Symbol-Pair Distances of a Class of Repeated-Root Constacyclic Codes of Length np^s over ${\mathbb{F}}_{p^m}$ and over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$

Abstract

Symbol-pair codes are a class of block codes with symbol-pair metrics designed to protect against pair errors that may occur in high-density data storage systems. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense that they can attain the highest pair-error correctability within the same code length and code size. Constructing MDS symbol-pair codes is one of the main topics in symbol-pair code research. In this paper, we investigate and characterize the symbol-pair distances of constacyclic codes of arbitrary lengths over finite fields and finite chain rings. Using the characterization of the symbol-pair distance, we present three new classes of MDS symbol-pair constacyclic codes that exhibit large minimum distances.

1. Introduction

Modern high-density data storage systems may fail to read the transmitted information individually as classical information transmission due to physical limitations. Motivated by this fact, Cassuto and Blaum [1] developed symbol-pair codes for a symbol-pair read channel whose outputs are overlapping pairs of symbols. Efficient decoding algorithms for cyclic codes over symbol-pair read channels were demonstrated in [2,3,4].

Let $\Xi$ be an alphabet of a size q with $q\ge 2$. The code $\mathcal{C}$ over $\Xi$ of a length n is a subset of ${\Xi }^n$. The elements in $\mathcal{C}$ are called codewords and will be referred to as vectors, denoted by bold letters. Let $\mathbf{x}=\left(x_0,x_1,\dots ,x_{n-1}\right)$, $\mathbf{y}=\left(y_0,y_1,\dots ,y_{n-1}\right)$ be vectors in ${\Xi }^n$. A vector $\mathbf{x}$ transmitted in the symbol-pair read channel reads

$$\pi \left(\mathbf{x}\right)=\left(\left(x_0,x_1\right),\left(x_1,x_2\right),\dots ,\left(x_{n-1},x_0\right)\right).$$

We refer to $\pi \left(\mathbf{x}\right)$ as the symbol-pair vector of $\mathbf{x}$. The symbol-pair distance between $\mathbf{x},\mathbf{y}\in \mathcal{C}$ is defined as the Hamming distance between $\pi \left(\mathbf{x}\right)$ and $\pi \left(\mathbf{y}\right)$; in other words, we have

$${\mathrm{d}}_{\mathrm{sp}}(\mathbf{x},\mathbf{y})={\mathrm{d}}_{\mathrm{H}}(\pi \left(\mathbf{x}\right),\pi \left(\mathbf{y}\right))=\left|\left\{i:\left(x_i,x_{i+1}\right)\ne \left(y_i,y_{i+1}\right)\right\}\right|.$$

The (minimum) symbol-pair distance of $\mathcal{C}$ is defined as follows:

$${\mathrm{d}}_{\mathrm{sp}}\left(\mathcal{C}\right)=min\left\{{\mathrm{d}}_{\mathrm{sp}}\right(\mathbf{x},\mathbf{y})\mid \mathbf{x},\mathbf{y}\in \mathcal{C}\mathrm{and}\mathbf{x}\ne \mathbf{y}\}.$$

For a code $\mathcal{C}\subset {\Xi }^n$ with a symbol-pair distance ${\mathrm{d}}_{\mathrm{sp}}={\mathrm{d}}_{\mathrm{sp}}\left(\mathcal{C}\right)$, the upper bound with a size of $\mathcal{C}$, known as the Singleton bound for symbol-pair codes [5], is

$$\left|\mathcal{C}\right|\le q^{n-{\mathrm{d}}_{\mathrm{sp}}+2}.$$

(1)

A symbol-pair code whose parameters satisfy Equation (1) with equality is called maximum distance separable (MDS). As indicated by Equation (1), MDS symbol-pair codes achieve the maximum possible symbol-pair distance for a given code length and size. This property establishes MDS symbol-pair codes as a class of optimal symbol-pair codes with superior pair error-correcting capability, since the symbol-pair distance directly determines the code’s ability to correct pair errors.

Constructing MDS symbol-pair codes is meaningful both in theory and practice. Research on the construction of MDS symbol-pair codes has been actively pursued in recent years [6,7,8,9,10,11]. Many MDS symbol-pair codes are obtained by analyzing the generator polynomials of constacyclic codes, as demonstrated in [6,10,11,12]. In [13], Li constructed two classes of MDS symbol-pair codes with larger symbol-pair distances of 12 and 16. Subsequently, Dinh et al. [14,15] constructed some almost MDS symbol-pair codes with larger symbol-pair distances of 7, 8, and 9.

The class of constacyclic codes has practical applications, as constacyclic codes possess a cyclic structure that supports efficient encoding and decoding algorithms using shift registers. An important research topic is to determine the symbol-pair distances of constacyclic codes and identify all MDS symbol-pair constacyclic codes. In [8], Dinh et al. characterized the symbol-pair distances of all constacyclic codes of a length $p^s$ over ${\mathbb{F}}_{p^m}$ and obtain all MDS symbol-pair codes of prime power lengths. These results have been extended to other specific lengths, such as $2p^s$ and $4p^s$ ([9,16,17]). However, it remains challenging to investigate the symbol-pair distances of repeated-root constacyclic codes for general lengths $n{p}^s$, where n is an arbitrary positive integer which is coprime to p.

Codes over finite rings have attracted significant attention due to the fact that many important yet seemingly nonlinear codes over finite fields are actually closely related to linear codes over finite rings. The class of finite rings of the form ${\mathbb{F}}_{p^m}\left[u\right]/\langle u^2\rangle ={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ and even the general form ${\mathbb{F}}_{p^m}\left[u\right]/\langle u^a\rangle ={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+\cdots +u^{a-1}{\mathbb{F}}_{p^m}$ have been used as alphabets for constacyclic codes. The research on symbol-pair distances of repeated-root constacyclic codes over the finite chain rings has attracted well-deserved attention. For example, progress has been achieved in studying constacyclic codes of a length $p^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ [7,18], ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+u^2{\mathbb{F}}_{p^m}$ [19,20], ${\mathbb{F}}_{p^m}\left[u\right]/\langle u^4\rangle$ [21], finite commutative chain rings [22], and the constacyclic codes of a length $2p^s$ [23], $3p^s$ [24] over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$. The characterization of symbol-pair distances of repeated-root constacyclic codes over finite rings has been resolved for special lengths. However, there is a scarcity of results regarding general lengths. In this paper, we aim to investigate the general lengths of a particular class of repeated-root constacyclic codes.

In this paper, we consider certain constacyclic codes of a length $n{p}^s$ over finite fields ${\mathbb{F}}_{p^m}$ and finite chain rings ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$, where p is a prime variable, m is a positive integer, and $u^2=0$. Let ${\alpha }_0$ be a nonzero element in ${\mathbb{F}}_{p^m}$ such that $x^n-{\alpha }_0$ is irreducible over ${\mathbb{F}}_{p^m}$. Let $\alpha ={\alpha }_0^{p^s}$ and $\beta$ be an element in ${\mathbb{F}}_{p^m}$. We provide a complete characterization of the symbol-pair distances of $\alpha$-constacyclic codes over ${\mathbb{F}}_{p^m}$ and $(\alpha +u\beta )$-constacyclic codes over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$. Furthermore, all MDS symbol-pair $\alpha$-constacyclic codes over ${\mathbb{F}}_{p^m}$ and all the MDS symbol-pair $(\alpha +u\beta )$-constacyclic codes over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ are determined. Additionally, three of these classes of MDS symbol-pair constacyclic codes exhibit new parameters with large minimum distances.

The remainder of this paper is organized as follows. In Section 2, we introduce some preliminaries and notations. In Section 3, we characterize the symbol-pair distances of all $\alpha$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}$ and identify all the MDS symbol-pair $\alpha$-constacyclic codes of a length $n{p}^s$ among these codes. In Section 4, we determine the symbol-pair distances of certain $(\alpha +u\beta )$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ and present all the MDS symbol-pair codes among these codes.

2. Preliminaries

In this section, we introduce the necessary notations and preliminary results that will be utilized in subsequent sections. Let R be a finite commutative ring with identity. A code $\mathcal{C}$ over R is called linear if $\mathcal{C}$ is a submodule of $R^n$. The symbol-pair weight of a vector $\mathbf{x}$ in $R^n$ is the symbol-pair distance between $\mathbf{x}$ and the all-zero vector $\mathbf{0}$ of $R^n$, denoted by ${\mathrm{wt}}_{\mathrm{sp}}\left(\mathbf{x}\right)$. The symbol-pair distance of a linear code is determined by the minimum symbol-pair weight among its nonzero codewords. For a unit $\lambda$ in R, the $\lambda$-constacyclic shift ${\tau }_{\lambda }$ on $R^n$ is defined as follows:

$${\tau }_{\lambda }\left(x_0,x_1,\dots ,x_{n-1}\right)=\left(\lambda x_{n-1},x_0,x_1,\dots ,x_{n-2}\right).$$

A linear code $\mathcal{C}$ is said to be $\lambda$-constacyclic if ${\tau }_{\lambda }\left(\mathcal{C}\right)=\mathcal{C}$. Each codeword $\mathbf{c}=\left(c_0,c_1,\dots ,c_{n-1}\right)$ in $\mathcal{C}$ is customarily identified with its polynomial representation $c\left(x\right)=c_0+c_{1}x+\cdots +c_{n-1}{x}^{n-1}$ in $R\left[x\right]/〈x^n-\lambda 〉$. In the quotient ring $R\left[x\right]/〈x^n-\lambda 〉$, the polynomial $xc\left(x\right)$ corresponds to performing a $\lambda$-constacyclic shift on the codeword $\mathbf{c}$. The subsequent theorem elucidates the algebraic properties of constacyclic codes.

Proposition 1

([25]). A linear code $\mathcal{C}$ of a length n over R is a λ-constacyclic code if and only if $\mathcal{C}$ is an ideal of the quotient ring $R\left[x\right]/〈x^n-\lambda 〉$.

In this paper, we focus on the constacyclic codes of a length $n{p}^s$, which are ideals in the quotient ring $R\left[x\right]/\langle x^{n{p}^s}-\lambda \rangle$. Assume that R is a Frobenius ring. Then, for any unit $\lambda \in R$, there exists a unit ${\lambda }_0$ such that ${\lambda }_0^{p^s}=\lambda$. Consequently, we have the factorization $x^{n{p}^s}-\lambda ={(x^n-{\lambda }_0)}^{p^s}$. Throughout this paper, we assume that the polynomial $x^n-{\lambda }_0$ is irreducible over R. The following proposition establishes the irreducibility criterion for binomials over finite fields.

Proposition 2

([26]). Let $n\ge 2$ be an integer and $\lambda \in {\mathbb{F}}_q^{*}$. Then, the binomial $x^n-\lambda$ is irreducible in ${\mathbb{F}}_q\left[x\right]$ if and only if the following two conditions are satisfied:

(1) 

Each prime factor of n divides the order e of λ in ${\mathbb{F}}_q^{*},$ but not ${\displaystyle \frac{q-1}{e}}$;

(2) 

$q\equiv 1(mod4)$ if $n\equiv 0(mod4)$.

According to Proposition 2, if n satisfies condition (1) of Proposition 2, and all prime factors of n divide $q-1$, then there exists $\lambda \in {\mathbb{F}}_q^{*}$ such that $x^n-\lambda$ is irreducible over ${\mathbb{F}}_q$.

2.1. Constacyclic Codes over ${\mathbb{F}}_{p^m}$

Let $\alpha$ be a nonzero element in ${\mathbb{F}}_{p^m}$. In this subsection, we present results for the properties of $\alpha$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}$. We denote with $\mathcal{F}$ the quotient ring ${\mathbb{F}}_{p^m}\left[x\right]/\langle x^{n{p}^s}-\alpha \rangle$. The structures and minimum (Hamming) distances of these codes are detailed in the following theorem.

Theorem 1

(Theorem 3.6 from [27]). Let ${\mathbb{F}}_{p^m}$ be a finite field and n be a positive integer with $gcd(n,p)=1$. Suppose that $x^n-{\alpha }_0$ is irreducible over ${\mathbb{F}}_{p^m}$ for ${\alpha }_0\in {\mathbb{F}}_{p^m}^{*}$ and $\alpha ={{\alpha }_0}^{p^s}$. Then, the α-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}$ are of the form ${\mathcal{C}}_i=\langle {\left(x^n-{\alpha }_0\right)}^i\rangle$, where $0\le i\le p^s$, and the minimum Hamming distance of ${\mathcal{C}}_i$ is given by

$$d_H\left({\mathcal{C}}_i\right)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}i=0,\hfill \\ (\theta +2)p^k,\hfill & \mathit{if}{p}^s-p^{s-k}+\theta p^{s-k-1}+1\le i\le p^s-p^{s-k}+(\theta +1)p^{s-k-1},\hfill \\ & \mathit{where}0\le \theta \le p-2\mathit{and}0\le k\le s-1,\hfill \\ 0\hfill & \mathit{if}i=p^s.\hfill \end{array}\right.$$

For simplicity, we denote the $\alpha$-constacyclic codes of a length $n{p}^s$ with the generator polynomial ${(x^n-{\alpha }_0)}^i$ as ${\mathcal{C}}_i$, where $0\le i\le p^s$. The following lemma provides a formula to compute the Hamming weight of the codeword ${(x^n-{\alpha }_0)}^i$ in ${\mathcal{C}}_i$.

Lemma 1

(Lemma 1 from [28]). For any nonnegative integer $i<p^s,$ let $i=i_{s-1}{p}^{s-1}+\cdots +i_{1}p+i_0$, where $0\le i_0,i_1,\dots ,i_{s-1}\le p-1$, which means that $(i_{s-1},\dots ,i_0)$ is the p-adic expansion of i. Then, it follows that

$${\mathrm{wt}}_{\mathrm{H}}\left({(x^n-{\alpha }_0)}^i\right)=\prod _{j=0}^{s-1}(i_j+1).$$

The subsequent lemma establishes the relationship between the symbol-pair distance and the Hamming distance.

Lemma 2

(Theorem 2 from [1]). For two codewords $\mathbf{x},\mathbf{y}$ in a code $\mathcal{C}$ of a length n with $0<{\mathrm{d}}_{\mathrm{H}}(\mathbf{x},\mathbf{y})<n$, define the set $S_H=\{j\mid x_j\ne y_j\}$. Let $S_H={\cup }_{l=1}^L{B}_l$ be a minimal partition of the set $S_H$ to subsets of consecutive indices (indices may wrap around modulo n). Then, it follows that

$${\mathrm{d}}_{\mathrm{sp}}(\mathbf{x},\mathbf{y})={\mathrm{d}}_{\mathrm{H}}(\mathbf{x},\mathbf{y})+L.$$

To calculate the symbol-pair distances, we will employ the concept of the coefficient weight of polynomials initially introduced by Dinh et al. in [29]. For a polynomial $f\left(x\right)=a_n{x}^n+\cdots +a_{1}x+a_0$ of a degree n, the coefficient weight of f, denoted by $cw\left(f\right)$, is

$$cw\left(f\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}f\mathrm{is}\mathrm{a}\mathrm{monomial}\hfill \\ min\left\{|i-j|:a_i\ne 0,a_j\ne 0,i\ne j\right\},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Intuitively, $cw\left(f\right)$ is the smallest distance among exponents of nonzero terms of $f\left(x\right)$. According to [8], if the conditions $0\le deg\left(g\right(x\left)\right)\le cw\left(f\right(x\left)\right)-2$ and $deg\left(f\right(x\left)\right)+deg\left(g\right(x\left)\right)\le n-2$ are satisfied, then the following equation holds:

$${\mathrm{wt}}_{\mathrm{sp}}\left(f\right(x\left)g\right(x\left)\right)={\mathrm{wt}}_{\mathrm{H}}\left(f\right(x\left)\right)·{\mathrm{wt}}_{\mathrm{sp}}\left(g\right(x\left)\right).$$

(2)

2.2. Constacyclic Codes over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$

Let $\alpha +u\beta$ be a unit in ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$. This subsection recalls the structures of $(\alpha +u\beta )$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$. We denote with $\mathcal{R}$ the quotient ring $({\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m})\left[x\right]/〈x^{n{p}^s}-\alpha -u\beta 〉$. Note that the structures of the ideals of $\mathcal{R}$ differ significantly depending on whether $\beta$ is zero or nonzero. The following two lemmas describe the ideals of $\mathcal{R}$ in the cases where $\beta \ne 0$ and $\beta =0$.

Lemma 3

(Theorem 3.3 of [30]). Let $x^n-{\alpha }_0$ be an irreducible polynomial in ${\mathbb{F}}_{p^m}\left[x\right]$, $\alpha ={\alpha }_0^{p^s}$, and β be a nonzero element in ${\mathbb{F}}_{p^m}$. Then, the ring $\mathcal{R}=({\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m})\left[x\right]/〈x^{n{p}^s}-\alpha -u\beta 〉$ is a chain ring whose ideal chain is as follows:

$$\mathcal{R}=\langle 1\rangle ⊋\langle x^n-{\alpha }_0\rangle ⊋\cdots ⊋\langle {\left(x^n-{\alpha }_0\right)}^{2p^s-1}\rangle ⊋\langle {\left(x^n-{\alpha }_0\right)}^{2p^s}\rangle =\langle 0\rangle .$$

In other words, $(\alpha +u\beta )$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ are precisely the ideals ${\mathcal{D}}_i=\langle {\left(x^n-{\alpha }_0\right)}^i\rangle$ of $\mathcal{R}$, where $0\le i\le 2p^s$. The number of codewords of $(\alpha +u\beta )$-constacyclic code ${\mathcal{D}}_i$ is $p^{mn(2p^s-i)}$. In particular, $\langle {\left(x^n-{\alpha }_0\right)}^{p^s}\rangle =\langle u\rangle$.

Lemma 4

(Corollary 3.10 of [31]). Let $x^n-{\alpha }_0$ be an irreducible polynomial in ${\mathbb{F}}_{p^m}\left[x\right]$, $\alpha ={\alpha }_0^{p^s}$, and $\mathcal{F}={\mathbb{F}}_{p^m}\left[x\right]/\langle x^{n{p}^s}-\alpha \rangle$. Then, all α-constacyclic codes over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ of a length $n{p}^s$ ($i.e.,$ all ideals of the ring $\mathcal{F}+u\mathcal{F}$) are given by the following three types:

(1) $\mathcal{D}=\langle {(x^n-{\alpha }_0)}^k\rangle$, where $0\le k\le p^s$, with $\left|\mathcal{D}\right|=p^{2mn(p^s-k)}$.

(2) $\mathcal{D}=\langle {\left(x^n-{\alpha }_0\right)}^{j}b\left(x\right)+u{\left(x^n-{\alpha }_0\right)}^k\rangle$, where $0\le k\le p^s-1$, $⌈{\displaystyle \frac{p^s+k}{2}}⌉\le j\le p^s-1$, and either $b\left(x\right)$ is zero or $b\left(x\right)$ is a unit in $\mathcal{F}$, with $\left|\mathcal{D}\right|=p^{mn(p^s-k)}$.

(3) $\mathcal{D}=\langle {\left(x^n-{\alpha }_0\right)}^{j}b\left(x\right)+u{\left(x^n-{\alpha }_0\right)}^k,{\left(x^n-{\alpha }_0\right)}^{k+t}\rangle$, where $0\le k\le p^s-2$, $1\le t\le p^s-k-1$, $k+⌈{\displaystyle \frac{t}{2}}⌉\le j\le k+t$, and either $b\left(x\right)$ is zero or $b\left(x\right)$ is a unit in $\mathcal{F}$, with $\left|\mathcal{D}\right|=p^{mn(2p^s-2k-t)}$.

Note that ${\mathbb{F}}_{p^m}$ is a subfield of ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$. The subfield subcode of codes over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ is defined as the set of codewords whose components belong to ${\mathbb{F}}_{p^m}$. We denote the subfield subcode of $\mathcal{D}$ by ${\mathcal{D}}_F$ and its symbol-pair distance by ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{D}}_F\right)$. A polynomial $c\left(x\right)$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ can be expressed as $c\left(x\right)=a\left(x\right)+ub\left(x\right)$, where $a\left(x\right),b\left(x\right)\in {\mathbb{F}}_{p^m}\left[x\right]$. It is observed that $c_i=a_i+u{b}_i=0$ if and only if $a_i=b_i=0$, with $c_i$, $a_i$, and $b_i$ being the coefficients of $x^i$ in polynomials $c\left(x\right)$, $a\left(x\right)$ and $b\left(x\right)$, respectively. Consequently, it follows that ${\mathrm{wt}}_{\mathrm{sp}}\left(c\left(x\right)\right)\ge max\left\{{\mathrm{wt}}_{\mathrm{sp}}\left(a\right(x\left)\right),{\mathrm{wt}}_{\mathrm{sp}}\left(b\right(x\left)\right)\right\}$.

3. Symbol-Pair Distances of Repeated-Root Constacyclic Codes over ${\mathbb{F}}_{p^m}$

We denote ${\mathcal{C}}_i=\langle {(x^n-\alpha )}^i\rangle$ as the $\alpha$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}$, where $0\le i\le p^s$. In this section, we provide a comprehensive characterization of the symbol-pair distances for the $\alpha$-constacyclic codes ${\mathcal{C}}_i$. Specifically, we focus on the case where $n\ge 2$. The analysis for the scenario where $n=1$ is distinct and can be found in [8].

We examine the symbol-pair distance of ${\mathcal{C}}_i$ for varying values of i. For the trivial cases where $i=0$ and $i=p^s$, it is observed that

$${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_0\right)={\mathrm{d}}_{\mathrm{sp}}\left(\mathcal{F}\right)=2$$

and

$${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_{p^s}\right)={\mathrm{d}}_{\mathrm{sp}}\left(\langle 0\rangle \right)=0.$$

In order to analyze the symbol-pair distances of ${\mathcal{C}}_i$ for $1\le i\le p^s-1$, we partition the set $\{i\in \mathbb{N},1\le i\le p^s-1\}$ into $s(p-1)$ parts such that

$$\begin{array}{cc}& {\cup }_{\begin{array}{c}0\le k\le s-1\\ 0\le \theta \le p-2\end{array}}\{i\in \mathbb{N},p^s-p^{s-k}+\theta p^{s-k-1}+1\le i\le p^s-p^{s-k}+(\theta +1)p^{s-k-1}\}\hfill \\ & =\{i\in \mathbb{N},1\le i\le p^s-1\}.\hfill \end{array}$$

If $i\le j$, then ${\mathcal{C}}_i\supseteq {\mathcal{C}}_j$, which implies that ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_i\right)\le {\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_j\right)$. To determine the symbol-pair distances of ${\mathcal{C}}_i$ for $p^s-p^{s-k}+\theta p^{s-k-1}+1\le i\le p^s-p^{s-k}+(\theta +1)p^{s-k-1}$, where $0\le k\le s-1$ and $0\le \theta \le p-2$, we establish an upper bound U on the symbol-pair distance of ${\mathcal{C}}_{p^s-p^{s-k}+(\theta +1)p^{s-k-1}}$ and a lower bound L on the symbol-pair distance of ${\mathcal{C}}_{p^s-p^{s-k}+\theta p^{s-k-1}+1}$. Given that ${\mathcal{C}}_{p^s-p^{s-k}+\theta p^{s-k-1}+1}\supseteq {\mathcal{C}}_{p^s-p^{s-k}+(\theta +1)p^{s-k-1}}$, it follows that

$$L\le {\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_{p^s-p^{s-k}+\theta p^{s-k-1}+1}\right)\le {\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_{p^s-p^{s-k}+(\theta +1)p^{s-k-1}}\right)\le U.$$

If $L=U$, then the symbol-pair distances of ${\mathcal{C}}_i$ are determined for all i in the interval $[p^s-p^{s-k}+\theta p^{s-k-1}+1,p^s-p^{s-k}+(\theta +1)p^{s-k-1}]$. The following lemma shows the corresponding upper bound.

Lemma 5.

Let $n,k,\theta$ be integers such that $n\ge 2$, $0\le k\le s-1$, and $0\le \theta \le p-2$. Then, ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_{p^s-p^{s-k}+(\theta +1)p^{s-k-1}}\right)\le 2(\theta +2)p^k$.

Proof. 

With Lemma 1, we have

$${\mathrm{wt}}_{\mathrm{H}}\left({\left(x^n-{\alpha }_0\right)}^{p^s-p^{s-k}+(\theta +1)p^{s-k-1}}\right)=(\theta +2)p^k.$$

Since

$$cw\left({\left(x^n-{\alpha }_0\right)}^{p^s-p^{s-k}+(\theta +1)p^{s-k-1}}\right)\ge n\ge 2,$$

it follows that

$${\mathrm{wt}}_{\mathrm{sp}}\left({\left(x^n-{\alpha }_0\right)}^{p^s-p^{s-k}+(\theta +1)p^{s-k-1}}\right)=2(\theta +2)p^k.$$

Thus

$${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_{p^s-p^{s-k}+(\theta +1)p^{s-k-1}}\right)\le 2(\theta +2)p^k.$$

□

The lower bounds of the symbol-pair distances of ${\mathcal{C}}_{p^s-p^{s-k}+\theta p^{s-k-1}+1}$ were more complicated, and we analyzed them in four subcases:

(1)

$\theta =0$, $k=0$;

(2)

$\theta =0$, $1\le k\le s-2$;

(3)

$1\le \theta \le p-2$, $0\le k\le s-2$;

(4)

$0\le \theta \le p-2$, $k=s-1$.

We started with the first case of $k=0$ and $\theta =0$.

Lemma 6.

The pair distance ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_1\right)$ of ${\mathcal{C}}_1=\langle x^n-{\lambda }_0\rangle \subseteq {\mathbb{F}}_{p^m}\left[x\right]/\langle x^{n{p}^s}-\lambda \rangle$ is greater than or equal to four.

Proof. 

We verify that a codeword with a symbol-pair weight of two must be of the form $u{x}^j$, which is invertible in $\mathcal{F}$. Hence, there is no codeword in ${\mathcal{C}}_1$ with a symbol-pair weight of two. Note that a codeword with a symbol-pair weight of three has the form $u_0{x}^j+u_1{x}^{j+1}$, where $0\le j\le n{p}^s-1$. It follows that $x^n-{\alpha }_0$ divides $u_0{x}^j+u_1{x}^{j+1}=(u_0+u_{1}x)x^j$, and hence $x^n-{\alpha }_0$ divides $u_0+u_{1}x$, which is impossible since the degree of $x^n-{\alpha }_0$ is greater than that of $u_0+u_{1}x$. Hence there is no codeword in ${\mathcal{C}}_1$ with a symbol-pair weight of three. Therefore, we obtain ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_1\right)\ge 4$. □

The following lemma provides the lower bound of the minimum symbol-pair distance of ${\mathcal{C}}_{p^s-p^{s-k}+1}$ in the case where $\theta =0$ and $1\le k\le s-2$.

Lemma 7.

Let $n,k$ be integers such that $n\ge 2$ and $1\le k\le s-2$. Then, ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_{p^s-p^{s-k}+1}\right)\ge 4p^k$.

Proof. 

Let $c\left(x\right)$ be any nonzero codeword in ${\mathcal{C}}_{p^s-p^{s-k}+1}$. Then, there is a nonzero element $f\left(x\right)$ in $\mathcal{F}$ such that $c\left(x\right)={\left(x^n-{\alpha }_0\right)}^{p^s-p^{s-k}+1}f\left(x\right)$ with $deg\left(f\right)<n{p}^s-n(p^s-p^{s-k}+1)=n(p^{s-k}-1)$. Let $g\left(x\right)=\left(x^n-{\alpha }_0\right)f\left(x\right)$. Then, $deg\left(g\right)<n{p}^{s-k}$, ${\mathrm{wt}}_{\mathrm{H}}\left(g\right(x\left)\right)\ge 2$, and

$$\begin{array}{cc}\hfill c\left(x\right)& ={\left(x^n-{\alpha }_0\right)}^{p^s-p^{s-k}}g\left(x\right)\hfill \\ & =\left[\sum _{j=0}^{p^k-1}\left({\displaystyle \genfrac{}{}{0pt}{}{p^k-1}{j}}\right){\left(-{\alpha }_0\right)}^{p^{s-k}\left(p^k-j-1\right)}{x}^{n{p}^{s-k}j}\right]g\left(x\right).\hfill \end{array}$$

We discuss the symbol-pair weight of $c\left(x\right)$ in the following three cases.

Case 1: If $deg\left(g\right)\le n{p}^{s-k}-2$, then

$$cw\left({\left(x^n-{\alpha }_0\right)}^{p^s-p^{s-k}}\right)=n{p}^{s-k}\ge deg\left(g\right)+2$$

and

$$deg\left({\left(x^n-{\alpha }_0\right)}^{p^s-p^{s-k}}\right)+deg\left(g\right)\le n{p}^s-2.$$

Under Equation (2), we have

$$\begin{array}{cc}\hfill {\mathrm{wt}}_{\mathrm{sp}}\left(c\left(x\right)\right)& ={\mathrm{wt}}_{\mathrm{H}}\left({\left(x^n-{\alpha }_0\right)}^{p^s-p^{s-k}}\right)·{\mathrm{wt}}_{\mathrm{sp}}\left(g\left(x\right)\right)\hfill \\ & =p^k{\mathrm{wt}}_{\mathrm{sp}}\left(g\left(x\right)\right).\hfill \end{array}$$

According to Lemma 6, ${\mathrm{wt}}_{\mathrm{sp}}\left(g\left(x\right)\right)\ge {\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_1\right)\ge 4$, which deduces that ${\mathrm{wt}}_{\mathrm{sp}}\left(c\left(x\right)\right)\ge 4p^k$.

Case 2: If $deg\left(g\right)=n{p}^{s-k}-1$ and $g\left(0\right)=0$, then there is an integer $l>0$ such that $g\left(x\right)=x^l{g}^{\prime }\left(x\right)$, where $deg\left(g^{\prime }\right)\le n{p}^{s-k}-2$. Clearly, this means that

$$\begin{array}{cc}\hfill {\mathrm{wt}}_{\mathrm{sp}}\left(c\right(x\left)\right)& ={\mathrm{wt}}_{\mathrm{sp}}\left({\left(x^n-{\alpha }_0\right)}^{p^s-p^{s-k}}g\left(x\right)\right)\hfill \\ & ={\mathrm{wt}}_{\mathrm{sp}}\left({\left(x^n-{\alpha }_0\right)}^{p^s-p^{s-k}}{x}^l{g}^{\prime }\left(x\right)\right)\hfill \\ & ={\mathrm{wt}}_{\mathrm{sp}}\left({\left(x^n-{\alpha }_0\right)}^{p^s-p^{s-k}}{g}^{\prime }\left(x\right)\right).\hfill \end{array}$$

Similar to the proof in Case 1, we have ${\mathrm{wt}}_{\mathrm{sp}}\left(c\left(x\right)\right)\ge 4p^k$.

Case 3: If $deg\left(g\right)=n{p}^{s-k}-1$ and $g\left(0\right)\ne 0$, then $g\left(x\right)=\left(x^n-{\alpha }_0\right)f\left(x\right)$ is an element in $\langle x^n-{\alpha }_0\rangle$ of the ring ${\mathbb{F}}_{p^m}\left[x\right]/\langle x^{n{p}^{s-k}}-{{\alpha }_0}^{n{p}^{s-k}}\rangle$ (i.e., a codeword of an ${\alpha }_0^{n{p}^{s-k}}$-constacyclic code of a length $n{p}^{s-k}$ over ${\mathbb{F}}_{p^m}$). According to Lemma 6, ${\mathrm{wt}}_{\mathrm{sp}}\left(g\right(x\left)\right)\ge 4$, which implies that $g\left(x\right)$ cannot be in the form $r_0+r_1{x}^{n{p}^{s-k}-1}$, where $r_0,r_1\ne 0$. Hence, ${\mathrm{wt}}_{\mathrm{H}}\left(g\right(x\left)\right)\ge 3$. When ${\mathrm{wt}}_{\mathrm{H}}\left(g\right(x\left)\right)\ge 4$, we have

$$\begin{array}{cc}\hfill {\mathrm{wt}}_{\mathrm{sp}}\left(c\right(x\left)\right)& \ge {\mathrm{wt}}_{\mathrm{H}}\left(c\right(x\left)\right)\hfill \\ & ={\mathrm{wt}}_{\mathrm{H}}\left({\left(x^n-{\alpha }_0\right)}^{p^s-p^{s-k}}\right)·{\mathrm{wt}}_{\mathrm{H}}\left(g\left(x\right)\right)\hfill \\ & \ge 4p^k.\hfill \end{array}$$

When ${\mathrm{wt}}_{\mathrm{H}}\left(g\right(x\left)\right)=3$, we assume that

$$g\left(x\right)=r_0+r_1{x}^l+r_2{x}^{n{p}^{s-k}-1},$$

where $0<l<n{p}^{s-k}-1$ and $r_0,r_1,r_2\ne 0$. Let $S_H$ be a set of the exponents of nonzero terms of $c\left(x\right)$. Then, the minimal partition of $S_H$ into subsets of consecutive indices may be the following three cases.

If $l=1$, then

$$S_H={\cup }_{1\le j\le p^k-1}\{n{p}^{s-k}j-1,n{p}^{s-k}j,n{p}^{s-k}j+1\}\cup \{0,1,n{p}^s-1\};$$

If $l=n{p}^{s-k}-2$, then

$$S_H={\cup }_{1\le j\le p^k-1}\{n{p}^{s-k}j-2,n{p}^{s-k}j-1,n{p}^{s-k}j\}\cup \{0,n{p}^s-1,n{p}^s-2\};$$

If $1<l<n{p}^{s-k}-2$, then

$$S_H={\cup }_{1\le j\le p^k-1}(\{n{p}^{s-k}j-1,n{p}^{s-k}j\}\cup \{n{p}^{s-k}j+l\})\cup \{0,n{p}^s-1\}\cup \left\{l\right\}.$$

Based on the above three cases, we conclude that ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_{p^s-p^{s-k}+1}\right)\ge 4p^k$. This completes the proof. □

The following lemma considers the case where $0\le k\le s-2$ and $1\le \theta \le p-2$.

Lemma 8.

Let $n,k,\theta$ be integers such that $n\ge 2$, $0\le k\le s-2$, and $1\le \theta \le p-2$. Then, ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_{p^s-p^{s-k}+\theta p^{s-k-1}+1}\right)\ge 2(\theta +2)p^k$.

Proof. 

Let $c\left(x\right)$ be any nonzero codeword in ${\mathcal{C}}_{p^s-p^{s-k}+\theta p^{s-k-1}+1}$. Then, there is a nonzero element $f\left(x\right)$ in $\mathcal{F}$ such that $c\left(x\right)={\left(x^n-{\alpha }_0\right)}^{p^s-p^{s-k}+\theta p^{s-k-1}+1}f\left(x\right)$ with $deg\left(f\right)<n[(p-\theta )p^{s-k-1}-1]$. Let $g\left(x\right)=\left(x^n-{\alpha }_0\right)f\left(x\right)$. Then, $deg\left(g\right)<n(p-\theta )p^{s-k-1}$, ${\mathrm{wt}}_{\mathrm{H}}\left(g\right(x\left)\right)\ge 2$, and

$$\begin{array}{cc}\hfill c\left(x\right)=& {\left(x^n-{\alpha }_0\right)}^{p^s-p^{s-k}+\theta p^{s-k-1}}g\left(x\right)\hfill \\ \hfill =& {(x^{n{p}^{s-k-1}}-{\alpha }_0^{p^{s-k-1}})}^{p^{k+1}-p+\theta }g\left(x\right).\hfill \end{array}$$

Suppose that $\mathcal{T}=\{i_1,\cdots ,i_{\eta }\}$ is the set of exponents of nonzero terms of $g\left(x\right)$. For an integer i, let ${\mathcal{S}}_i$ be a set of integers congruent to i modulo $n{p}^{s-k-1}$ (i.e., ${\mathcal{S}}_i=\{j\mid j\equiv i(modn{p}^{s-k-1})\}$). We consider two cases where $\mathcal{T}\subset {\mathcal{S}}_{i_1}$ and $\mathcal{T}\not\subset {\mathcal{S}}_{i_1}$.

Case 1 has $\mathcal{T}\subset {\mathcal{S}}_{i_1}$. We assume that $g\left(x\right)={\sum }_{t=1}^{\eta }{r}_t{x}^{i_1+n{p}^{s-k-1}{u}_t}$, where $0=u_1<\dots <u_{\eta }$. Thus, we have

$$\begin{array}{cc}\hfill c\left(x\right)=& {(x^{n{p}^{s-k-1}}-{\alpha }_0^{p^{s-k-1}})}^{p^{k+1}-p+\theta }g\left(x\right)\hfill \\ \hfill =& \left[{(x^{n{p}^{s-k-1}}-{\alpha }_0^{p^{s-k-1}})}^{p^{k+1}-p+\theta }\sum _{t=1}^{\eta }{r}_t{x}^{n{p}^{s-k-1}{u}_t}\right]x^{i_1}.\hfill \end{array}$$

It follows that $cw\left(c\left(x\right)\right)\ge n{p}^{s-k-1}\ge np\ge 4$, and hence

$${\mathrm{wt}}_{\mathrm{sp}}\left(c\left(x\right)\right)=2·{\mathrm{wt}}_{\mathrm{H}}\left(c\left(x\right)\right)\ge 2·{\mathrm{d}}_{\mathrm{H}}\left({\mathcal{C}}_{p^s-p^{s-k}+\theta p^{s-k-1}+1}\right)\ge 2(\theta +2)p^k.$$

Case 2 has $\mathcal{T}\not\subset {\mathcal{S}}_{i_1}$. We only demonstrate that when $\mathcal{T}\subset {\mathcal{S}}_{i_1}\cup {\mathcal{S}}_{i_2}$ with $i_1\not\equiv i_2(modn{p}^{s-k-1})$, the rest is similar. Let $g\left(x\right)=g_1\left(x\right)+g_2\left(x\right)$, $g_1\left(x\right)={\sum }_{t=1}^{{\eta }_1}{r}_t^{\left(1\right)}{x}^{i_1+n{p}^{s-k-1}{u}_t}$, and $g_2\left(x\right)={\sum }_{t=1}^{{\eta }_2}{r}_t^{\left(2\right)}{x}^{i_2+n{p}^{s-k-1}{v}_t}$, where $0=u_1<\dots <u_{{\eta }_1}$ and $0=v_1<\dots <v_{{\eta }_2}$. Then, we have

$$\begin{array}{cc}\hfill c\left(x\right)=& {(x^{n{p}^{s-k-1}}-{\alpha }_0^{p^{s-k-1}})}^{p^{k+1}-p+\theta }\left(g_1\left(x\right)+g_2\left(x\right)\right)\hfill \\ \hfill =& \left[{(x^{n{p}^{s-k-1}}-{\alpha }_0^{p^{s-k-1}})}^{p^{k+1}-p+\theta }\sum _{t=1}^{{\eta }_1}{r}_t^{\left(1\right)}{x}^{n{p}^{s-k-1}{u}_t}\right]x^{i_1}\hfill \\ & +\left[{(x^{n{p}^{s-k-1}}-{\alpha }_0^{p^{s-k-1}})}^{p^{k+1}-p+\theta }\sum _{t=1}^{{\eta }_2}{r}_t^{\left(2\right)}{x}^{n{p}^{s-k-1}{v}_t}\right]x^{i_2}.\hfill \end{array}$$

Let $S_H$ be a set of the exponents of nonzero terms of $c\left(x\right)$. Then, we have

$$\begin{array}{cc}\hfill S_H=& \{i_1+n{p}^{s-k-1}{w}_j\mid w_j^{\left(1\right)}\in \mathbb{N},1\le j\le l_1\}\hfill \\ & \cup \{i_2+n{p}^{s-k-1}{w}_j\mid w_j^{\left(2\right)}\in \mathbb{N},1\le j\le l_2\},\hfill \end{array}$$

where $l_t={\mathrm{wt}}_{\mathrm{H}}\left({\left(x^n-{\alpha }_0\right)}^{p^s-p^{s-k}+\theta p^{s-k-1}}{g}_t\left(x\right)\right)$ for $t=1,2$. According to Theorem 1, ${\mathrm{d}}_{\mathrm{H}}\left({\mathcal{C}}_{p^s-p^{s-k}+\theta p^{s-k-1}}\right)\ge (\theta +1)p^k$, and hence $l_1,l_2\ge (\theta +1)p^k$. Since $n{p}^{s-k-1}\ge 4$, $S_H$ is at least partitioned into $(\theta +1)p^k$ subsets of consecutive indices. Under Theorem 2, we have

$${\mathrm{wt}}_{\mathrm{sp}}\left(c\left(x\right)\right)\ge 3(\theta +1)p^k\ge 2(\theta +2)p^k.$$

Therefore, we have proven that ${\mathrm{wt}}_{\mathrm{sp}}\left(c\left(x\right)\right)\ge 2(\theta +2)p^k$ holds in all cases; that is, ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_{p^s-p^{s-k}+\theta p^{s-k-1}+1}\right)\ge 2(\theta +2)p^k$. □

The following lemma is about the case of $k=s-1$ and $0\le \theta \le p-2$.

Lemma 9.

Let $n,\theta$ be integers such that $n\ge 2$ and $1\le \theta \le p-1$. Then, ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_{p^s-p+\theta }\right)\ge 2(\theta +1)p^{s-1}$.

Proof. 

Let $c\left(x\right)$ be any nonzero codeword in ${\mathcal{C}}_{p^s-p+\theta }$. Then, there is a nonzero element $f\left(x\right)$ in $\mathcal{F}$ such that $c\left(x\right)={\left(x^n-{\alpha }_0\right)}^{p^s-p+\theta }f\left(x\right)$ with $deg\left(f\right)<n(p-\theta )$. Suppose that $\mathcal{T}=\{i_1,\cdots ,i_{\eta }\}$ is a set of the exponents of nonzero terms of $f\left(x\right)$. For an integer i, let ${\mathcal{S}}_i$ be a set of integers congruent to i modulo n (i.e., ${\mathcal{S}}_i=\{j\mid j\equiv i(modn)\}$). We consider the set $\mathcal{T}$ in two cases.

Case 1 has $\mathcal{T}\subset {\mathcal{S}}_{i_1}$. We may assume that $f\left(x\right)={\sum }_{t=1}^{\eta }{r}_t{x}^{i_1+n{u}_t}$, where $0=u_1<\dots <u_{\eta }$. Then, we have

$$c\left(x\right)=\left[{\left(x^n-{\alpha }_0\right)}^{p^s-p+\theta }\sum _{t=1}^{\eta }{r}_t{x}^{n{u}_t}\right]x^{i_1}.$$

It follows that $cw\left(c\right(x\left)\right)\ge n\ge 2$, and hence

$${\mathrm{wt}}_{\mathrm{sp}}\left(c\left(x\right)\right)=2·{\mathrm{wt}}_{\mathrm{H}}\left(c\left(x\right)\right)\ge 2·{\mathrm{d}}_{\mathrm{H}}\left({\mathcal{C}}_{p^s-p+\theta }\right)\ge 2(\theta +1)p^{s-1}.$$

Case 2 has $\mathcal{T}\not\subset {\mathcal{S}}_{i_1}$. We may assume that $\mathcal{T}\subset {\mathcal{S}}_{i_1}\cup {\mathcal{S}}_{i_2}$, where $i_1\not\equiv i_2(modn)$. Let $f\left(x\right)=f_1\left(x\right)+f_2\left(x\right)$, $f_1\left(x\right)={\sum }_{t=1}^{{\eta }_1}{r}_t^{\left(1\right)}{x}^{i_1+n{u}_t}$, and $f_2\left(x\right)={\sum }_{t=1}^{{\eta }_2}{r}_t^{\left(2\right)}{x}^{i_2+n{v}_t}$, where $0=u_1<\dots <u_{{\eta }_1}$ and $0=v_1<\dots <v_{{\eta }_2}$. Then, we have

$$\begin{array}{cc}\hfill c\left(x\right)=& \left[{\left(x^n-{\alpha }_0\right)}^{p^s-p+\theta }\sum _{t=1}^{{\eta }_1}{r}_t^{\left(1\right)}{x}^{n{u}_t}\right]x^{i_1}\hfill \\ & +\left[{\left(x^n-{\alpha }_0\right)}^{p^s-p+\theta }\sum _{t=1}^{{\eta }_2}{r}_t^{\left(2\right)}{x}^{n{v}_t}\right]x^{i_2}.\hfill \end{array}$$

Since $i_1\not\equiv i_2(modn)$, we have

$$\begin{array}{cc}\hfill {\mathrm{wt}}_{\mathrm{H}}\left(c\right(x\left)\right)=& {\mathrm{wt}}_{\mathrm{H}}\left(\left[{\left(x^n-{\alpha }_0\right)}^{p^s-p+\theta }\sum _{t=1}^{{\eta }_1}{r}_t^{\left(1\right)}{x}^{n{u}_t}\right]x^{i_1}\right)\hfill \\ & +{\mathrm{wt}}_{\mathrm{H}}\left(\left[{\left(x^n-{\alpha }_0\right)}^{p^s-p+\theta }\sum _{t=1}^{{\eta }_2}{r}_t^{\left(2\right)}{x}^{n{v}_t}\right]x^{i_2}\right)\hfill \\ \hfill \ge & 2·{\mathrm{d}}_{\mathrm{H}}\left({\mathcal{C}}_{p^s-p+\theta }\right)\hfill \\ \hfill =& 2(\theta +1)p^{s-1},\hfill \end{array}$$

which implies that ${\mathrm{wt}}_{\mathrm{sp}}\left(c\left(x\right)\right)\ge {\mathrm{wt}}_{\mathrm{H}}\left(c\left(x\right)\right)\ge 2(\theta +1)p^{s-1}$. When combining the two cases discussed above, it follows that ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_{p^s-p+\theta }\right)\ge 2(\theta +1)p^{s-1}$. □

By integrating the upper bound established in Lemma 5 with the lower bounds derived in Lemmas 6, 7, 8, and 9, we can fully determine the symbol-pair distances of $\alpha$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}$. To ensure the completeness of this theorem, we provide the symbol-pair distances for both the case where $n=1$ and the case where $n\ge 2$.

Theorem 2.

Let ${\alpha }_0$ be a nonzero element in ${\mathbb{F}}_{p^m}$ and $\alpha ={\alpha }_0^{p^s}$. Let ${\mathcal{C}}_i=\langle {\left(x^n-{\alpha }_0\right)}^i\rangle \subseteq \mathcal{F}$ for $i\in \left\{0,1,\dots ,p^s\right\}$ be a given α-constacyclic code of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}$. Then, the symbol-pair distance ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_i\right)$ is completely determined as follows:

(1) 

When $n\ge 1$, ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_0\right)=2$ and ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_{p^s}\right)=0$.

(2) 

When $n=1$ and $1\le i\le p^s-1$, the following is true:

$${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_i\right)=\left\{\begin{array}{cc}3p^k,\hfill & \mathit{if}i=p^s-p^{s-k}+1\mathit{and}0\le k\le s-2;\hfill \\ 4p^k,\hfill & \mathit{if}{p}^s-p^{s-k}+2\le i\le p^s-p^{s-k}+p^{s-k-1}\hfill \\ & \mathit{and}0\le k\le s-2;\hfill \\ 2(\theta +2)p^k,\hfill & \mathit{if}{p}^s-p^{s-k}+\theta p^{s-k-1}+1\le i\le \hfill \\ & p^s-p^{s-k}+(\theta +1)p^{s-k-1},\hfill \\ & 0\le k\le s-2\mathit{and}1\le \theta \le p-2;\hfill \\ (\theta +2)p^{s-1},\hfill & \mathit{if}i=p^s-p+\theta \mathit{and}1\le \theta \le p-2;\hfill \\ p^s,\hfill & \mathit{if}i=p^s-1.\hfill \end{array}\right.$$

(3) 

When $n\ge 2$ and $1\le i\le p^s-1$, the following is true:

$${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_i\right)=2(\theta +2)p^k,$$

where $p^s-p^{s-k}+\theta p^{s-k-1}+1\le i\le p^s-p^{s-k}+(\theta +1)p^{s-k-1}$, $0\le k\le s-1$, and $0\le \theta \le p-2$.

Example 1.

We present several examples of symbol-pair α-constacyclic codes of a length $n{\left(13\right)}^s$ over ${\mathbb{F}}_{13}$, where α is set to three and eight, n is set to three and four, and s is set to one and two. We denote k and $d_{sp}$ as the dimension and symbol-pair distance of a symbol-pair α-constacyclic code, respectively. In

Table 1. $\alpha$-constacyclic codes of a length $n{\left(13\right)}^s$ over ${\mathbb{F}}_{13}$.

n	s	$\mathit{\alpha }$	Generator Polymial	$[\mathit{n}{\left(13\right)}^{\mathit{s}},\mathit{k},{\mathit{d}}_{\mathit{s}\mathit{p}}]$
3	1	3	${(x^3-3)}^4$	$[39,27,10]$
3	2	3	${(x^3-3)}^{18}$	$[507,453,6]$
3	2	3	${(x^3-3)}^{160}$	$[507,27,130]$
4	1	8	${(x^4-8)}^4$	$[52,36,10]$
4	2	8	${(x^4-8)}^{18}$	$[676,604,6]$
4	2	8	${(x^4-8)}^{160}$	$[676,36,130]$

, we calculate the symbol-pair distances for these codes using Theorem 2.

By leveraging the results for the symbol-pair distances from the preceding analysis, we provide a comprehensive characterization of all MDS symbol-pair codes when $x^n-{\alpha }_0$ is irreducible over ${\mathbb{F}}_{p^m}$. While it is important to note that these MDS symbol-pair codes have been previously documented in the existing literature, our work offers a thorough examination within the framework of repeated-root constacyclic codes of arbitrary lengths under the irreducibility condition.

Theorem 3.

Let ${\alpha }_0$ be a nonzero element of ${\mathbb{F}}_{p^m}$ and $\alpha ={\alpha }_0^{p^s}$. Assume that $x^n-{\alpha }_0$ is irreducible over ${\mathbb{F}}_{p^m}$. All nontrivial MDS symbol-pair α-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}$ are summarized in

Table 2. All the MDS symbol-pair $\alpha$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}$.

Length	Generator Polymial	Dimension	Pair Distance	Remark
$p^s$	$x-{\alpha }_0$	$p^s-1$	3
$p^s$	${(x-{\alpha }_0)}^2$	$p^s-2$	4
$p^s$	${(x-{\alpha }_0)}^4$	5	6	$p=3$, $s=2$
$p^s$	${(x-{\alpha }_0)}^k$	$p-k$	$k+2$	$s=1$, $1\le k\le p-2$
$p^s$	${(x-{\alpha }_0)}^{p^s-2}$	2	$p^s$
$2p^s$	$x^2-{\alpha }_0$	$2p^s-2$	4
$2p^s$	${(x^2-{\alpha }_0)}^k$	$2p-2k$	$2k+2$	$s=1$, $1\le k\le p-2$
$2p^s$	${(x^2-{\alpha }_0)}^{p^s-1}$	2	$2p^s$

.

Proof. 

When $x^n-{\alpha }_0$ is irreducible over ${\mathbb{F}}_{p^m}$, the $\alpha$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}$ are ${\mathcal{C}}_i=\langle {\left(x^n-{\alpha }_0\right)}^i\rangle$, where $0\le i\le p^s$. Note that $|{\mathcal{C}}_i|=|\langle {(x^n-{\alpha }_0)}^i\rangle |=p^{m(n{p}^s-ni)}$, and the Singleton bound for symbol-pair constacyclic codes implies that $|{\mathcal{C}}_i|\le p^{m(n{p}^s-{\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_i\right)+2)}$ (i.e., $ni\ge {\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_i\right)-2$ for $i\in \{0,1,\dots ,p^s-1\}$). Therefore, ${\mathcal{C}}_i$ is an MDS symbol-pair code if and only if

$$ni-{\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_i\right)+2=0.$$

(3)

If $n\ge 3$, then let $i=p^s-p^{s-k}+\theta p^{s-k-1}+\gamma$ with $0\le k\le s-1$, $0\le \theta \le p-2$, and $1\le \gamma \le p^{s-k-1}$. According to Theorem 2, we have ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_i\right)=2(\theta +2)p^k$, and hence

$$\begin{array}{cc}& ni-{\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_i\right)+2\hfill \\ \hfill =& n(p^s-p^{s-k}+\theta p^{s-k-1}+\gamma )-2(\theta +2)p^k+2\hfill \\ \hfill =& [n{p}^{s-k}-2(\theta +2)](p^k-1)+(n{p}^{s-k-1}-2)\theta +n\gamma -2\hfill \\ \hfill \ge & n-2>0.\hfill \end{array}$$

Consequently, in this scenario, no additional MDS symbol-pair $\alpha$-constacyclic code exists.

If n equals one or two, then by applying Theorem 2, it is possible to determine all values of i for which the aforementioned Equation (3) holds. These values are summarized in Table 2. □

4. Symbol-Pair Distances of Repeated-Root Constacyclic Codes over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$

Let ${\alpha }_0,\beta \in {\mathbb{F}}_{p^m}$ with ${\alpha }_0\ne 0$, and define $\alpha ={\alpha }_0^{p^s}$. In this section, we investigate the relationship between the symbol-pair distances of $\alpha$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}$ and the symbol-pair distances of $(\alpha +u\beta )$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$, where $u^2=0$, n is a positive integer which is coprime to p, and $x^n-{\alpha }_0$ is irreducible over ${\mathbb{F}}_{p^m}$. We examine the symbol-pair distances for both cases when $\beta \ne 0$ and $\beta =0$, as detailed in the subsequent four theorems.

Theorem 4.

Let ${\alpha }_0$ be a nonzero element in ${\mathbb{F}}_{p^m}$ satisfying that $x^n-{\alpha }_0$ is irreducible over ${\mathbb{F}}_{p^m}$. Denote $\alpha ={\alpha }_0^{p^s}$. Let β be a nonzero element in ${\mathbb{F}}_{p^m}$. Let $\mathcal{D}$ be an $(\alpha +u\beta )$-constacyclic code of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ ($i.e.$, $\mathcal{D}=\langle {\left(x^n-{\alpha }_0\right)}^i\rangle$ for $0\le i\le 2p^s$). The symbol-pair distance of $\mathcal{D}$ is

$${\mathrm{d}}_{\mathrm{sp}}\left(\mathcal{D}\right)=\left\{\begin{array}{cc}2,\hfill & \mathit{if}0\le i\le p^s;\hfill \\ {\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x^n-{\alpha }_0)}^{i-p^s}\rangle }_F\right),\hfill & \mathit{if}{p}^s+1\le i\le 2p^s.\hfill \end{array}\right.$$

Proof. 

When $0\le i\le p^s$, we have $u\beta ={\left(x^n-{\alpha }_0\right)}^{p^s}\in \mathcal{D}$, and the symbol-pair weight of $u\beta$ is two. When combined with ${\mathrm{d}}_{\mathrm{sp}}\left(\mathcal{D}\right)\ge 2$, we find that the symbol-pair distance of $\mathcal{D}$ is two.

When $p^s+1\le i\le 2p^s$, we have

$$\langle {(x^n-{\alpha }_0)}^i\rangle =\langle u{(x^n-{\alpha }_0)}^{i-p^s}\rangle ,$$

which implies that the codewords in the code $\langle {(x^n-{\alpha }_0)}^i\rangle$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ are precisely the codewords in the code $\langle {(x^n-{\alpha }_0)}^{i-p^s}\rangle$ over ${\mathbb{F}}_{p^m}$ multiplied by u. Consequently, the symbol-pair distance of $\mathcal{D}$ is equal to that of ${\langle {(x^n-{\alpha }_0)}^{i-p^s}\rangle }_F$. □

The symbol-pair distance of $\mathcal{D}$ becomes more intricate when $\beta =0$. We analyze this scenario in three distinct cases corresponding to the three types of $\alpha$-constacyclic codes described in Lemma 4.

Theorem 5.

Let $\mathcal{D}$ be an α-constacyclic code of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ with type I in Lemma 4 ($i.e.$, $\mathcal{D}=\langle {\left(x^n-{\alpha }_0\right)}^k\rangle$ for $0\le k\le p^s$). Then, ${\mathrm{d}}_{\mathrm{sp}}\left(\mathcal{D}\right)={\mathrm{d}}_{\mathrm{sp}}\left({\langle {\left(x^n-{\alpha }_0\right)}^k\rangle }_F\right)$.

Proof. 

Notice that $\mathcal{D}\supseteq \langle u{\left(x^n-{\alpha }_0\right)}^k\rangle$, and hence

$${\mathrm{d}}_{\mathrm{sp}}\left(\mathcal{D}\right)\le {\mathrm{d}}_{\mathrm{sp}}\left(\langle u{\left(x^n-{\alpha }_0\right)}^k\rangle \right)={\mathrm{d}}_{\mathrm{sp}}\left({\langle {\left(x^n-{\alpha }_0\right)}^k\rangle }_F\right).$$

(4)

Next, for any nonzero codeword $c\left(x\right)$ in $\mathcal{D}$, there are $f_0\left(x\right),f_u\left(x\right)$ in ${\mathbb{F}}_{p^m}\left[x\right]$ such that

$$\begin{array}{cc}\hfill c\left(x\right)& =[f_0\left(x\right)+u{f}_u\left(x\right)]{\left(x^n-{\alpha }_0\right)}^k\hfill \\ & =f_0\left(x\right){\left(x^n-{\alpha }_0\right)}^k+u{f}_u\left(x\right){\left(x^n-{\alpha }_0\right)}^k.\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill {\mathrm{wt}}_{\mathrm{sp}}\left(c\right(x\left)\right)& \ge max\{{\mathrm{wt}}_{\mathrm{sp}}\left(f_0\left(x\right){\left(x^n-{\alpha }_0\right)}^k\right),{\mathrm{wt}}_{\mathrm{sp}}\left(f_u\left(x\right){\left(x^n-{\alpha }_0\right)}^k\right)\}\hfill \\ \hfill & \ge {\mathrm{d}}_{\mathrm{sp}}\left({\langle {\left(x^n-{\alpha }_0\right)}^k\rangle }_F\right).\hfill \end{array}$$

(5)

By combining Equations (4) and (5), we obtain the symbol-pair distance of $\mathcal{D}$. □

Theorem 6.

Let $\mathcal{D}$ be an α-constacyclic code of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ with type II in Lemma 4 ($i.e.$, $\mathcal{D}=\langle {(x^n-{\alpha }_0)}^{j}b\left(x\right)+u{(x^n-{\alpha }_0)}^k\rangle$, where $0\le k\le p^s-1$, $⌈{\displaystyle \frac{p^s+k}{2}}⌉\le j\le p^s-1$ and either $b\left(x\right)$ is zero or $b\left(x\right)$ is a unit in $\mathcal{F}$). Then, it follows that

$${\mathrm{d}}_{\mathrm{sp}}\left(\mathcal{D}\right)=\left\{\begin{array}{cc}{\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x^n-{\alpha }_0)}^k\rangle }_F\right),\hfill & \mathit{if}b\left(x\right)=0;\hfill \\ {\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x^n-{\alpha }_0)}^{p^s-j+k}\rangle }_F\right),\hfill & \mathit{if}b\left(x\right)\mathit{is}\mathit{a}\mathit{unit}\mathit{in}\mathcal{F}.\hfill \end{array}\right.$$

Proof. 

If $b\left(x\right)=0$, then $\mathcal{D}=\langle u{\left(x^n-{\alpha }_0\right)}^k\rangle$, and hence

$${\mathrm{d}}_{\mathrm{sp}}\left(\mathcal{D}\right)={\mathrm{d}}_{\mathrm{sp}}\left(\langle u{\left(x^n-{\alpha }_0\right)}^k\rangle \right)={\mathrm{d}}_{\mathrm{sp}}\left({\langle {\left(x^n-{\alpha }_0\right)}^k\rangle }_F\right).$$

Assume that $b\left(x\right)$ is a unit in $\mathcal{F}$. Since

$${\left(x^n-{\alpha }_0\right)}^{p^s-j}\left[{\left(x^n-{\alpha }_0\right)}^{j}b\left(x\right)+u{\left(x^n-{\alpha }_0\right)}^k\right]=u{\left(x^n-{\alpha }_0\right)}^{p^s-j+k},$$

it follows that

$$\langle u{\left(x^n-{\alpha }_0\right)}^{p^s-j+k}\rangle \subseteq \mathcal{D},$$

and hence

$$\begin{array}{cc}\hfill {\mathrm{d}}_{\mathrm{sp}}\left(\mathcal{D}\right)\le & {\mathrm{d}}_{\mathrm{sp}}\left(\langle u{\left(x^n-{\alpha }_0\right)}^{p^s-j+k}\rangle \right)\hfill \\ \hfill =& {\mathrm{d}}_{\mathrm{sp}}\left({\langle {\left(x^n-{\alpha }_0\right)}^{p^s-j+k}\rangle }_F\right).\hfill \end{array}$$

(6)

For any nonzero codeword $c\left(x\right)$ in $\mathcal{D}$, there are $f_0\left(x\right)$, $f_u\left(x\right)$ in ${\mathbb{F}}_{p^m}\left[x\right]$ such that

$$\begin{array}{cc}\hfill c\left(x\right)& =[f_0\left(x\right)+u{f}_u\left(x\right)]\left[{\left(x^n-{\alpha }_0\right)}^{j}b\left(x\right)+u{\left(x^n-{\alpha }_0\right)}^k\right]\hfill \\ & =f_0\left(x\right){\left(x^n-{\alpha }_0\right)}^{j}b\left(x\right)+u\left[f_0\left(x\right){\left(x^n-{\alpha }_0\right)}^k+f_u\left(x\right){\left(x^n-{\alpha }_0\right)}^{j}b\left(x\right)\right].\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill {\mathrm{wt}}_{\mathrm{sp}}\left(c\right(x\left)\right)\ge & max\{{\mathrm{wt}}_{\mathrm{sp}}\left(f_0\left(x\right){\left(x^n-{\alpha }_0\right)}^{j}b\left(x\right)\right),\hfill \\ & {\mathrm{wt}}_{\mathrm{sp}}(f_0\left(x\right){\left(x^n-{\alpha }_0\right)}^k+f_u\left(x\right){\left(x^n-{\alpha }_0\right)}^{j}b\left(x\right))\}.\hfill \end{array}$$

If ${\left(x^n-{\alpha }_0\right)}^{p^s-j}\mid f_0\left(x\right)$, let $f_0\left(x\right)={\left(x^n-{\alpha }_0\right)}^{p^s-j}{f}_0^{\prime }\left(x\right)$, and then

$$\begin{array}{cc}\hfill {\mathrm{wt}}_{\mathrm{sp}}\left(c\right(x\left)\right)& \ge {\mathrm{wt}}_{\mathrm{sp}}\left(r\left(x\right){\left(x^n-{\alpha }_0\right)}^{p^s-j+k}\right)\hfill \\ \hfill & \ge {\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x^n-{\alpha }_0)}^{p^s-j+k}\rangle }_F\right),\hfill \end{array}$$

(7)

where

$$r\left(x\right)=f_0^{\prime }\left(x\right)+f_u\left(x\right){\left(x^n-{\alpha }_0\right)}^{2j-p^s-k}b\left(x\right)$$

If ${\left(x^n-{\alpha }_0\right)}^{p^s-j}\nmid f_0\left(x\right)$, then

$${\mathrm{wt}}_{\mathrm{sp}}\left(c\left(x\right)\right)\ge {\mathrm{wt}}_{\mathrm{sp}}\left(f_0\left(x\right){\left(x^n-{\alpha }_0\right)}^{j}b\left(x\right)\right).$$

Since $j\ge p^s-j+k$, we have

$${\mathrm{wt}}_{\mathrm{sp}}\left(c\left(x\right)\right)\ge {\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x^n-{\alpha }_0)}^{p^s-j+k}\rangle }_F\right).$$

(8)

According to Equations (6), (7), and (8), we have ${\mathrm{d}}_{\mathrm{sp}}\left(\mathcal{D}\right)={\mathrm{d}}_{\mathrm{sp}}\left({\langle {\left(x^n-{\alpha }_0\right)}^{p^s-j+k}\rangle }_F\right)$. □

Remark 1.

Our results for the symbol-pair distances of constacyclic codes over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ generalize the results of [18], which focus on the constacyclic codes under the condition of $n=1$.

Remark 2.

According to Theorem 6, when $b\left(x\right)$ is a unit, the symbol-pair distance of the constacyclic code $\mathcal{D}=\langle {(x-{\alpha }_0)}^{j}b\left(x\right)+u{(x-{\alpha }_0)}^k\rangle$ with $k<2j-p^s$ is given by ${\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x-{\alpha }_0)}^{p^s-j+k}\rangle }_F\right)$. This finding contradicts the conclusions in [18], which claimed that the symbol-pair distance is equal to ${\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x-{\alpha }_0)}^j\rangle }_F\right)$. To validate our findings, we provide an example demonstrating the correctness of our results.

Consider a cyclic code $\mathcal{D}=\langle {(x-1)}^7+u(x-1)\rangle$ with a length of nine over the finite ring ${\mathbb{F}}_3+u{\mathbb{F}}_3$, where $u^2=0$. According to Theorem 12 in [18], it can be calculated that

$${\mathrm{d}}_{\mathrm{sp}}\left(\mathcal{D}\right)={\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x-1)}^7\rangle }_F\right)=9.$$

However, there exists a codeword $u{(x-1)}^3={(x-1)}^2[{(x-1)}^7+u(x-1)]\in \mathcal{D}$. It is verified that

$${\mathrm{wt}}_{\mathrm{sp}}\left(u{(x-1)}^3\right)=4.$$

This implies that the symbol-pair distance of $\mathcal{D}$ is at most four, which contradicts the conclusion derived from Theorem 12 in [18]. In fact, according to Theorem 6, we have

$${\mathrm{d}}_{\mathrm{sp}}\left(\mathcal{D}\right)={\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x-1)}^3\rangle }_F\right)=4.$$

The following theorem shows the symbol-pair distances of the constacyclic codes corresponding to type III in Lemma 4. The proof follows a similar approach to that of the previous theorem and has therefore been omitted here.

Theorem 7.

Let $\mathcal{D}$ be an α-constacyclic code of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ with type III in Lemma 4 ($i.e.$, $\mathcal{D}=\langle {(x^n-{\alpha }_0)}^{j}b\left(x\right)+u{(x^n-{\alpha }_0)}^k,{(x^n-{\alpha }_0)}^{k+t}\rangle$, where $0\le k\le p^s-2$, $1\le t\le p^s-k-1$, $k+⌈{\displaystyle \frac{t}{2}}⌉\le j\le k+t$, and either $b\left(x\right)$ is zero or $b\left(x\right)$ is a unit in $\mathcal{F}$). Then, it follows that

$${\mathrm{d}}_{\mathrm{sp}}\left(\mathcal{D}\right)=\left\{\begin{array}{cc}{\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x^n-{\alpha }_0)}^k\rangle }_F\right),\hfill & \mathit{if}b\left(x\right)=0;\hfill \\ {\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x^n-{\alpha }_0)}^{2k+t-j}\rangle }_F\right),\hfill & \mathit{if}b\left(x\right)\mathit{is}\mathit{a}\mathit{unit}\mathit{in}\mathcal{F}.\hfill \end{array}\right.$$

The formulas for symbol-pair distances of $(\alpha +u\beta )$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ are summarized in

Table 3. The symbol-pair distance of $(\alpha +u\beta )$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$.

Generator Polymial	Pair Distance	Remark
${(x^n-{\alpha }_0)}^i$	2	$\beta \ne 0$, and $0\le i\le p^s$
${(x^n-{\alpha }_0)}^i$	${\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x^n-{\alpha }_0)}^{i-p^s}\rangle }_F\right)$	$\beta \ne 0$, $p^s<i\le 2p^s$
${(x^n-{\alpha }_0)}^k$	${\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x^n-{\alpha }_0)}^k\rangle }_F\right)$	$\beta =0$, and $0\le k\le p^s$
$u{(x^n-{\alpha }_0)}^k$	${\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x^n-{\alpha }_0)}^k\rangle }_F\right)$	$\beta =0$, and $0\le k<p^s$
${(x^n-{\alpha }_0)}^{j}b\left(x\right)+u{(x^n-{\alpha }_0)}^k$	${\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x^n-{\alpha }_0)}^{p^s-j+k}\rangle }_F\right)$	$\beta =0$, $0\le k<p^s$, $⌈{\displaystyle \frac{p^s+k}{2}}⌉\le j<p^s,$ and $b\left(x\right)$ is a unit in $\mathcal{F}$
$u{(x^n-{\alpha }_0)}^k$, ${(x^n-{\alpha }_0)}^{k+t}$	${\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x^n-{\alpha }_0)}^k\rangle }_F\right)$	$\beta =0$, $0\le k\le p^s-2$, and $1\le t\le p^s-k-1$
${(x^n-{\alpha }_0)}^{j}b\left(x\right)+u{(x^n-{\alpha }_0)}^k$, ${(x^n-{\alpha }_0)}^{k+t}$	${\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x^n-{\alpha }_0)}^{2k+t-j}\rangle }_F\right)$	$\beta =0$, $0\le k\le p^s-2$, $1\le t\le p^s-k-1$, $k+⌈{\displaystyle \frac{t}{2}}⌉\le j\le k+t,$ and $b\left(x\right)$ is a unit in $\mathcal{F}$

.

Example 2.

We present several examples of symbol-pair $(\alpha +u\beta )$-constacyclic codes with a length of 507 over ${\mathbb{F}}_{13}+u{\mathbb{F}}_{13}$, where $\alpha +u\beta$ is set to $3+u$ and 3. In

Table 4. $(\alpha +u\beta )$-constacyclic codes of a length $N=507$ over ${\mathbb{F}}_{13}+u{\mathbb{F}}_{13}$.

$\mathit{\alpha }+\mathit{u}\mathit{\beta }$	Generator Polymial	$(\mathit{N},\mathit{M},{\mathit{d}}_{\mathbf{sp}})$
3+u	${(x^3-3)}^{12}$	$(507,{13}^{978},2)$
3+u	${(x^3-3)}^{200}$	$(507,{13}^{414},8)$
3	${(x^3-3)}^{26}$	$(507,{13}^{858},6)$
3	$u{(x^3-3)}^{26}$	$(507,{13}^{429},6)$
3	${(x^3-3)}^{100}+u{(x^3-3)}^{26}$	$(507,{13}^{429},18)$
3	$u{(x^3-3)}^{26},{(x^3-3)}^{30}$	$(507,{13}^{846},6)$
3	${(x^3-3)}^{28}+u{(x^3-3)}^{26},{(x^3-3)}^{30}$	$(507,{13}^{846},8)$

, we calculate the symbol-pair distances for these codes.

In the following, we utilize the symbol-pair distances of $(\alpha +u\beta )$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ as established in the previous subsection to derive MDS symbol-pair codes. Specifically, we first investigated the MDS symbol-pair $(\alpha +u\beta )$-constacyclic codes under the condition that $\beta \ne 0$.

Theorem 8.

Let ${\alpha }_0,\beta$ be nonzero elements in ${\mathbb{F}}_{p^m}$. Define $\alpha ={\alpha }_0^{p^s}$. Suppose that $x^n-{\alpha }_0$ is irreducible over ${\mathbb{F}}_{p^m}$. Let ${\mathcal{D}}_i=\langle {(x^n-{\alpha }_0)}^i\rangle \subseteq \mathcal{R}$ be an $(\alpha +u\beta )$-constacyclic code of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$, where $0\le i\le 2p^s$. Then, ${\mathcal{D}}_i$ is an MDS symbol-pair code if and only if $i=0$.

Proof. 

Under Theorem 3, we have

$$|{\mathcal{D}}_i|=|\langle {(x^n-{\alpha }_0)}^i\rangle |=p^{mn(2p^s-i)}.$$

The Singleton bound shows that

$$|{\mathcal{D}}_i{|\le |R|}^{n{p}^s-{\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{C}}_i\right)+2},$$

which is equivalent to

$$ni\ge 2{\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{D}}_i\right)-4.$$

Therefore, ${\mathcal{D}}_i$ is an MDS symbol-pair code if and only if

$$ni=2{\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{D}}_i\right)-4.$$

(9)

If $0\le i\le p^s$, then we have ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{D}}_i\right)=2$ under Theorem 4. According to Equation (9), we obtain $i=0$. If $p^s+1\le i\le 2p^s-1$, then ${\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{D}}_i\right)={\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x^n-{\alpha }_0)}^{i-p^s}\rangle }_F\right)$. By applying the Singleton bound to the constacyclic code ${\langle {(x^n-{\alpha }_0)}^{i-p^s}\rangle }_F$, we obtain

$$n(i-p^s)\ge {\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x^n-{\alpha }_0)}^{i-p^s}\rangle }_F\right)-2.$$

(10)

By reformulating Equation (10) we have

$$\begin{array}{cc}\hfill ni\ge & {\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{D}}_i\right)-2+n{p}^s\ge 2{\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{D}}_i\right)-2\hfill \\ \hfill >& 2{\mathrm{d}}_{\mathrm{sp}}\left({\mathcal{D}}_i\right)-4,\hfill \end{array}$$

which implies no MDS symbol-pair constacyclic code when i is in the range of $p^s+1\le i\le 2p^s-1$. □

The following theorem addresses the remaining case where $\beta =0$, leading to the discovery of three new classes of MDS symbol-pair $\alpha$-constacyclic codes over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$.

Theorem 9.

Let ${\alpha }_0$ be a nonzero element in ${\mathbb{F}}_{p^m}$ and $\alpha ={\alpha }_0^{p^s}$. There are three classes of MDS symbol-pair α-constacyclic codes of a length $2p^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$, and they are as follows:

(1) 

$\mathcal{D}=\langle (x^2-{\alpha }_0)+ub\left(x\right)\rangle$, where $b\left(x\right)$ is either zero or a unit in $\mathcal{F}$;

(2) 

$\mathcal{D}=\langle {(x^2-{\alpha }_0)}^{p^s-1}+u{(x^2-{\alpha }_0)}^{p^s-2}b\left(x\right)\rangle$, where $b\left(x\right)$ is either zero or a unit in $\mathcal{F}$;

(3) 

$\mathcal{D}=\langle {(x^2-{\alpha }_0)}^j+u{(x^2-{\alpha }_0)}^{k}b\left(x\right)\rangle$, where $s=1$, $1\le j\le p-1$, $max\{0,2j-p\}\le k<j$, and $b\left(x\right)$ is either zero or a unit in $\mathcal{F}$.

Proof. 

(1) According to Lemma 4, the size of $\mathcal{D}$ is $p^{4m(p^s-1)}$. According to Theorems 5 and 7, the symbol-pair distance of $\mathcal{D}$ is four. It achieves the Singleton bound

$$\left|\mathcal{D}\right|=p^{4m(p^s-1)}=p^{2m(2p^s-4+2)}$$

with equality. Therefore, $\mathcal{D}$ is an MDS code.

(2) Under Lemma 4, the size of $\mathcal{D}$ is $p^{4m}$. According to Theorems 5 and 6, the symbol-pair distance of $\mathcal{D}$ is $2p^s$. It achieves the Singleton bound

$$\left|\mathcal{D}\right|=p^{4m}=p^{2m(2p^s-2p^s+2)}$$

with equality. Therefore, $\mathcal{D}$ is an MDS code.

(3) According to Lemma 4, the size of $\mathcal{D}$ is $p^{4m(p-j)}$. According to Theorems 5, 6, and 7, the symbol-pair distance of $\mathcal{D}$ is $2j+2$. It achieves the Singleton bound

$$\left|\mathcal{D}\right|=p^{4m(p-j)}=p^{2m(2p-2j-2+2)}$$

with equality. Therefore, $\mathcal{D}$ is an MDS code. □

Remark 3.

In [7], Dinh et al. introduced two additional classes of MDS symbol-pair codes with the parameters $(2^s,2^{m(2^{s-1}+4)},3·2^{s-2})$ and $(3^s,3^{m(2·3^{s-1}+4)},2·3^{s-1})$. The first α-constacyclic code of a length $2^s$ over ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ is

$$\mathcal{D}=\langle {\left(x^n-{\alpha }_0\right)}^{2^s-3}+u{\left(x^n-{\alpha }_0\right)}^{2^{s-1}-4}\rangle ,$$

where $s\ge 3$. According to Remark 2, the symbol-pair distance of $\mathcal{C}$ is ${\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x-{\alpha }_0)}^{2^{s-1}-1}\rangle }_F\right)=4$, rather than ${\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x-{\alpha }_0)}^{2^s-3}\rangle }_F\right)=3·2^{s-2}$. Consequently, this code does not satisfy the MDS property.

The second α-constacyclic code of a length $3^s$ over ${\mathbb{F}}_{3^m}+u{\mathbb{F}}_{3^m}$ is

$$\mathcal{D}=\langle {\left(x^n-{\alpha }_0\right)}^{3^s-5}+u{\left(x^n-{\alpha }_0\right)}^{3^{s-1}-4}\rangle ,$$

where $s\ge 3$. According to 2, the symbol-pair distance of $\mathcal{C}$ is ${\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x-{\alpha }_0)}^{3^{s-1}+1}\rangle }_F\right)=6$, rather than ${\mathrm{d}}_{\mathrm{sp}}\left({\langle {(x-{\alpha }_0)}^{3^s-5}\rangle }_F\right)=2·3^{s-1}$. Therefore, this code also does not satisfy the MDS property.

According to Theorem 9, the MDS symbol-pair $\alpha$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ are listed in

Table 5. MDS symbol-pair $\alpha$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$, where b(x) is either zero or a unit in ${\mathbb{F}}_{p^m}\left[x\right]/\langle x^{n{p}^s}-\alpha \rangle$.

Generator Polynomial	Size	Pair Distance	Remark	References
$(x-{\alpha }_0)+ub\left(x\right)$	$p^{2m(p^s-1)}$	3		[18,22]
${(x-{\alpha }_0)}^2+u{(x-{\alpha }_0)}^{k}b\left(x\right)$	$p^{2m(p^s-2)}$	4	$s\ge 2$ $k=0,1$	[18,22]
${(x-{\alpha }_0)}^4+u{(x-{\alpha }_0)}^{k}b\left(x\right)$	$p^{10m}$	6	$p=3$ $s=2$ $0\le k\le 3$	[18,22]
${(x-{\alpha }_0)}^j+u{(x-{\alpha }_0)}^{k}b\left(x\right)$	$p^{2m(p-k)}$	$j+2$	$s=1$ $1\le j\le p-2$ $max\{0,2j-p\}$ $\le k<j$	[18,22]
${(x-{\alpha }_0)}^{p^s-2}+u{(x-{\alpha }_0)}^{k}b\left(x\right)$	$p^{4m}$	$p^s$	$k=p^s-4,p^s-3$	[18,22]
$(x^2-{\alpha }_0)+ub\left(x\right)$	$p^{4m(p^s-1)}$	4		Theorem 9
${(x^2-{\alpha }_0)}^j+u{(x^2-{\alpha }_0)}^{k}b\left(x\right)$	$p^{4m(p-k)}$	$2j+2$	$s=1$ $1\le j\le p-2$ $2j-p\le k<j$	Theorem 9
${(x^2-{\alpha }_0)}^{p^s-1}+u{(x^2-{\alpha }_0)}^{p^s-2}b\left(x\right)$	$p^{4m}$	$2p^s$		Theorem 9

. When the polynomial $x^n-{\alpha }_0$ is irreducible over ${\mathbb{F}}_{p^m}$, we draw the following conclusion. The proof, which follows a similar approach to that of Theorem 3, has been omitted for brevity.

Theorem 10.

Let ${\alpha }_0$ be a nonzero element of ${\mathbb{F}}_{p^m}$ and $\alpha ={\alpha }_0^{p^s}$. Let β be an element of ${\mathbb{F}}_{p^m}$. All nontrivial MDS symbol-pair $(\alpha +u\beta )$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ for when $x^n-{\alpha }_0$ is irreducible over ${\mathbb{F}}_{p^m}$ are presented in Table 5.

Remark 4.

Table 5 presents all the MDS symbol-pair $(\alpha +u\beta )$-constacyclic codes over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$. Notice that the codes considered in [18] are a subcase of the codes we considered in this paper, which confine $n=1$. Compared with the known results, we obtained three classes of MDS symbol-pair $(\alpha +u\beta )$-constacyclic codes with new parameters, which are exhibited in the above table.

5. Conclusions

Let ${\mathbb{F}}_{p^m}$ be a finite field and ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ be a finite ring with $u^2=0$. We completely characterized the symbol-pair distances of $\alpha$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}$ and $(\alpha +u\beta )$-constacyclic codes of a length $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$, where $n,s$ are positive integers with $gcd(n,p)=1$, $\beta \in {\mathbb{F}}_{p^m}$, and $\alpha ={{\alpha }_0}^{p^s}\in {\mathbb{F}}_{p^m}$ such that $x^n-{\alpha }_0$ is irreducible over ${\mathbb{F}}_{p^m}$. Using the characterization of symbol-pair distances, we systematically presented all MDS symbol-pair $\alpha$-constacyclic codes over ${\mathbb{F}}_{p^m}$ and all MDS symbol-pair $(\alpha +u\beta )$-constacyclic codes over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$. Additionally, we introduced three new classes of MDS symbol-pair codes that exhibited relatively large pair distances. It is an interesting problem to consider the case where $x^n-{\alpha }_0$ is reducible over ${\mathbb{F}}_{p^m}$.

Codes over Gaussian integers equipped with the Mannheim metric and over Eisenstein integers with the hexagonal metric can be used for coding over two-dimensional signal spaces [32,33,34]. The symbol-pair codes over Gaussian integers and over Eisenstein integers would be an interesting topic that has not been investigated yet.