $v^2=0$, $uv=vu$). An element $\alpha ={\alpha }_1+{\alpha }_{2}u+{\alpha }_{3}v+{\alpha }_{4}uv\in \mathcal{R}$ is a unit if and only if ${\alpha }_1$ is non zero in ${\mathbb{F}}_{p^m}$. Let ${\alpha }_1\in {\mathbb{F}}_{p^m}^{*}$. Now, ${\alpha }_1^{p^{tm}}={\alpha }_1$ for any positive integer t. For positive integers m and s, by division algorithm, $s=q_{0}m+r_0$, where $q_0$ and $r_0$ are non-negative integers with $0\le r_0\le m-1$. Let ${\alpha }_0={\alpha }_1^{p^{(q_0+1)m-s}}={\alpha }_1^{p^{m-r_0}}$. Then, ${{\alpha }_0}^{p^s}={\alpha }_1^{p^{(q_0+1)m}}={\alpha }_1$.

For any unit $\alpha$ of $\mathcal{R}$, let

$${\mathcal{R}}_{\alpha }={\displaystyle \frac{\mathcal{R}\left[x\right]}{\langle x^{2p^s}-\alpha \rangle }}.$$

Proposition 1 indicates that $\alpha$-constacyclic codes with length $2p^s$ over $\mathcal{R}$ are ideals of ${\mathcal{R}}_{\alpha }$.

In [24], Ahendouz and Akharraz studied the $({\alpha }_1+{\alpha }_{2}u+{\alpha }_{3}v+{\alpha }_{4}uv)$-constacyclic code of length $2p^s$ over the ring $\mathcal{R}$, where ${\alpha }_1\in {\mathbb{F}}_{p^m}^{*}$ and ${\alpha }_2,{\alpha }_3,{\alpha }_4\in {\mathbb{F}}_{p^m}$ with ${\alpha }_2$ and ${\alpha }_3$ are not both zero. Also, their duals are determined.

Theorem 3

([24]). Let α be a square in $\mathcal{R}$, say $\alpha ={\gamma }^2$ for some $\gamma \in \mathcal{R}$. A subset $\mathcal{C}$ of ${\mathcal{R}}_{\alpha }$ is a α-constacyclic code of length $2p^s$ over $\mathcal{R}$ if and only if $\mathcal{C}\cong {\mathcal{C}}^{\prime }\oplus {\mathcal{C}}^{″}$, where ${\mathcal{C}}^{\prime }$ is a γ-constacyclic code of length $p^s$ over $\mathcal{R}$ and ${\mathcal{C}}^{\prime \prime }$ is a $-\gamma$-constacyclic code of length $p^s$ over $\mathcal{R}$.

Theorem 4

([24]). Let $\alpha ={\alpha }_1+{\alpha }_{2}u+{\alpha }_{3}v+{\alpha }_{4}uv$ be a non-square in $\mathcal{R}$, where ${\alpha }_1\in {\mathbb{F}}_{p^m}^{*}$ and ${\alpha }_2,{\alpha }_3,{\alpha }_4\in {\mathbb{F}}_{p^m}$ with ${\alpha }_2$ and ${\alpha }_3$ are not both zero. Then, α-constacyclic codes of length $2p^s$ over $\mathcal{R}$, that is, the ideals of the quotient ring $R_{\alpha }$, are

• Type A: $\langle 0\rangle$, $\langle 1\rangle$.
• Type B: $\langle u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\rangle$, where $0\le \mathsf{\ell }\le 2p^s-1$.
• Type C: $\langle {(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right)\rangle$, where $0\le \mathsf{\ell }\le 2p^s-1$, $0\le t<\mathsf{\ell }$ and either $z\left(x\right)$ is 0 or $z\left(x\right)$ is a unit that can be represented as $z\left(x\right)={\displaystyle \sum _{\kappa }}(z_{0\kappa }x+z_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }$ with $z_{0\kappa },z_{1\kappa }\in {\mathbb{F}}_{p^m}$ and $z_{00}x+z_{10}\ne 0$.
• Type D: $\langle {(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right),u{(x^2-{\alpha }_0)}^{\mu }\rangle$, where $0\le \mathsf{\ell }\le 2p^s-1$, $0\le t<\mu \le \Im$ with $z\left(x\right)$ as in Type C.
  Here, $\Im =min\left\{i\right|u{(x^2-{\alpha }_0)}^i\in \langle {(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right)\rangle \}$. Also,

  $$\Im =\left\{\begin{array}{cc}\mathsf{\ell }\hfill & ifz\left(x\right)=0;\hfill \\ min\{\mathsf{\ell },\theta \}\hfill & ifz\left(x\right)\ne 0.\hfill \end{array}\right.$$

  where $\theta =max\{\mathsf{\ell }|{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\mathrm{divides}{(x^2-{\alpha }_0)}^{2p^s+t-\mathsf{\ell }}+2{\alpha }_2{(x^2-{\alpha }_0)}^{p^s}\}$.

Let $\alpha ={\alpha }_1+{\alpha }_{2}u+{\alpha }_{3}v+{\alpha }_{4}uv$, where ${\alpha }_1,{\alpha }_4\in {\mathbb{F}}_{p^m}^{*}$ and ${\alpha }_2,{\alpha }_3\in {\mathbb{F}}_{p^m}$ with ${\alpha }_2$ and ${\alpha }_3$ are not both zero. In the following sections, we determine the Hamming and symbol-pair distances of the $\alpha$-constacyclic code of length $2p^s$ over the ring $\mathcal{R}$.

4. Hamming Distance

The Hamming distances of $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$ are determined in this section, where $\alpha$ is an arbitrary unit of $\mathcal{R}$. As shown in [24], the types of $\alpha$ determine the construction of $\alpha$-constacyclic codes. Therefore, we must take into account two scenarios: $\alpha$ is a square and $\alpha$ is a non-square.

By Theorem 2 and 3, when $\alpha$ is a square in $\mathcal{R}$, the Hamming distance of the $\alpha$-constacyclic code of length $2p^s$ is the minimum of the Hamming distance of the $\gamma$-constacyclic code of length $p^s$ and the Hamming distance of the −$\gamma$-constacyclic code of length $p^s$.

Permouth et al. [46] determined the Hamming distances of the $\alpha$-constacyclic code of length $2p^s$ over ${\mathbb{F}}_{p^m}$.

Theorem 5

([46]). Let $\mathcal{C}=\langle {(x^2-{\alpha }_0)}^{\mathsf{\ell }}\rangle$⊆${\displaystyle \frac{{\mathbb{F}}_{p^m}\left[x\right]}{\langle x^{2p^s}-\alpha \rangle }}$, for $0\le \mathsf{\ell }\le p^s$. Then,

$$d_H\left(\mathcal{C}\right)=\left\{\begin{array}{cc}1\hfill & if\mathsf{\ell }=0;\hfill \\ 2\hfill & if1\le \mathsf{\ell }\le p^{s-1};\hfill \\ ({\beta }_0+2)\hfill & if{\beta }_0{p}^{s-1}+1\le \mathsf{\ell }\le ({\beta }_0+1)p^{s-1},where1\le {\beta }_0\le p-2;\hfill \\ (\Gamma +1)p^{\gamma }\hfill & if{p}^s-p^{s-\gamma }+(\Gamma -1)\omega +1\le \mathsf{\ell }\le p^s-p^{s-\gamma }+\Gamma \omega ;\hfill \\ 0\hfill & if\mathsf{\ell }=p^s,\hfill \end{array}\right.$$

where $\omega =p^{s-\gamma -1}$, $1\le \Gamma \le p-1$ and $0\le \gamma \le s-1$.

Now, we determine the Hamming distances of the $\alpha$-constacyclic code of length $2p^s$ over $\mathcal{R}$ when $\alpha$ is not a square in $\mathcal{R}$. The Hamming distance of Type A ideals $\langle 0\rangle$ and $\langle 1\rangle$ of $R_{\alpha }$ are obviously 0 and 1, respectively. Now, we calculate the Hamming distance of Type B, C and Type D $\alpha$-constacyclic code of length $2p^s$ when $\alpha$ is not a square in $\mathcal{R}$. Note that Section 3 of Theorem 4 provides the structure of $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$ when $\alpha$ is not a square in $\mathcal{R}$. The Hamming distance of Type B $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$ is discussed in the following theorem.

Theorem 6.

Let ${\mathcal{C}}_2=\langle u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\rangle$, where $0\le \mathsf{\ell }\le 2p^s-1$. Then,

$$d_H\left({\mathcal{C}}_2\right)=\left\{\begin{array}{cc}1\hfill & if0\le \mathsf{\ell }\le p^s;\hfill \\ (\Gamma +1)p^{\gamma }\hfill & if2p^s-p^{s-\gamma }+(\Gamma -1)\omega +1\le \mathsf{\ell }\le 2p^s-p^{s-\gamma }+\Gamma \omega ,\hfill \end{array}\right.$$

where $\omega =p^{s-\gamma -1}$, $1\le \Gamma \le p-1$ and $0\le \gamma \le s-1$.

Proof. 

From Theorem 4, Type B $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$ are given by ${\mathcal{C}}_2=\langle u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\rangle$, where $0\le \mathsf{\ell }\le 2p^s-1$.

• Case (i): If $0\le \mathsf{\ell }\le p^s$, then $d_H\left({\mathcal{C}}_2\right)=1$.
• Case (ii): If $p^s+1\le \mathsf{\ell }\le 2p^s-1$, then ${\mathcal{C}}_2=\langle u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\rangle =\langle uv{(x^2-{\alpha }_0)}^{\mathsf{\ell }-p^s}\rangle$. Thus, ${\mathcal{C}}_2$ is precisely the $\alpha$-constacyclic code $\langle {(x^2-{\alpha }_0)}^{\mathsf{\ell }-p^s}\rangle$ in ${\displaystyle \frac{{\mathbb{F}}_{p^m}\left[x\right]}{\langle x^{2p^s}-\alpha \rangle }}$ multiplied by $uv$. Therefore, $d_H\left({\mathcal{C}}_2\right)=d_H\left({\langle {(x^2-{\alpha }_0)}^{\kappa }\rangle }_{{\mathbb{F}}_{p^m}}\right)$, and the proof follows from Theorem 5.

□

The following result calculates the Hamming distance of Type C $\alpha$-constacyclic codes.

Theorem 7.

Let ${\mathcal{C}}_3=\langle {(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right)\rangle$, where $0\le \mathsf{\ell }\le 2p^s-1$, $0\le t<\mathsf{\ell }$ and $z\left(x\right)$ is 0 or a unit. Then,

$$d_H\left({\mathcal{C}}_3\right)=\left\{\begin{array}{cc}1\hfill & if0\le \Im \le p^s;\hfill \\ (\Gamma +1)p^{\gamma }\hfill & if2p^s-p^{s-\gamma }+(\Gamma -1)\omega +1\le \Im \le 2p^s-p^{s-\gamma }+\Gamma \omega ,\hfill \end{array}\right.$$

where $\omega =p^{s-\gamma -1}$, $1\le \Gamma \le p-1$ and $0\le \gamma \le s-1$.

Proof. 

From Theorem 4, Type C $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$ are given by ${\mathcal{C}}_3=\langle {(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right)\rangle$, where $0\le \mathsf{\ell }\le 2p^s-1$, $0\le t<\mathsf{\ell }$, and $z\left(x\right)$ is 0 or a unit. Let ℑ be the smallest integer such that $u{(x^2-{\alpha }_0)}^{\Im }\in {\mathcal{C}}_3$. Then, $d_H\left({\mathcal{C}}_3\right)\le d_H\left(\langle u{(x^2-{\alpha }_0)}^{\Im }\rangle \right)$. Let $a\left(x\right)\in {\mathcal{C}}_3$. Then, $b\left(x\right)={\displaystyle \sum _{\kappa =0}^{2p^s-1}}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }+u{\displaystyle \sum _{\kappa =0}^{2p^s-1}}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\in R_{\alpha }$ exists, where $a_{0\kappa },b_{0\kappa },a_{1\kappa },b_{1\kappa }\in {\mathbb{F}}_{p^m}$ such that $a\left(x\right)=b\left(x\right)[{(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right)]$. Thus,

$$\begin{array}{cc}\hfill a\left(x\right)=& \left[\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }+u\sum _{\kappa =0}^{2p^s-1}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\right]\hfill \\ \hfill & \times [{(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right)]\hfill \\ \hfill =& {(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-1}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{t}z\left(x\right)\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \end{array}$$

(2)

Then,

$$\begin{array}{cc}\hfill w_H\left(a\left(x\right)\right)& \ge w_H\left(ua\left(x\right)\right)\hfill \\ \hfill & =w_H\left(u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\right)\hfill \\ \hfill & \ge d_H\left(〈u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}〉\right)\hfill \\ \hfill & \ge d_H\left(〈u{(x^2-{\alpha }_0)}^{\Im }〉\right)(\mathrm{since}\Im \le \mathsf{\ell })\hfill \end{array}$$

By simplifying Equation (2),

$$\begin{array}{cc}\hfill a\left(x\right)=& 2{\alpha }_{2}u{(x^2-{\alpha }_0)}^{p^s}\sum _{\kappa =0}^{\mathsf{\ell }-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-\mathsf{\ell }-1}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{2p^s+t-\mathsf{\ell }}z\left(x\right)\sum _{\kappa =0}^{\mathsf{\ell }-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }.\hfill \end{array}$$

Then, we have $a\left(x\right)\in \langle u{(x^2-{\alpha }_0)}^{\Im }\rangle$. Thus, $d_H\left(\langle u{(x^2-{\alpha }_0)}^{\Im }\rangle \right)\le d_H\left({\mathcal{C}}_3\right)$. Hence, $d_H\left({\mathcal{C}}_3\right)=d_H\left(\langle u{(x^2-{\alpha }_0)}^{\Im }\rangle \right)$. The proof follows from Theorem 6.

□

We examine the Hamming distance of Type D $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$ in the following theorem.

Theorem 8.

Let ${\mathcal{C}}_4=\langle {(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right),u{(x^2-{\alpha }_0)}^{\mu }\rangle$, where $0\le \mathsf{\ell }\le 2p^s-1$, $0\le t<\mathsf{\ell }$ and $z\left(x\right)$ is 0 or a unit. Then,

$$d_H\left({\mathcal{C}}_4\right)=\left\{\begin{array}{cc}1\hfill & if0\le \mu \le p^s;\hfill \\ (\Gamma +1)p^{\gamma }\hfill & if2p^s-p^{s-\gamma }+(\Gamma -1)\omega +1\le \mu \le 2p^s-p^{s-\gamma }+\Gamma \omega ,\hfill \end{array}\right.$$

where $\omega =p^{s-\gamma -1}$, $1\le \Gamma \le p-1$ and $0\le \gamma \le s-1$.

Proof. 

From Theorem 4, Type D $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$ are given by ${\mathcal{C}}_4=\langle {(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right),u{(x^2-{\alpha }_0)}^{\mu }\rangle$, where $0\le \mathsf{\ell }\le 2p^s-1$, $0\le t<\mathsf{\ell }$ and $z\left(x\right)$ is 0 or a unit. We have $u{(x^2-{\alpha }_0)}^{\mu }\in {\mathcal{C}}_4$. Then, $d_H\left({\mathcal{C}}_4\right)\le d_H\left(\langle u{(x^2-{\alpha }_0)}^{\mu }\rangle \right)$. Let $a\left(x\right)\in {\mathcal{C}}_4$. Then, $b\left(x\right)={\displaystyle \sum _{\kappa =0}^{2p^s-1}}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }+u{\displaystyle \sum _{\kappa =0}^{2p^s-1}}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\in R_{\alpha }$ exists, where $a_{0\kappa },b_{0\kappa },a_{1\kappa },b_{1\kappa }\in {\mathbb{F}}_{p^m}$ and $b^{\prime }\left(x\right)={\displaystyle \sum _{\kappa =0}^{2p^s-1}}(a_{0\kappa }^{\prime }x+b_{0\kappa }^{\prime }){(x^2-{\alpha }_0)}^{\kappa }+u{\displaystyle \sum _{\kappa =0}^{2p^s-1}}(a_{1\kappa }^{\prime }x+b_{1\kappa }^{\prime }){(x^2-{\alpha }_0)}^{\kappa }$, where $a_{0\kappa }^{\prime },b_{0\kappa }^{\prime },a_{1\kappa }^{\prime },b_{1\kappa }^{\prime }\in {\mathbb{F}}_{p^m}$, such that $a\left(x\right)=b\left(x\right)[{(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right)]+b^{\prime }\left(x\right)u{(x^2-{\alpha }_0)}^{\mu }$. Thus,

$$\begin{array}{cc}\hfill a\left(x\right)=& \left[\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }+u\sum _{\kappa =0}^{2p^s-1}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\right][{(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right)]\hfill \\ \hfill & +\left[\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }^{\prime }x+b_{0\kappa }^{\prime }){(x^2-{\alpha }_0)}^{\kappa }+u\sum _{\kappa =0}^{2p^s-1}(a_{1\kappa }^{\prime }x+b_{1\kappa }^{\prime }){(x^2-{\alpha }_0)}^{\kappa }\right]u{(x^2-{\alpha }_0)}^{\mu }\hfill \\ \hfill =& {(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-1}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{t}z\left(x\right)\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{\mu }\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }^{\prime }x+b_{0\kappa }^{\prime }){(x^2-{\alpha }_0)}^{\kappa }.\hfill \end{array}$$

(3)

Then,

$$\begin{array}{cc}\hfill w_H\left(a\left(x\right)\right)& \ge w_H\left(ua\left(x\right)\right)\hfill \\ \hfill & =w_H\left(u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\right)\hfill \\ \hfill & \ge d_H\left(〈u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}〉\right)\hfill \\ \hfill & \ge d_H\left(〈u{(x^2-{\alpha }_0)}^{\mu }〉\right),as\mu <\Im \le \mathsf{\ell }.\hfill \end{array}$$

By simplifying Equation (3),

$$\begin{array}{cc}\hfill a\left(x\right)=& 2{\alpha }_{2}u{(x^2-{\alpha }_0)}^{p^s}\sum _{\kappa =0}^{\mathsf{\ell }-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-\mathsf{\ell }-1}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{2p^s+t-\mathsf{\ell }}z\left(x\right)\sum _{\kappa =0}^{\mathsf{\ell }-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{\mu }\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }^{\prime }x+b_{0\kappa }^{\prime }){(x^2-{\alpha }_0)}^{\kappa }\hfill \end{array}$$

Then, we have $\mu <\Im$ and $a\left(x\right)\in \langle u{(x^2-{\alpha }_0)}^{\mu }\rangle$. Then, $d_H\left(\langle u{(x^2-{\alpha }_0)}^{\mu }\rangle \right)\le d_H\left({\mathcal{C}}_4\right)$. Hence, $d_H\left({\mathcal{C}}_4\right)=d_H\left(\langle u{(x^2-{\alpha }_0)}^{\mu }\rangle \right)$. The proof follows from Theorem 6. □

5. Symbol-Pair Distance

The symbol-pair distances of all $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$ are determined in this section, where $\alpha$ is an arbitrary unit of $\mathcal{R}$. As shown in [24], the types of $\alpha$ determine the construction of $\alpha$-constacyclic codes. Therefore, we must take into account two scenarios: $\alpha$ is a square and $\alpha$ is a non-square.

By Theorem 2 and 3, when $\alpha$ is a square in $\mathcal{R}$, the symbol-pair distance of the $\alpha$-constacyclic code of length $2p^s$ is a minimum of the symbol-pair distance of $\gamma$-constacyclic code of length $p^s$ and the symbol-pair distance of the $-\gamma$-constacyclic code of length $p^s$.

Dinh et al. [35] determined the symbol-pair distances of the $\alpha$-constacyclic code of length $2p^s$ over ${\mathbb{F}}_{p^m}$.

Theorem 9

([35]). Let $\mathcal{C}=\langle {(x^2-{\alpha }_0)}^{\mathsf{\ell }}\rangle$⊆${\displaystyle \frac{{\mathbb{F}}_{p^m}\left[x\right]}{\langle x^{2p^s}-\alpha \rangle }}$, for $0\le \mathsf{\ell }\le p^s$. Then,

$$d_{sp}\left(\mathcal{C}\right)=\left\{\begin{array}{cc}2\hfill & if\mathsf{\ell }=0;\hfill \\ 2(\Gamma +1)p^{\gamma }\hfill & if{p}^s-p^{s-\gamma }+(\Gamma -1)\omega +1\le \mathsf{\ell }\le p^s-p^{s-\gamma }+\Gamma \omega ,\hfill \\ 0\hfill & if\mathsf{\ell }=p^s,\hfill \end{array}\right.$$

where $\omega =p^{s-\gamma -1}$, $1\le \Gamma \le p-1$ and $0\le \gamma \le s-1$.

Now, we determine the symbol-pair distances of the $\alpha$-constacyclic code of length $2p^s$ over $\mathcal{R}$ when $\alpha$ is not a square in $\mathcal{R}$. The symbol-pair distance of Type A ideals $\langle 0\rangle$ and $\langle 1\rangle$ of $R_{\alpha }$ are obviously 0 and 2, respectively. Now, we calculate the symbol-pair distance of Type B, C and Type D of length $2p^s$ when $\alpha$ is not a square in $\mathcal{R}$. Note that Section 3 of Theorem 4 provides the structure of $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$ when $\alpha$ is not a square in $\mathcal{R}$.

The symbol-pair distance distance of Type B $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$ is discussed in the following theorem.

Theorem 10.

Let ${\mathcal{C}}_2=\langle u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\rangle$, where $0\le \mathsf{\ell }\le 2p^s-1$. Then,

$$d_{sp}\left({\mathcal{C}}_2\right)=\left\{\begin{array}{cc}2\hfill & if0\le \mathsf{\ell }\le p^s;\hfill \\ 2(\Gamma +1)p^{\gamma }\hfill & if2p^s-p^{s-\gamma }+(\Gamma -1)\omega +1\le \mathsf{\ell }\le 2p^s-p^{s-\gamma }+\Gamma \omega ;\hfill \end{array}\right.$$

where $\omega =p^{s-\gamma -1}$, $1\le \Gamma \le p-1$ and $0\le \gamma \le s-1$.

Proof. 

From Theorem 4, Type B $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$ are given by ${\mathcal{C}}_2=\langle u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\rangle$, where $0\le \mathsf{\ell }\le 2p^s-1$.

• Case (i): If $0\le \mathsf{\ell }\le p^s$, then $d_{sp}\left({\mathcal{C}}_2\right)=2$.
• Case (ii): If $p^s+1\le \mathsf{\ell }\le 2p^s-1$, then ${\mathcal{C}}_2=\langle u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\rangle =\langle uv{(x^2-{\alpha }_0)}^{\mathsf{\ell }-p^s}\rangle$. Thus, ${\mathcal{C}}_2$ is precisely the $\alpha$-constacyclic code $\langle {(x^2-{\alpha }_0)}^{\mathsf{\ell }-p^s}\rangle$ in ${\displaystyle \frac{{\mathbb{F}}_{p^m}\left[x\right]}{\langle x^{2p^s}-\alpha \rangle }}$ multiplied by $uv$. Therefore, $d_{sp}\left({\mathcal{C}}_2\right)=d_{sp}\left({\langle {(x^2-{\alpha }_0)}^{\kappa }\rangle }_{{\mathbb{F}}_{p^m}}\right)$ and the proof follows from Theorem 9.

□

The following result calculates the symbol-pair distance of Type C $\alpha$-constacyclic codes.

Theorem 11.

Let ${\mathcal{C}}_3=\langle {(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right)\rangle$, where $0\le \mathsf{\ell }\le 2p^s-1$, $0\le t<\mathsf{\ell }$ and $z\left(x\right)$ is 0 or a unit. Then,

$$d_{sp}\left({\mathcal{C}}_3\right)=\left\{\begin{array}{cc}2\hfill & if0\le \Im \le p^s;\hfill \\ 2(\Gamma +1)p^{\gamma }\hfill & if2p^s-p^{s-\gamma }+(\Gamma -1)\omega +1\le \Im \le 2p^s-p^{s-\gamma }+\Gamma \omega ,\hfill \end{array}\right.$$

where $\omega =p^{s-\gamma -1}$, $1\le \Gamma \le p-1$ and $0\le \gamma \le s-1$.

Proof. 

From Theorem 4, Type C $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$ are given by ${\mathcal{C}}_3=\langle {(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right)\rangle$, where $0\le \mathsf{\ell }\le 2p^s-1$, $0\le t<\mathsf{\ell }$ and $z\left(x\right)$ is 0 or a unit. Let ℑ be the smallest integer such that $u{(x^2-{\alpha }_0)}^{\Im }\in {\mathcal{C}}_3$. Then, $d_{sp}\left({\mathcal{C}}_3\right)\le d_{sp}\left(\langle u{(x^2-{\alpha }_0)}^{\Im }\rangle \right)$. Let $a\left(x\right)\in {\mathcal{C}}_3$. Then, $b\left(x\right)={\displaystyle \sum _{\kappa =0}^{2p^s-1}}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }+u{\displaystyle \sum _{\kappa =0}^{2p^s-1}}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\in R_{\alpha }$ exists, where $a_{0\kappa },b_{0\kappa },a_{1\kappa },b_{1\kappa }\in {\mathbb{F}}_{p^m}$ such that $a\left(x\right)=b\left(x\right)[{(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right)]$. Thus,

$$\begin{array}{cc}\hfill a\left(x\right)=& \left[\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }+u\sum _{\kappa =0}^{2p^s-1}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\right]\hfill \\ \hfill & \times \left[{(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right)\right]\hfill \\ \hfill =& {(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-1}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{t}z\left(x\right)\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }.\hfill \end{array}$$

(4)

Then,

$$\begin{array}{cc}\hfill w_{sp}\left(a\left(x\right)\right)& \ge w_{sp}\left(ua\left(x\right)\right)\hfill \\ \hfill & =w_{sp}\left(u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\right)\hfill \\ \hfill & \ge d_{sp}\left(〈u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}〉\right)\hfill \\ \hfill & \ge d_{sp}\left(〈u{(x^2-{\alpha }_0)}^{\Im }〉\right),\mathrm{as}\Im \le \mathsf{\ell }.\hfill \end{array}$$

By simplifying Equation (4),

$$\begin{array}{cc}\hfill a\left(x\right)=& 2{\alpha }_{2}u{(x^2-{\alpha }_0)}^{p^s}\sum _{\kappa =0}^{\mathsf{\ell }-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-\mathsf{\ell }-1}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{2p^s+t-\mathsf{\ell }}z\left(x\right)\sum _{\kappa =0}^{\mathsf{\ell }-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }.\hfill \end{array}$$

Then, we have $a\left(x\right)\in \langle u{(x^2-{\alpha }_0)}^{\Im }\rangle$. Then, we obtain $d_{sp}\left(\langle u{(x^2-{\alpha }_0)}^{\Im }\rangle \right)\le d_{sp}\left({\mathcal{C}}_3\right)$. Hence, $d_{sp}\left({\mathcal{C}}_3\right)=d_{sp}\left(\langle u{(x^2-{\alpha }_0)}^{\Im }\rangle \right)$. The proof follows from Theorem 10. □

We examine the symbol-pair distance of Type D $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$ in the following theorem.

Theorem 12.

Let ${\mathcal{C}}_4=\langle {(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right),u{(x^2-{\alpha }_0)}^{\mu }\rangle$, where $0\le \mathsf{\ell }\le 2p^s-1$, $0\le t<\mathsf{\ell }$ and $z\left(x\right)$ is 0 or a unit. Then,

$$d_{sp}\left({\mathcal{C}}_4\right)=\left\{\begin{array}{cc}2\hfill & if0\le \mu \le p^s;\hfill \\ 2(\Gamma +1)p^{\gamma }\hfill & if2p^s-p^{s-\gamma }+(\Gamma -1)\omega +1\le \mu \le 2p^s-p^{s-\gamma }+\Gamma \omega ,\hfill \end{array}\right.$$

where $\omega =p^{s-\gamma -1}$, $1\le \Gamma \le p-1$ and $0\le \gamma \le s-1$.

Proof. 

From Theorem 4, Type D $\alpha$-constacyclic codes of length $2p^s$ over $\mathcal{R}$ are given by ${\mathcal{C}}_4=\langle {(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right),u{(x^2-{\alpha }_0)}^{\mu }\rangle$, where $0\le \mathsf{\ell }\le 2p^s-1$, $0\le t<\mathsf{\ell }$ and $z\left(x\right)$ is 0 or a unit. We have $u{(x^2-{\alpha }_0)}^{\mu }\in {\mathcal{C}}_4$. Then, $d_{sp}\left({\mathcal{C}}_4\right)\le d_{sp}\left(\langle u{(x^2-{\alpha }_0)}^{\mu }\rangle \right)$. Let $a\left(x\right)\in {\mathcal{C}}_4$. Then, $b\left(x\right)={\displaystyle \sum _{\kappa =0}^{2p^s-1}}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }+u{\displaystyle \sum _{\kappa =0}^{2p^s-1}}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\in R_{\alpha }$ exists, where $a_{0\kappa },b_{0\kappa },a_{1\kappa },b_{1\kappa }\in {\mathbb{F}}_{p^m}$ and $b^{\prime }\left(x\right)={\displaystyle \sum _{\kappa =0}^{2p^s-1}}(a_{0\kappa }^{\prime }x+b_{0\kappa }^{\prime }){(x^2-{\alpha }_0)}^{\kappa }+u{\displaystyle \sum _{\kappa =0}^{2p^s-1}}(a_{1\kappa }^{\prime }x+b_{1\kappa }^{\prime }){(x^2-{\alpha }_0)}^{\kappa }$, where $a_{0\kappa }^{\prime },b_{0\kappa }^{\prime },a_{1\kappa }^{\prime },b_{1\kappa }^{\prime }\in {\mathbb{F}}_{p^m}$, such that $a\left(x\right)=b\left(x\right)[{(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right)]+b^{\prime }\left(x\right)u{(x^2-{\alpha }_0)}^{\mu }$. Thus,

$$\begin{array}{cc}\hfill a\left(x\right)=& \left[\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }+u\sum _{\kappa =0}^{2p^s-1}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\right][{(x^2-{\alpha }_0)}^{\mathsf{\ell }}+u{(x^2-{\alpha }_0)}^{t}z\left(x\right)]\hfill \\ \hfill & +\left[\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }^{\prime }x+b_{0\kappa }^{\prime }){(x^2-{\alpha }_0)}^{\kappa }+u\sum _{\kappa =0}^{2p^s-1}(a_{1\kappa }^{\prime }x+b_{1\kappa }^{\prime }){(x^2-{\alpha }_0)}^{\kappa }\right]u{(x^2-{\alpha }_0)}^{\mu }\hfill \\ \hfill =& {(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-1}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{t}z\left(x\right)\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{\mu }\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }^{\prime }x+b_{0\kappa }^{\prime }){(x^2-{\alpha }_0)}^{\kappa }.\hfill \end{array}$$

(5)

Then,

$$\begin{array}{cc}\hfill w_{sp}\left(a\left(x\right)\right)& \ge w_{sp}\left(ua\left(x\right)\right)\hfill \\ \hfill & =w_{sp}\left(u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\right)\hfill \\ \hfill & \ge d_{sp}\left(〈u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}〉\right)\hfill \\ \hfill & \ge d_{sp}\left(〈u{(x^2-{\alpha }_0)}^{\mu }〉\right),as\mu <\Im \le \mathsf{\ell }.\hfill \end{array}$$

By simplifying Equation (5),

$$\begin{array}{cc}\hfill a\left(x\right)=& 2{\alpha }_{2}u{(x^2-{\alpha }_0)}^{p^s}\sum _{\kappa =0}^{\mathsf{\ell }-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{\mathsf{\ell }}\sum _{\kappa =0}^{2p^s-\mathsf{\ell }-1}(a_{1\kappa }x+b_{1\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{2p^s+t-\mathsf{\ell }}z\left(x\right)\sum _{\kappa =0}^{\mathsf{\ell }-1}(a_{0\kappa }x+b_{0\kappa }){(x^2-{\alpha }_0)}^{\kappa }\hfill \\ \hfill & +u{(x^2-{\alpha }_0)}^{\mu }\sum _{\kappa =0}^{2p^s-1}(a_{0\kappa }^{\prime }x+b_{0\kappa }^{\prime }){(x^2-{\alpha }_0)}^{\kappa }.\hfill \end{array}$$

We have $\mu <\Im$ and by Theorem 4, $a\left(x\right)\in \langle u{(x^2-{\alpha }_0)}^{\mu }\rangle$. Then, $d_{sp}\left(\langle u{(x^2-{\alpha }_0)}^{\mu }\rangle \right)\le d_{sp}\left({\mathcal{C}}_4\right)$. Hence, $d_{sp}\left({\mathcal{C}}_4\right)=d_{sp}\left(\langle u{(x^2-{\alpha }_0)}^{\mu }\rangle \right)$. The proof follows from Theorem 10. □

Example 1.

Let $p=3$, $m=1$, $s=1$, and $\alpha =2+v+uv$. It is simple to verify that 2 in ${\mathbb{F}}_3$ is not a square. Let us calculate the Hamming and symbol pair distances of Type B $(2+v+uv)-$ constacyclic codes of length 6 over ${\mathbb{F}}_3+u{\mathbb{F}}_3+v{\mathbb{F}}_3+uv{\mathbb{F}}_3$. Let ${\mathcal{C}}_2=\langle u{(x^2-2)}^{\mathsf{\ell }}\rangle$, where $0\le \mathsf{\ell }\le 5$. For $\mathsf{\ell }=0,1,2,3$ by Theorem 6, $d_H\left({\mathcal{C}}_2\right)=1$. And by Theorem 10, $d_{sp}\left({\mathcal{C}}_2\right)=1$. Also, $1\le \Im \le 2$ and $\mu =0$. For $\Im =1$, $\mathsf{\ell }=4$, and for $\Im =2$, $\mathsf{\ell }=5$. Then, by Theorem 6, $d_H\left({\mathcal{C}}_2\right)=2$ and $d_H\left({\mathcal{C}}_2\right)=3$, respectively. And by Theorem 10, $d_{sp}\left({\mathcal{C}}_2\right)=4$ and $d_{sp}\left({\mathcal{C}}_2\right)=6$, respectively. The Hamming and symbol-pair distance of the $(2+v+uv)-$ Constacyclic code of length 6 over ${\mathbb{F}}_3+u{\mathbb{F}}_3+v{\mathbb{F}}_3+uv{\mathbb{F}}_3$ are determined in

Table 1. Hamming and symbol-pair distance of $(2+v+uv)-$ Constacyclic code of length 6 over ${\mathbb{F}}_3+u{\mathbb{F}}_3+v{\mathbb{F}}_3+uv{\mathbb{F}}_3$.

Ideal ($\mathcal{C}$)	${\mathit{d}}_{\mathit{H}}$	${\mathit{d}}_{\mathbf{sp}}$
Type A
$\langle 0\rangle$	0	0
$\langle 1\rangle$	1	2
Type B
$\langle u\rangle$	1	2
$\langle u(x^2-2)\rangle$	1	2
$\langle u{(x^2-2)}^2\rangle$	1	2
$\langle u{(x^2-2)}^3\rangle$	1	2
$\langle u{(x^2-2)}^4\rangle$	2	4
$\langle u{(x^2-2)}^5\rangle$	3	6
Type C
$\langle (x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^2\rangle$	1	2
$\langle {(x^2-2)}^3\rangle$	1	2
$\langle {(x^2-2)}^4\rangle$	2	4
$\langle {(x^2-2)}^5\rangle$	3	6
$\langle (x^2-2)+uz\left(x\right)\rangle$	1	2
$\langle {(x^2-2)}^2+uz\left(x\right)\rangle$	1	2
$\langle {(x^2-2)}^2+u(x^2-2)z\left(x\right)\rangle$	1	2
$\langle {(x^2-2)}^3+uz\left(x\right)\rangle$	1	2
$\langle {(x^2-2)}^3+u(x^2-2)z\left(x\right)\rangle$	1	2
$\langle {(x^2-2)}^3+u{(x^2-2)}^{2}z\left(x\right)\rangle$	1	2
$\langle {(x^2-2)}^4+uz\left(x\right)\rangle$	1	2
$\langle {(x^2-2)}^4+u(x^2-2)z\left(x\right)\rangle$	1	2
$\langle {(x^2-2)}^4+u{(x^2-2)}^{2}z\left(x\right)\rangle$	2	4
$\langle {(x^2-2)}^4+u{(x^2-2)}^{3}z\left(x\right)\rangle$	2	4
Type C
$\langle {(x^2-2)}^5+uz\left(x\right)\rangle$	1	2
$\langle {(x^2-2)}^5+u(x^2-2)z\left(x\right)\rangle$	1	2
$\langle {(x^2-2)}^5+u{(x^2-2)}^{2}z\left(x\right)\rangle$	1	2
$\langle {(x^2-2)}^5+u{(x^2-2)}^{3}z\left(x\right)\rangle$	2	4
$\langle {(x^2-2)}^5+u{(x^2-2)}^{4}z\left(x\right)\rangle$	3	6
Type D
$\langle (x^2-2),u\rangle$	1	2
$\langle {(x^2-2)}^2,u\rangle$	1	2
$\langle {(x^2-2)}^2,u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^3,u\rangle$	1	2
$\langle {(x^2-2)}^3,u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^3,u{(x^2-2)}^2\rangle$	1	2
$\langle {(x^2-2)}^4,u\rangle$	1	2
$\langle {(x^2-2)}^4,u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^4,u{(x^2-2)}^2\rangle$	1	2
$\langle {(x^2-2)}^4,u{(x^2-2)}^3\rangle$	1	2
$\langle {(x^2-2)}^5,u\rangle$	1	2
$\langle {(x^2-2)}^5,u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^5,u{(x^2-2)}^2\rangle$	1	2
$\langle {(x^2-2)}^5,u{(x^2-2)}^3\rangle$	1	2
$\langle {(x^2-2)}^5,u{(x^2-2)}^4\rangle$	2	4
$\langle (x^2-2)+uz\left(x\right),u\rangle$	1	2
$\langle {(x^2-2)}^2+uz\left(x\right),u\rangle$	1	2
$\langle {(x^2-2)}^2+uz\left(x\right),u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^2+u(x^2-2)z\left(x\right),u\rangle$	1	2
$\langle {(x^2-2)}^2+u(x^2-2)z\left(x\right),u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^3+uz\left(x\right),u\rangle$	1	2
$\langle {(x^2-2)}^3+uz\left(x\right),u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^3+uz\left(x\right),u{(x^2-2)}^2\rangle$	1	2
$\langle {(x^2-2)}^3+u(x^2-2)z\left(x\right),u\rangle$	1	2
$\langle {(x^2-2)}^3+u(x^2-2)z\left(x\right),u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^3+u(x^2-2)z\left(x\right),u{(x^2-2)}^2\rangle$	1	2
$\langle {(x^2-2)}^3+u{(x^2-2)}^{2}z\left(x\right),u\rangle$	1	2
$\langle {(x^2-2)}^3+u{(x^2-2)}^{2}z\left(x\right),u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^3+u{(x^2-2)}^{2}z\left(x\right),u{(x^2-2)}^2\rangle$	1	2
$\langle {(x^2-2)}^4+uz\left(x\right),u\rangle$	1	2
$\langle {(x^2-2)}^4+uz\left(x\right),u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^4+u(x^2-2)z\left(x\right),u\rangle$	1	2
$\langle {(x^2-2)}^4+u(x^2-2)z\left(x\right),u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^4+u(x^2-2)z\left(x\right),u{(x^2-2)}^2\rangle$	1	2
$\langle {(x^2-2)}^4+u{(x^2-2)}^{2}z\left(x\right),u\rangle$	1	2
$\langle {(x^2-2)}^4+u{(x^2-2)}^{2}z\left(x\right),u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^4+u{(x^2-2)}^{2}z\left(x\right),u{(x^2-2)}^2\rangle$	1	2
$\langle {(x^2-2)}^4+u{(x^2-2)}^{2}z\left(x\right),u{(x^2-2)}^3\rangle$	1	2
$\langle {(x^2-2)}^4+u{(x^2-2)}^{3}z\left(x\right),u\rangle$	1	2
$\langle {(x^2-2)}^4+u{(x^2-2)}^{3}z\left(x\right),u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^4+u{(x^2-2)}^{3}z\left(x\right),u{(x^2-2)}^2\rangle$	1	2
$\langle {(x^2-2)}^4+u{(x^2-2)}^{3}z\left(x\right),u{(x^2-2)}^3\rangle$	1	2
$\langle {(x^2-2)}^5+uz\left(x\right),u\rangle$	1	2
$\langle {(x^2-2)}^5+u(x^2-2)z\left(x\right),u\rangle$	1	2
$\langle {(x^2-2)}^5+u(x^2-2)z\left(x\right),u(x^2-2)\rangle$	1	2
Type D
$\langle {(x^2-2)}^5+u{(x^2-2)}^{2}z\left(x\right),u\rangle$	1	2
$\langle {(x^2-2)}^5+u{(x^2-2)}^{2}z\left(x\right),u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^5+u{(x^2-2)}^{2}z\left(x\right),u{(x^2-2)}^2\rangle$	1	2
$\langle {(x^2-2)}^5+u{(x^2-2)}^{3}z\left(x\right),u\rangle$	1	2
$\langle {(x^2-2)}^5+u{(x^2-2)}^{3}z\left(x\right),u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^5+u{(x^2-2)}^{3}z\left(x\right),u{(x^2-2)}^2\rangle$	1	2
$\langle {(x^2-2)}^5+u{(x^2-2)}^{3}z\left(x\right),u{(x^2-2)}^3\rangle$	1	2
$\langle {(x^2-2)}^5+u{(x^2-2)}^{4}z\left(x\right),u\rangle$	1	2
$\langle {(x^2-2)}^5+u{(x^2-2)}^{4}z\left(x\right),u(x^2-2)\rangle$	1	2
$\langle {(x^2-2)}^5+u{(x^2-2)}^{4}z\left(x\right),u{(x^2-2)}^2\rangle$	1	2
$\langle {(x^2-2)}^5+u{(x^2-2)}^{4}z\left(x\right),u{(x^2-2)}^3\rangle$	1	2
$\langle {(x^2-2)}^5+u{(x^2-2)}^{4}z\left(x\right),u{(x^2-2)}^4\rangle$	2	4

.

6. Conclusions

In the case of error-correcting codes, distances of constacyclic codes are crucial. Determining the Hamming and symbol-pair distances of constacyclic codes is an interesting field of study. Let p be an odd prime and m and s be positive integers, and let $\mathcal{R}={\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2,v^2,uv-vu\rangle }}$, a finite commutative non-chain ring with identity. In this article, we computed the Hamming and symbol-pair distances of $\alpha$-constacyclic codes $\mathcal{C}$ of length $2p^s$ over $\mathcal{R}$. By examining the results obtained, we can conclude that for constacyclic codes of length $2p^s$ over $\mathcal{R}$, we have $d_{sp}\left(\mathcal{C}\right)=2d_H\left(\mathcal{C}\right)$. The objective in designing codes for the symbol-pair read channel is to attain a high minimum symbol-pair distance with respect to the minimum Hamming distance. From Theorem 1 and since $d_{sp}\left(\mathcal{C}\right)=2d_H\left(\mathcal{C}\right)$, these codes have the largest possible minimum symbol-pair distance with respect to their minimum Hamming distance. Also, we provided an example of constacyclic codes of length 6 over ${\mathbb{F}}_3+u{\mathbb{F}}_3+v{\mathbb{F}}_3+uv{\mathbb{F}}_3$ along with the Hamming and symbol-pair distances.