Symbol-Triple Distance of Repeated-Root Constacyclic Codes of Prime Power Lengths over ${\mathbb{F}}_q+u{\mathbb{F}}_q+u^2{\mathbb{F}}_q$

Abstract

Let p be an odd prime, where $\vartheta$ and m are positive integers. Let $\psi$ be a nonzero element of the finite field ${\mathbb{F}}_q$, where $q=p^m$, and $\mathfrak{R}={\mathbb{F}}_q+u{\mathbb{F}}_q+u^2{\mathbb{F}}_q(u^3=0)$. In this paper, we determine completely the symbol-triple distances of all $\psi$-constacyclic codes of length $p^{\vartheta }$ over $\mathfrak{R}$.

1. Introduction

High-density data storage systems are very popular nowadays due to their great advantages in simplifying and improving storage, saving money, and saving time. However, these systems still have restrictions on the combination of high density, rewriting capability, fast response, and long retention time. To overcome these restrictions in high-density data storage systems, a new metric, called symbol-pair metric, was introduced by Cassuto and Blaum in [1,2]. This metric is reasonable for channels for which outputs are overlapping pairs of symbols. After that, for the read channel output larger than 2, $b\ge 3$, Yaakobi et al. [3] introduced b-symbol metrics, which generalize the symbol-pair metric, and provided extensions of many results and code constructions. By putting $b=3$, we obtained the symbol-triple metric.

Consider R a finite commutative ring with unity. For a positive integer n, the number of nonzero entries of a codeword $\zeta =({\zeta }_0,{\zeta }_1,\dots ,{\zeta }_{n-1})\in R^n$ is the Hamming weight of $\zeta$, denoted by ${\mathtt{w}}_{\mathbf{H}}(\zeta )$. Let $\zeta$ and $\xi$ be two codewords, and their Hamming distance denoted by ${\mathtt{d}}_{\mathbf{H}}(\zeta ,\xi )$ is the number of their components which differ.

A codeword $\mathit{\alpha }=({\alpha }_0,{\alpha }_1,\dots ,{\alpha }_{n-1})$ is represented in symbol-triple read channels as

$$\begin{array}{c}\hfill \mathit{\pi }(\mathit{\alpha })=[({\alpha }_0,{\alpha }_1,{\alpha }_2),({\alpha }_1,{\alpha }_2,{\alpha }_3),\dots ,({\alpha }_{n-1},{\alpha }_0,{\alpha }_1)]\in {(R^3)}^n.\end{array}$$

The Hamming weight of the symbol-triple vector $\mathit{\pi }(\mathit{\alpha })$ is called the symbol-triple weight of the vector $\mathit{\alpha }$:

$$\begin{array}{cc}\hfill {\mathtt{w}}_{\mathbf{st}}(\mathit{\alpha })& ={\mathtt{w}}_{\mathbf{H}}(\mathit{\pi }(\mathit{\alpha }))\hfill \\ & =\left|\right\{0\le i\le n-1:({\alpha }_i,{\alpha }_{i+1},{\alpha }_{i+2})\ne (0,0,0),{\alpha }_n={\alpha }_0\left\}\right|.\hfill \end{array}$$

For two vectors $\mathit{\alpha }=({\alpha }_0,{\alpha }_1,\dots ,{\alpha }_{n-1})$ and $\mathit{\beta }=({\beta }_0,{\beta }_1,\dots ,{\beta }_{n-1})$ in $R^n$, the symbol-triple distance between them is:

$$\begin{array}{cc}\hfill {\mathtt{d}}_{\mathbf{st}}(\mathit{\alpha },\mathit{\beta })& ={\mathtt{d}}_{\mathbf{H}}(\mathit{\pi }(\mathit{\alpha }),\mathit{\pi }(\mathit{\beta }))\hfill \\ & =\left|\right\{0\le i\le n-1:({\alpha }_i,{\alpha }_{i+1},{\alpha }_{i+2})\ne ({\beta }_i,{\beta }_{i+1},{\beta }_{i+2})\left\}\right|.\hfill \end{array}$$

Then, for a code C, the symbol-triple distance is defined as

$${\mathtt{d}}_{\mathbf{st}}(C)=\min \{{\mathtt{d}}_{\mathbf{st}}(\mathit{\alpha },\mathit{\beta })\mid \mathit{\alpha },\mathit{\beta }\in C,\mathit{\alpha }\ne \mathit{\beta }\}.$$

For a linear code C, the symbol-triple distance and symbol-triple weight are coincided. They are defined as:

$${\mathtt{d}}_{\mathbf{st}}(C)=\min \{{\mathtt{w}}_{\mathbf{st}}(\mathit{c})\mid \mathit{c}\ne 0,\mathit{c}\in C\}.$$

Constacyclic codes are linear codes that have an important role in error-correcting codes theory, which can be viewed as a generalization of cyclic codes. They are preferred in engineering because of their rich algebraic structures that make them more practical, since they are efficiently encoded, and they provide efficient error detection and correction. Castagnoli et al. [4] and Van Lint [5] were the first to investigate constacyclic codes. For a unit $\varphi$ of R, $\varphi$-constacyclic codes of length n over R are in one-to-one correspondence with ideals of the polynomial ring $R\left[x\right]/⟨x^n-\varphi ⟩$. If n is divisible by the characteristic p of R, then we obtain the so-called repeated-root codes.

Let ${\mathbb{F}}_q$ be a finite field of $q=p^m$ elements, where p is a prime, m is a positive integer and let $\lambda \ge 2$ be an integer. Then, the ring $R={\mathbb{F}}_q\left[u\right]/⟨u^{\lambda }⟩$ is a finite commutative chain ring. Many authors [6,7,8] studied algebraic structures of constacyclic codes over R:

For $\lambda =2$, many authors studied them (see, e.g., [9,10,11,12,13,14,15,16,17,18]). In particular, for a prime power length, their structure and their symbol-pair distance were completely established in [12,19].

For $\lambda =3$, DNA cyclic codes were studied in [20], which are of deep importance for biology. In [21], Laaouine et al. described the structure of all $\psi$-constacyclic codes of length $p^{\vartheta }$ over $\mathfrak{R}={\mathbb{F}}_q+u{\mathbb{F}}_q+u^2{\mathbb{F}}_q$ for a nonzero element $\psi$ of ${\mathbb{F}}_q$ and classified them into eight types. Then, their Hamming distances have been computed by Dinh et al. [22]. In [23], Charkani et al. computed their symbol-pair distances.

Accordingly, we resolve to compute the symbol-triple distance of all $\psi$-constacyclic codes of length $p^{\vartheta }$ over $\mathfrak{R}$. After giving some preliminaries and notations in Section 2, Section 3 establishes their symbol-triple distance. Section 4 contains some examples by fixing some values for p and $\vartheta$. Finally, a conclusion is given in Section 5.

2. Some Preliminaries

From now on in this paper, ${\mathbb{F}}_q$ is the field of order q with $q=p^m$ where p is a prime positive integer, m is a positive integer, and we denote

$$\mathfrak{R}={\mathbb{F}}_q+u{\mathbb{F}}_q+u^2{\mathbb{F}}_q.$$

$\mathfrak{R}$ is a finite chain ring, $⟨u⟩=u\mathfrak{R}$ is its maximal ideal, and its invertible elements in are of the form: $\psi +\rho u+\chi u^2$ where $\psi ,\rho ,\chi \in {\mathbb{F}}_q$ and $\psi \ne 0$.

Linear codes $\mathfrak{C}$ of length n over $\mathfrak{R}$ are defined as $\mathfrak{R}$-submodules of ${\mathfrak{R}}^n$. For a unit $\varphi$ in $\mathfrak{R}$, $\varphi$-constacyclic codes $\mathfrak{C}$ of length n over $\mathfrak{R}$ are the linear codes of length n over $\mathfrak{R}$, which satisfy the following property: $(a_0,a_1,\dots ,a_{n-1})\in \mathfrak{C}$ implies that $(\varphi a_{n-1},a_0,a_1,\dots ,a_{n-2})\in \mathfrak{C}$. Then, in order to study them, it suffices to study ideals of the ring $\mathfrak{R}\left[x\right]/⟨x^n-\varphi ⟩$ (cf. [24,25]).

So, for a unit $\psi$ of ${\mathbb{F}}_q$, $\psi$-constacyclic codes of length $p^{\vartheta }$ over $\mathfrak{R}$ are exactly the ideals of

$${\mathfrak{R}}_{\psi }=\mathfrak{R}\left[x\right]/⟨x^{p^{\vartheta }}-\psi ⟩.$$

In [21], Laaouine et al. classified and determined their structure:

Theorem 1

(cf. [21]). The ring ${\mathfrak{R}}_{\psi }$ is local finite non-chain ring, its maximal ideal is $⟨u,x-\phi ⟩$, where $\phi \in {\mathfrak{F}}_q$ such that ${\phi }^{p^{\vartheta }}=\psi$. ψ-constacyclic codes of length $p^{\vartheta }$ over $\mathfrak{R}$ (i.e., ideals of the ring ${\mathfrak{R}}_{\psi }$) are

$\mathtt{Type}\mathtt{1}({\mathfrak{C}}_1):$

$$⟨0⟩,⟨1⟩.$$

$\mathtt{Type}\mathtt{2}({\mathfrak{C}}_2):$

$${\mathfrak{C}}_2=⟨u^2{(x-\phi )}^a⟩,$$

where $0\le a\le p^{\vartheta }-1$.

$\mathtt{Type}\mathtt{3}({\mathfrak{C}}_3):$

$${\mathfrak{C}}_3=⟨u{(x-\phi )}^b+u^2{(x-\phi )}^{t}f(x)⟩,$$

where $0\le \mathsf{A}\le b\le p^{\vartheta }-1$, $0\le t<\mathsf{A}$, either $f(x)$ is 0 or $f(x)$ is a unit in ${\mathfrak{R}}_{\psi }$. Here, $\mathsf{A}$ is the smallest integer such that $u^2{(x-\phi )}^{\mathsf{A}}\in {\mathfrak{C}}_3$.

$\mathtt{Type}\mathtt{4}({\mathfrak{C}}_4):$

$${\mathfrak{C}}_4=⟨u{(x-\phi )}^b+u^2{(x-\phi )}^{t}f(x),u^2{(x-\phi )}^c⟩,$$

where $0\le c<\mathsf{A}\le b\le p^{\vartheta }-1$, $0\le t<c$, either $f(x)$ is 0 or $f(x)$ is a unit in ${\mathfrak{R}}_{\psi }$, and $\mathsf{A}$ is the same as in $\mathtt{Type}\mathtt{3}$.

$\mathtt{Type}\mathtt{5}({\mathfrak{C}}_5):$

$${\mathfrak{C}}_5=⟨{(x-\phi )}^{\alpha }+u{(x-\phi )}^{t_1}{f}_1(x)+u^2{(x-\phi )}^{t_2}{f}_2(x)⟩,$$

where $0<\mathsf{C}\le \mathsf{B}\le \alpha \le p^{\vartheta }-1$, $0\le t_1<\mathsf{B}$, $0\le t_2<\mathsf{C}$ and $f_1(x),f_2(x)$ are 0 or are units in ${\mathfrak{R}}_{\psi }$. Here, $\mathsf{B}$ being the smallest integer such that $u{(x-\phi )}^{\mathsf{B}}+u^{2}g(x)\in {\mathfrak{C}}_5$, for some $g(x)\in {\mathfrak{R}}_{\psi }$ and $\mathsf{C}$ is the smallest integer satisfying $u^2{(x-\phi )}^{\mathsf{C}}\in {\mathcal{C}}_5$.

$\mathtt{Type}\mathtt{6}({\mathfrak{C}}_6):$

$${\mathfrak{C}}_6=⟨{(x-\phi )}^{\alpha }+u{(x-\phi )}^{t_1}{f}_1(x)+u^2{(x-\phi )}^{t_2}{f}_2(x),u^2{(x-\phi )}^{\gamma }⟩,$$

where $0\le \gamma <\mathsf{C}\le \mathsf{B}\le \alpha \le p^{\vartheta }-1$, $0\le t_1<\mathsf{B}$, $0\le t_2<\gamma$ and $f_1(x),f_2(x)$ are 0 or are units in ${\mathfrak{R}}_{\psi }$. Here, $\mathsf{B}$, $\mathsf{C}$ are the same as in $\mathtt{Type}\mathtt{5}$.

$\mathtt{Type}\mathtt{7}({\mathfrak{C}}_7):$

$${\mathfrak{C}}_7=⟨{(x-\phi )}^{\alpha }+u{(x-\phi )}^{t_1}{f}_1(x)+u^2{(x-\phi )}^{t_2}{f}_2(x),u{(x-\phi )}^{\beta }+u^2{(x-\phi )}^{t_3}{f}_3(x)⟩,$$

where $0\le \mathsf{D}\le \beta <\mathsf{B}\le \alpha \le p^{\vartheta }-1$, $0\le t_1<\beta$, $0\le t_2<\mathsf{D}$, $0\le t_3<\mathsf{D}$ and $f_1(x),f_2(x),f_3(x)$ are 0 or are units in ${\mathfrak{R}}_{\psi }$. Here, $\mathsf{D}$ is the smallest integer such that $u^2{(x-\phi )}^{\mathsf{D}}\in {\mathfrak{C}}_7$ and $\mathsf{B}$ is the same as in $\mathtt{Type}\mathtt{5}$.

$\mathtt{Type}\mathtt{8}({\mathfrak{C}}_8):$

$$\begin{array}{c}{\mathfrak{C}}_8{=⟨(x-\phi )}^{\alpha }+u{(x-\phi )}^{t_1}{f}_1(x)+u^2{(x-\phi )}^{t_2}{f}_2(x),u{(x-\phi )}^{\beta }+u^2{(x-\phi )}^{t_3}{f}_3(x),u^2{(x-\phi )}^{\gamma }⟩,\hfill \end{array}$$

where $0\le \gamma <\mathsf{D}\le \beta <\mathsf{B}\le \alpha \le p^{\vartheta }-1$, $0\le t_1<\beta$, $0\le t_2<\gamma$, $0\le t_3<\gamma$ and $f_1(x),f_2(x),f_3(x)$ are 0 or are units in ${\mathfrak{R}}_{\psi }$. Here, $\mathsf{B}$ as in $\mathtt{Type}\mathtt{5}$ and $\mathsf{D}$ is the same as in $\mathtt{Type}\mathtt{7}$.

Proposition 1

(cf. [21]). We have

$$\mathsf{A}=\left\{\begin{array}{cc}b,\hfill & iff(x)=0,\hfill \\ min\{b,p^{\vartheta }-b+t\},\hfill & iff(x)\ne 0.\hfill \end{array}\right.,$$

$$\mathsf{B}=\left\{\begin{array}{cc}\alpha ,\hfill & if{f}_1(x)=0,\hfill \\ min\{\alpha ,p^{\vartheta }-\alpha +t_1\},\hfill & if{f}_1(x)\ne 0.\hfill \end{array}\right.,$$

$$\mathsf{C}=\left\{\begin{array}{cc}\alpha ,\hfill & if{f}_1(x)=f_2(x)=0,\hfill \\ min\{\alpha ,p^{\vartheta }-\alpha +t_2\},\hfill & if{f}_1(x)=0and{f}_2(x)\ne 0,\hfill \\ min\{\alpha ,p^{\vartheta }-\alpha +t_1\},\hfill & if{f}_1(x)\ne 0.\hfill \end{array}\right.$$

$$\mathsf{D}=\left\{\begin{array}{cc}\beta ,\hfill & if{f}_1(x)=f_2(x)=f_3(x)=0\hfill \\ & or{f}_1(x)\ne 0and{f}_3(x)=0,\hfill \\ min\{\beta ,p^{\vartheta }-\alpha +t_2\},\hfill & if{f}_1(x)=f_3(x)=0,f_2(x)\ne 0,\hfill \\ min\{\beta ,p^{\vartheta }-\beta +t_3\},\hfill & if{f}_1(x)=f_2(x)=0,f_3(x)\ne 0\hfill \\ & or{f}_1(x)\ne 0and{f}_3(x)\ne 0,\hfill \\ min\{\beta ,p^{\vartheta }-\alpha +t_2,p^{\vartheta }-\beta +t_3\},\hfill & if{f}_1(x)=0,f_2(x)\ne 0,f_3(x)\ne 0.\hfill \end{array}\right.$$

3. Symbol-Triple Distance

This section will be devoted to determining the symbol-triple distances of all types of $\psi$-constacyclic codes of length $p^{\vartheta }$ over $\mathfrak{R}$. Therefore, we recall the following result.

Theorem 2

(cf. [26]). Let $\mathfrak{C}$ be a ψ-constacyclic code of length $p^{\vartheta }$ over ${\mathbb{F}}_q$. Then, $\mathfrak{C}=⟨{(x-\phi )}^{\iota }⟩$ $\subseteq {\mathbb{F}}_q\left[x\right]/⟨x^{p^{\vartheta }}-\psi ⟩$, $0\le \iota \le p^{\vartheta }$, and its symbol-triple distance ${\mathbf{\Delta }}_{\mathit{\iota }}:=d_{\mathit{st}}(\mathfrak{C})$ is determined by:

$${\mathbf{\Delta }}_{\mathit{\iota }}=\left\{\begin{array}{cc}3,\hfill & if\iota =0;\hfill \\ 7,\hfill & if\iota =4and\mathit{where}p=3,s=2;\hfill \\ 4p^{\eta },\hfill & if\iota =p^{\vartheta }-p^{\vartheta -\eta }+1,\mathit{where}0\le \eta \le \vartheta -2;\hfill \\ 5p^{\eta },\hfill & if\iota =p^{\vartheta }-p^{\vartheta -\eta }+2,\mathit{where}0\le \eta \le \vartheta -2;\hfill \\ 6p^{\eta },\hfill & if{p}^{\vartheta }-p^{\vartheta -\eta }+3\le \iota \le p^{\vartheta }-p^{\vartheta -\eta }+p^{\vartheta -\eta -1},\hfill \\ \hfill & \mathit{where}0\le \eta \le \vartheta -2;\hfill \\ 8p^{\eta },\hfill & if\iota =p^{\vartheta }-p^{\vartheta -\eta }+p^{\vartheta -\eta -1}+1,\hfill \\ \hfill & \mathit{where}0\le \eta \le \vartheta -2ands\ge 3;\hfill \\ 9p^{\eta },\hfill & if{p}^{\vartheta }-pr+r+2\le \iota \le p^{\vartheta }-pr+2r,\hfill \\ \hfill & \mathit{where}r=p^{\vartheta -\eta -1},0\le \eta \le \vartheta -2andp\ge 5;\hfill \\ 3(\tau +2)p^{\eta },\hfill & if{p}^{\vartheta }-pr+\tau r+1\le \iota \le p^{\vartheta }-pr+(\tau +1)r,\hfill \\ & \mathit{where}r=p^{\vartheta -\eta -1},0\le \eta \le \vartheta -2,and2\le \tau \le p-2;\hfill \\ (\varrho +3)p^{\vartheta -1},\hfill & if\iota =p^{\vartheta }-p+\varrho ,\mathit{where}1\le \varrho \le p-3;\hfill \\ p^{\vartheta },\hfill & if{p}^{\vartheta }-2\le \iota \le p^{\vartheta }-1;\hfill \\ 0,\hfill & if\iota =p^{\vartheta }.\hfill \end{array}\right.$$

For a code $\mathfrak{C}$ over $\mathfrak{R}$, the symbol-triple distance is denoted by ${\mathbf{\Delta }}_{\mathit{\iota }}:={\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_{\mathbb{F}})$.

Now, for each type of $\psi$-constacyclic codes of length $p^{\vartheta }$ over $\mathfrak{R}$, we compute the symbol-triple distance one by one.

$\mathtt{Type}\mathtt{1}$ consists only of the trivial ideals $⟨0⟩$, $⟨1⟩$. Hence, they have symbol-triple distances of 0 and 3, respectively.

The codewords of a code ${\mathfrak{C}}_2=⟨u^2{(x-\phi )}^a⟩$ of $\mathtt{Type}\mathtt{2}$ with $0\le a\le p^s-1$ are exactly those of the $\psi$-constacyclic codes $⟨{(x-\phi )}^a⟩$ in ${\mathbb{F}}_q\left[x\right]/⟨x^{p^{\vartheta }}-\psi ⟩$ multiplied by $u^2$. Thus, we obtain ${\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_2)={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^a⟩}_{\mathbb{F}})$ and Theorem 2 gives it.

Theorem 3.

For a ψ-constacyclic code ${\mathfrak{C}}_2=⟨u^2{(x-\phi )}^a⟩$, $0\le a\le p^{\vartheta }-1$ of $\mathtt{Type}\mathtt{2}$. The symbol-triple distance $d_{\mathit{st}}({\mathfrak{C}}_2)$ is given by

$$d_{\mathit{st}}({\mathfrak{C}}_2)={\mathbf{\Delta }}_{\mathit{a}},\mathit{where}\mathbf{0}\le a\le p^{\vartheta }-1.$$

Now, we are going to calculate the symbol-triple distances of codes of $\mathtt{Type}\mathtt{3}$, $\mathtt{4}$, $\mathtt{5}$, $\mathtt{6}$, $\mathtt{7}$ and $\mathtt{8}$. To do this, notice that

$${\mathtt{wt}}_{\mathbf{st}}(\mathsf{\Gamma }(x))\ge {\mathtt{wt}}_{\mathbf{st}}(u\mathsf{\Gamma }(x)),$$

(1)

where $\mathsf{\Gamma }(x)\in {\mathfrak{R}}_{\psi }$.

The symbol-triple distance $\psi$-constacyclic codes of $\mathtt{Type}\mathtt{3}$ can be calculated by the next theorem:

Theorem 4.

Let ${\mathfrak{C}}_3=⟨u{(x-\phi )}^b+u^2{(x-\phi )}^{t}f(x)⟩$ $\subseteq {\mathfrak{R}}_{\psi }$ be of $\mathtt{Type}\mathtt{3}$. The symbol-triple distance $d_{\mathit{st}}({\mathfrak{C}}_3)$ of ${\mathfrak{C}}_3$ is given by

$$d_{\mathit{st}}({\mathfrak{C}}_3)={\mathbf{\Delta }}_{\mathsf{A}},where\mathbf{0}\le \mathsf{A}\le p^{\vartheta }-1.$$

Proof. 

Let ${\rm Y}(x)$ be an arbitrary nonzero element of ${\mathfrak{C}}_3$. That means that there exists ${{\rm Y}}_0(x),{{\rm Y}}_u(x),$

• ${{\rm Y}}_{u^2}(x)\in {\mathbb{F}}_q\left[x\right]$ verifying

$${\rm Y}(x)=[{{\rm Y}}_0(x)+u{{\rm Y}}_u(x)+u^2{{\rm Y}}_{u^2}(x)][u{(x-\phi )}^b+u^2{(x-\phi )}^{t}f(x)].$$

Thus,

$$u{\rm Y}(x)=u^2{{\rm Y}}_0(x){(x-\phi )}^b.$$

By (1), we obtain that

$$\begin{array}{cc}\hfill {\mathtt{wt}}_{\mathbf{st}}({\rm Y}(x))& \ge {\mathtt{wt}}_{\mathbf{st}}(u{\rm Y}(x))\hfill \\ & ={\mathtt{wt}}_{\mathbf{st}}(u^2{{\rm Y}}_0(x)){(x-\phi )}^b)\hfill \\ & \ge {\mathtt{d}}_{\mathbf{st}}(⟨u^2{(x-\phi )}^b⟩)\hfill \\ & ={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^b⟩}_{\mathbb{F}}).\hfill \end{array}$$

Since $⟨{(x-\phi )}^b⟩\subseteq ⟨{(x-\phi )}^{\mathsf{A}}⟩$, we have

$${\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^b⟩}_{\mathbb{F}})\ge {\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\mathsf{A}}⟩}_{\mathbb{F}}).$$

It follows that ${\mathtt{wt}}_{\mathbf{st}}({\rm Y}(x))\ge {\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\mathsf{A}}⟩}_{\mathbb{F}})$ for any nonzero element ${\rm Y}(x)$ of ${\mathfrak{C}}_3$. Then,

$${\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_3)\ge {\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\mathsf{A}}⟩}_{\mathbb{F}}).$$

(2)

Moreover, we have

$$⟨u^2{(x-\phi )}^{\mathsf{A}}⟩\subseteq {\mathfrak{C}}_3,$$

it follows that

$${\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\mathsf{A}}⟩}_{\mathbb{F}})={\mathtt{d}}_{\mathbf{st}}(⟨u^2{(x-\phi )}^{\mathsf{A}}⟩)\ge {\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_3).$$

(3)

Then, by combining (2) and (3), we obtain

$$\begin{array}{cc}\hfill {\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_3)& ={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\mathsf{A}}⟩}_{\mathbb{F}})\hfill \\ & ={\mathbf{\Delta }}_{\mathsf{A}},\mathrm{where}\mathbf{0}\le \mathsf{A}\le p^{\vartheta }-1.\hfill \end{array}$$

This proves the theorem. □

Now, let us show the symbol-triple distance for $\mathtt{Type}\mathtt{4}$.

Theorem 5.

For ψ-constacyclic code ${\mathfrak{C}}_4=⟨u{(x-\phi )}^b+u^2{(x-\phi )}^{t}f(x),u^2{(x-\phi )}^c⟩$ of $\mathtt{Type}\mathtt{4}$, the symbol-triple distance $d_{\mathit{st}}({\mathfrak{C}}_4)$ is given by

$$d_{\mathit{st}}({\mathfrak{C}}_4)={\mathbf{\Delta }}_c,\mathit{where}\mathbf{0}\le c<p^{\vartheta }-1.$$

Proof. 

At first, since $u^2{(x-\phi )}^c\in {\mathfrak{C}}_4$, we obtain

$${\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_4)\le {\mathtt{d}}_{\mathbf{st}}(⟨u^2{(x-\phi )}^c⟩)={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^c⟩}_{\mathbb{F}}).$$

To prove that ${\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^c⟩}_{\mathbb{F}})\le {\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_4)$, let ${\rm Y}(x)\in {\mathfrak{C}}_4\backslash ⟨u^2{(x-\phi )}^c⟩$, we should prove that ${\mathtt{wt}}_{\mathbf{st}}({\rm Y}(x))\ge {\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^c⟩}_{\mathbb{F}}).$

By (1), we obtain that

$$\begin{array}{cc}\hfill {\mathtt{wt}}_{\mathbf{st}}({\rm Y}(x))& \ge {\mathtt{wt}}_{\mathbf{st}}(u{\rm Y}(x))\hfill \\ & \ge {\mathtt{d}}_{\mathbf{st}}(⟨u^2{(x-\phi )}^b⟩)\hfill \\ & ={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^b⟩}_{\mathbb{F}})\hfill \\ & \ge {\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^c⟩}_{\mathbb{F}})(\mathrm{because}⟨{(x-\phi )}^b⟩\subseteq ⟨{(x-\phi )}^c⟩).\hfill \end{array}$$

Hence, ${\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^c⟩}_{\mathbb{F}})\le {\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_4)$, forcing

$$\begin{array}{cc}\hfill {\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_4)& ={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^c⟩}_{\mathbb{F}})\hfill \\ & ={\mathbf{\Delta }}_c,\mathrm{where}\mathbf{0}\le c<p^{\vartheta }-1.\hfill \end{array}$$

This proves the theorem. □

Now, we calculate the symbol-triple distance for $\mathtt{Type}\mathtt{5}$:

Theorem 6.

For a ψ-constacyclic code ${\mathfrak{C}}_5=⟨{(x-\phi )}^{\alpha }+u{(x-\phi )}^{t_1}{f}_1(x)+u^2{(x-\phi )}^{t_2}{f}_2(x)⟩$ $\subseteq {\mathfrak{R}}_{\psi }$ of $\mathtt{Type}\mathtt{5}$, the symbol-triple distance $d_{\mathit{st}}({\mathfrak{C}}_5)$ is given by

$$d_{\mathit{st}}({\mathfrak{C}}_5)={\mathbf{\Delta }}_{\mathsf{C}},\mathit{where}\mathbf{0}<\mathsf{C}\le p^{\vartheta }-1.$$

Proof. 

Let ${\mathfrak{C}}_5=⟨{(x-\phi )}^{\alpha }+u{(x-\phi )}^{t_1}{f}_1(x)+u^2{(x-\phi )}^{t_2}{f}_2(x)⟩$ be of $\mathtt{Type}\mathtt{5}$. Now, for each nonzero ${\rm Y}(x)\in {\mathfrak{C}}_5$, there exist ${{\rm Y}}_0(x),{{\rm Y}}_u(x),{{\rm Y}}_{u^2}(x)\in {\mathbb{F}}_q\left[x\right]$ such that

$${\rm Y}(x)=[{{\rm Y}}_0(x)+u{{\rm Y}}_u(x)+u^2{{\rm Y}}_{u^2}(x)][{(x-\phi )}^{\alpha }+u{(x-\phi )}^{t_1}{f}_1(x)+u^2{(x-\phi )}^{t_2}{f}_2(x)].$$

Thus,

$$u^2{\rm Y}(x)=u^2{{\rm Y}}_0(x){(x-\phi )}^{\alpha }.$$

By (1), we see that

$$\begin{array}{cc}\hfill {\mathtt{wt}}_{\mathbf{st}}({\rm Y}(x))& \ge {\mathtt{wt}}_{\mathbf{st}}(u^2{\rm Y}(x))\hfill \\ & ={\mathtt{wt}}_{\mathbf{st}}(u^2{{\rm Y}}_0(x){(x-\phi )}^{\alpha })\hfill \\ & \ge {\mathtt{d}}_{\mathbf{st}}(⟨u^2{(x-\phi )}^{\alpha }⟩)\hfill \\ & ={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\alpha }⟩}_{\mathbb{F}}).\hfill \end{array}$$

The fact that $⟨{(x-\phi )}^{\alpha }⟩\subseteq ⟨{(x-\phi )}^{\mathsf{C}}⟩$ follows that

$${\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\alpha }⟩}_{\mathbb{F}})\ge {\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\mathsf{C}}⟩}_{\mathbb{F}}).$$

This implies that ${\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\mathsf{C}}⟩}_{\mathbb{F}})\le {\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_5)$.

Moreover, we have that

$$u^2{(x-\phi )}^{\mathsf{C}}\in {\mathfrak{C}}_5,$$

then ${\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_5)\le {\mathtt{d}}_{\mathbf{st}}(⟨u^2{(x-\phi )}^{\mathsf{C}}⟩)={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\mathsf{C}}⟩}_{\mathbb{F}})$ and we obtain

$$\begin{array}{cc}\hfill {\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_5)& ={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\mathsf{C}}⟩}_{\mathbb{F}})\hfill \\ & ={\mathbf{\Delta }}_{\mathsf{C}},\mathrm{where}\mathbf{0}<\mathsf{C}\le p^{\vartheta }-1.\hfill \end{array}$$

This proves the theorem. □

The symbol-triple distance for $\mathtt{Type}\mathtt{6}$ can be established by the next theorem:

Theorem 7.

For a ψ-constacyclic code${\mathfrak{C}}_6{=⟨(x-\phi )}^{\alpha }+u{(x-\phi )}^{t_1}{f}_1(x)+u^2{(x-\phi )}^{t_2}{f}_2(x),$$u^2{(x-\phi )}^{\gamma }⟩$$\subseteq {\mathfrak{R}}_{\psi }$of$\mathtt{Type}\mathtt{6}$, the symbol-triple distance$d_{\mathit{st}}({\mathfrak{C}}_6)$is given by

$$d_{\mathit{st}}({\mathfrak{C}}_6)={\mathbf{\Delta }}_{\gamma },where\mathbf{0}\le \gamma <p^{\vartheta }-1.$$

Proof. 

At first, since $u^2{(x-\phi )}^{\gamma }\in {\mathfrak{C}}_6$, we obtain

$${\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_6)\le {\mathtt{d}}_{\mathbf{st}}(⟨u^2{(x-\phi )}^{\gamma }⟩)={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\gamma }⟩}_{\mathbb{F}}).$$

Now, let ${\rm Y}(x)\in {\mathfrak{C}}_6\backslash ⟨u^2{(x-\phi )}^{\gamma }⟩$. Thus, by (1), we find that

$$\begin{array}{cc}\hfill {\mathtt{wt}}_{\mathbf{st}}({\rm Y}(x))& \ge {\mathtt{wt}}_{\mathbf{st}}(u^2{\rm Y}(x))\hfill \\ & \ge {\mathtt{d}}_{\mathbf{st}}(⟨u^2{(x-\phi )}^{\alpha }⟩)\hfill \\ & ={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\alpha }⟩}_{\mathbb{F}})\hfill \\ & \ge {\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\gamma }⟩}_{\mathbb{F}})(\mathrm{because}⟨{(x-\phi )}^{\alpha }⟩\subseteq ⟨{(x-\phi )}^{\gamma }⟩).\hfill \end{array}$$

Hence, ${\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\gamma }⟩}_{\mathbb{F}})\le {\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_6)$, forcing

$$\begin{array}{cc}\hfill {\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_6)& ={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\gamma }⟩}_{\mathbb{F}})\hfill \\ & ={\mathbf{\Delta }}_{\gamma },\mathrm{where}\mathbf{0}\le \gamma <p^{\vartheta }-1.\hfill \end{array}$$

This proves the theorem. □

Now, we show the symbol-triple distance for $\mathtt{Type}\mathtt{7}$.

Theorem 8.

For a ψ-constacyclic code ${\mathfrak{C}}_7{=⟨(x-\phi )}^{\alpha }+u{(x-\phi )}^{t_1}{f}_1(x)+u^2{(x-\phi )}^{t_2}{f}_2(x),$$u{(x-\phi )}^{\beta }+u^2{(x-\phi )}^{t_3}{f}_3(x)⟩$ of $\mathtt{Type}\mathtt{7}$, the symbol-triple distance $d_{\mathit{st}}({\mathfrak{C}}_7)$ is given by

$$d_{\mathbf{st}}({\mathfrak{C}}_7)={\mathbf{\Delta }}_{\mathsf{D}},\mathit{where}\mathbf{0}\le \mathsf{D}<p^{\vartheta }-1.$$

Proof. 

At first, since $u^2{(x-\phi )}^{\mathsf{D}}\in {\mathfrak{C}}_7$, we obtain

$${\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_7)\le {\mathtt{d}}_{\mathbf{st}}(⟨u^2{(x-\phi )}^{\mathsf{D}}⟩)={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\mathsf{D}}⟩}_{\mathbb{F}}).$$

Now, let ${\rm Y}(x)\in {\mathfrak{C}}_7$. Then, we distinguish two different cases.

$\ast \mathsf{Case}\mathsf{1}$: if ${\rm Y}(x)\in ⟨u⟩$, then by (1), we obtain

$$\begin{array}{cc}\hfill {\mathtt{wt}}_{\mathbf{st}}({\rm Y}(x))& \ge {\mathtt{wt}}_{\mathbf{st}}(u{\rm Y}(x))\hfill \\ & \ge {\mathtt{d}}_{\mathbf{st}}(⟨u^2{(x-\phi )}^{\beta }⟩)\hfill \\ & ={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\beta }⟩}_{\mathbb{F}}).\hfill \end{array}$$

$\ast \mathsf{Case}\mathsf{2}$: if ${\rm Y}(x)\notin ⟨u⟩$, then by (1), we obtain

$$\begin{array}{cc}\hfill {\mathtt{wt}}_{\mathbf{st}}({\rm Y}(x))& \ge {\mathtt{wt}}_{\mathbf{st}}(u^2{\rm Y}(x))\hfill \\ & \ge {\mathtt{d}}_{\mathbf{st}}(⟨u^2{(x-\phi )}^{\alpha }⟩)\hfill \\ & ={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\alpha }⟩}_{\mathbb{F}}).\hfill \end{array}$$

We have $⟨{(x-\phi )}^{\alpha }⟩\subseteq ⟨{(x-\phi )}^{\beta }⟩\subseteq ⟨{(x-\phi )}^{\mathsf{D}}⟩$, then, we obtain

$${\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\alpha }⟩}_{\mathbb{F}})\ge {\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\beta }⟩}_{\mathbb{F}})\ge {\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\mathsf{D}}⟩}_{\mathbb{F}}).$$

Therefore, ${\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\mathsf{D}}⟩}_{\mathbb{F}})\le {\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_7)$, forcing

$$\begin{array}{cc}\hfill {\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_7)& ={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\mathsf{D}}⟩}_{\mathbb{F}})\hfill \\ & ={\mathbf{\Delta }}_{\mathsf{D}},\mathrm{where}\mathbf{0}\le \mathsf{D}<p^{\vartheta }-1.\hfill \end{array}$$

This completes the proof. □

Finally, we give the symbol-triple distance for $\mathtt{Type}\mathtt{8}$.

Theorem 9.

For a ψ-constacyclic code ${\mathfrak{C}}_8{=⟨(x-\phi )}^{\alpha }+u{(x-\phi )}^{t_1}{f}_1(x)+u^2{(x-\phi )}^{t_2}{f}_2(x),$$u{(x-\phi )}^{\beta }+u^2{(x-\phi )}^{t_3}{f}_3(x),u^2{(x-\phi )}^{\gamma }⟩$$\subseteq {\mathfrak{R}}_{\psi }$ of $\mathtt{Type}\mathtt{8}$, the symbol-triple distance $d_{\mathit{st}}({\mathfrak{C}}_8)$ is given by

$$d_{\mathit{st}}({\mathfrak{C}}_8)={\mathbf{\Delta }}_{\gamma },\mathit{where}\mathbf{0}\le \gamma <p^{\vartheta }-2.$$

Proof. 

Let ${\mathfrak{C}}_8{=⟨(x-\phi )}^{\alpha }+u{(x-\phi )}^{t_1}{f}_1(x)+u^2{(x-\phi )}^{t_2}{f}_2(x),u{(x-\phi )}^{\beta }+u^2{(x-\phi )}^{t_3}{f}_3$ $(x),u^2{(x-\phi )}^{\gamma }⟩$ be of $\mathtt{Type}\mathtt{8}$. In addition, consider an arbitrary polynomial ${\rm Y}(x)\in {\mathfrak{C}}_8\backslash ⟨u^2{(x-\phi )}^{\gamma }⟩$. Now, we distinguish two cases:

$\ast \mathsf{Case}\mathsf{1}$: if ${\rm Y}(x)\in ⟨u⟩$ then by (1), we obtain

$$\begin{array}{cc}\hfill {\mathtt{wt}}_{\mathbf{st}}({\rm Y}(x))& \ge {\mathtt{wt}}_{\mathbf{st}}(u{\rm Y}(x))\hfill \\ & \ge {\mathtt{d}}_{\mathbf{st}}(⟨u^2{(x-\phi )}^{\beta }⟩)\hfill \\ & ={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\beta }⟩}_{\mathbb{F}})\hfill \\ & \ge {\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\gamma }⟩}_{\mathbb{F}})(\mathrm{because}⟨{(x-\phi )}^{\beta }⟩\subseteq ⟨{(x-\phi )}^{\gamma }⟩).\hfill \end{array}$$

$\ast \mathsf{Case}\mathsf{2}$: if ${\rm Y}(x)\notin ⟨u⟩$, then by (1), we obtain

$$\begin{array}{cc}\hfill {\mathtt{wt}}_{\mathbf{st}}({\rm Y}(x))& \ge {\mathtt{wt}}_{\mathbf{st}}(u^2{\rm Y}(x))\hfill \\ & \ge {\mathtt{d}}_{\mathbf{st}}(⟨u^2{(x-\phi )}^{\alpha }⟩)\hfill \\ & ={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\alpha }⟩}_{\mathbb{F}})\hfill \\ & \ge {\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\gamma }⟩}_{\mathbb{F}})(\mathrm{because}⟨{(x-\phi )}^{\alpha }⟩\subseteq ⟨{(x-\phi )}^{\gamma }⟩).\hfill \end{array}$$

This implies that ${\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\gamma }⟩}_{\mathbb{F}})\le {\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_8)$.

Moreover, we have that

$$⟨u^2{(x-\phi )}^{\gamma }⟩\subseteq {\mathfrak{C}}_8,$$

then ${\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_8)\le {\mathtt{d}}_{\mathbf{st}}(⟨u^2{(x-\phi )}^{\gamma }⟩)={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\gamma }⟩}_{\mathbb{F}})$ and we obtain

$$\begin{array}{cc}\hfill {\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_8)& ={\mathtt{d}}_{\mathbf{st}}({⟨{(x-\phi )}^{\gamma }⟩}_{\mathbb{F}})\hfill \\ & ={\mathbf{\Delta }}_{\gamma },\mathrm{where}\mathbf{0}\le \gamma <p^{\vartheta }-2.\hfill \end{array}$$

This completes the proof. □

4. Examples

This section is devoted to show some examples of symbol-triple distances of constacyclic codes of length $p^{\vartheta }$ over ${\mathbb{F}}_q+u{\mathbb{F}}_q+u^2{\mathbb{F}}_q(u^3=0)$.

Example 1.

Consider the ring $\mathfrak{R}={\mathbb{F}}_7+u{\mathbb{F}}_7+u^2{\mathbb{F}}_7$, with $p=7$, $m=1$ and $\psi \in {\mathbb{F}}_7^{*}$. There are eight types of generators of ψ-constacyclic codes of length 49 over $\mathfrak{R}$. Then, for each type of generator, we give the symbol-triple distances:

▹

$\mathtt{Type}\mathtt{1}$$({\mathfrak{C}}_1)$: $⟨0⟩$, $⟨1⟩$,

$$d_{\mathit{st}}(⟨0⟩)=0,d_{\mathit{st}}(⟨1⟩)=3.$$

▹

$\mathtt{Type}\mathtt{2}$$({\mathfrak{C}}_2)$: ${\mathfrak{C}}_2=⟨u^2{(x-\psi )}^a⟩$, $0\le a\le 48$; then, the symbol-triple distance, $d_{\mathit{st}}({\mathfrak{C}}_2)={\mathbf{\Delta }}_{\mathit{a}}$, is presented in Table 1.

Table 1. Symbol-triple distance of $\mathbf{\psi }$-constacyclic codes over ${\mathbb{F}}_{\mathbf{7}}+\mathit{u}{\mathbb{F}}_{\mathbf{7}}+{\mathit{u}}^{\mathbf{2}}{\mathbb{F}}_{\mathbf{7}}$ of $\mathtt{Type}\mathtt{2}$.

Range of a	${\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_2)$
$a=0$	3
$a=1$	4
$a=2$	5
$3\le a\le 7$	6
$a=8$	8
$9\le a\le 14$	9
$15\le a\le 21$	12
$22\le a\le 28$	15
$29\le a\le 35$	18
$36\le a\le 42$	21
$a=43$	28
$a=44$	35
$a=45$	42
$46\le a\le 48$	49

▹

$\mathtt{Type}\mathtt{3}$$({\mathfrak{C}}_3)$: ${\mathfrak{C}}_3=⟨u{(x-\psi )}^b+u^2{(x-\psi )}^{t}f(x)⟩$, where $f(x)$ is 0 or $f(x)$ is a unit in $\mathfrak{R}\left[x\right]/⟨x^{49}-\psi ⟩$, $0\le b\le 48$, $0\le t<b$, and then the symbol-triple distance, $d_{\mathit{st}}({\mathfrak{C}}_3)={\mathbf{\Delta }}_{\mathsf{A}},\mathit{where}\mathbf{0}\le \mathsf{A}\le 48$, is presented in Table 2.

Table 2. Symbol-triple distance of $\mathbf{\psi }$-constacyclic codes over ${\mathbb{F}}_{\mathbf{7}}+\mathit{u}{\mathbb{F}}_{\mathbf{7}}+{\mathit{u}}^{\mathbf{2}}{\mathbb{F}}_{\mathbf{7}}$ of $\mathtt{Type}\mathtt{3}$.

Range of $\mathsf{A}$	${\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_3)$
$\mathsf{A}=0$	3
$\mathsf{A}=1$	4
$\mathsf{A}=2$	5
$3\le \mathsf{A}\le 7$	6
$\mathsf{A}=8$	8
$9\le \mathsf{A}\le 14$	9
$15\le \mathsf{A}\le 21$	12
$22\le \mathsf{A}\le 28$	15
$29\le \mathsf{A}\le 35$	18
$36\le \mathsf{A}\le 42$	21
$\mathsf{A}=43$	28
$\mathsf{A}=44$	35
$\mathsf{A}=45$	42
$46\le \mathsf{A}\le 48$	49

▹

$\mathtt{Type}\mathtt{4}$$({\mathfrak{C}}_4)$: ${\mathfrak{C}}_4=⟨u{(x-\psi )}^b+u^2{(x-\psi )}^{t}f(x),u^2{(x-\psi )}^c⟩$, where $f(x)$ is a unit in $\mathfrak{R}\left[x\right]/⟨x^{49}-\psi ⟩$ or 0, $0\le c<b\le 48$, $0\le t<c$, then, the symbol-triple distance, $d_{\mathit{st}}({\mathfrak{C}}_4)={\mathbf{\Delta }}_c,\mathit{where}\mathbf{0}\le c\le 47$, is presented in Table 3.

Table 3. Symbol-triple distance of $\mathbf{\psi }$-constacyclic codes over ${\mathbb{F}}_{\mathbf{7}}+\mathit{u}{\mathbb{F}}_{\mathbf{7}}+{\mathit{u}}^{\mathbf{2}}{\mathbb{F}}_{\mathbf{7}}$ of $\mathtt{Type}\mathtt{4}$.

Range of c	${\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_4)$
$c=0$	3
$c=1$	4
$c=2$	5
$3\le c\le 7$	6
$c=8$	8
$9\le c\le 14$	9
$15\le c\le 21$	12
$22\le c\le 28$	15
$29\le c\le 35$	18
$36\le c\le 42$	21
$c=43$	28
$c=44$	35
$c=45$	42
$46\le c\le 47$	49

▹

$\mathtt{Type}\mathtt{5}$$({\mathfrak{C}}_5)$: ${\mathfrak{C}}_5=⟨{(x-\psi )}^{\alpha }+u{(x-\psi )}^{t_1}{f}_1(x)+u^2{(x-\psi )}^{t_2}{f}_2(x)⟩$, where $f_1(x),f_2(x)$ are 0 or are units in $\mathfrak{R}\left[x\right]/⟨x^{49}-\psi ⟩$, $1\le \alpha \le 48$, $0\le t_1<\alpha$, $0\le t_2<\alpha$. Then, the symbol-triple distance, $d_{\mathit{st}}({\mathfrak{C}}_5)={\mathbf{\Delta }}_{\mathsf{C}},\mathit{where}\mathbf{1}\le \mathsf{C}\le 48$, is presented in Table 4.

Table 4. Symbol-triple distance of $\mathbf{\psi }$-constacyclic codes over ${\mathbb{F}}_{\mathbf{7}}+\mathit{u}{\mathbb{F}}_{\mathbf{7}}+{\mathit{u}}^{\mathbf{2}}{\mathbb{F}}_{\mathbf{7}}$ of $\mathtt{Type}\mathtt{5}$.

Range of $\mathsf{C}$	${\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_5)$
$\mathsf{C}=1$	4
$\mathsf{C}=2$	5
$3\le \mathsf{C}\le 7$	6
$\mathsf{C}=8$	8
$9\le \mathsf{C}\le 14$	9
$15\le \mathsf{C}\le 21$	12
$22\le \mathsf{C}\le 28$	15
$29\le \mathsf{C}\le 35$	18
$36\le \mathsf{C}\le 42$	21
$\mathsf{C}=43$	28
$\mathsf{C}=44$	35
$\mathsf{C}=45$	42
$46\le \mathsf{C}\le 48$	49

▹

$\mathtt{Type}\mathtt{6}$$({\mathfrak{C}}_6)$: ${\mathfrak{C}}_6=⟨{(x-\psi )}^{\alpha }+u{(x-\psi )}^{t_1}{f}_1(x)+u^2{(x-\psi )}^{t_2}{f}_2(x),u^2{(x-\psi )}^{\gamma }⟩$, where $f_1(x),f_2(x)$ are 0 or are units in $\mathfrak{R}\left[x\right]/⟨x^{49}-\psi ⟩$, $0\le \gamma <\alpha \le 48$, $0\le t_1<\alpha$, $0\le t_2<\gamma$, then, the symbol-triple distance, $d_{\mathit{st}}({\mathfrak{C}}_6)={\mathbf{\Delta }}_{\gamma },\mathit{where}\mathbf{0}\le \gamma \le 47$, is presented in Table 5.

Table 5. Symbol-triple distance of $\mathbf{\psi }$-constacyclic codes over ${\mathbb{F}}_{\mathbf{7}}+\mathit{u}{\mathbb{F}}_{\mathbf{7}}+{\mathit{u}}^{\mathbf{2}}{\mathbb{F}}_{\mathbf{7}}$ of $\mathtt{Type}\mathtt{6}$.

Range of $\mathit{\gamma }$	${\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_6)$
$\gamma =0$	3
$\gamma =1$	4
$\gamma =2$	5
$3\le \gamma \le 7$	6
$\gamma =8$	8
$9\le \gamma \le 14$	9
$15\le \gamma \le 21$	12
$22\le \gamma \le 28$	15
$29\le \gamma \le 35$	18
$36\le \gamma \le 42$	21
$\gamma =43$	28
$\gamma =44$	35
$\gamma =45$	42
$46\le \gamma \le 47$	49

▹

$\mathtt{Type}\mathtt{7}$$({\mathfrak{C}}_7)$: ${\mathfrak{C}}_7=⟨{(x-\psi )}^{\alpha }+u{(x-\psi )}^{t_1}{f}_1(x)+u^2{(x-\psi )}^{t_2}{f}_2(x),u{(x-\psi )}^{\beta }+u^2{(x-\psi )}^{t_3}{f}_3(x)⟩$, where $f_1(x),f_2(x),f_3(x)$ are 0 or are units in $\mathfrak{R}\left[x\right]/⟨x^{49}-\psi ⟩$, $0\le \beta <\alpha \le 48$, $0\le t_1<\beta$, $0\le t_2<\beta$, $0\le t_3<\beta$. Then, the symbol-triple distance, $d_{\mathit{st}}={\mathbf{\Delta }}_{\mathsf{D}},\mathit{where}\mathbf{0}\le \mathsf{D}\le 47$, is presented in Table 6.

Table 6. Symbol-triple distance of $\mathbf{\psi }$-constacyclic codes over ${\mathbb{F}}_{\mathbf{7}}+\mathit{u}{\mathbb{F}}_{\mathbf{7}}+{\mathit{u}}^{\mathbf{2}}{\mathbb{F}}_{\mathbf{7}}$ of $\mathtt{Type}\mathtt{7}$.

Range of $\mathsf{D}$	${\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_7)$
$\mathsf{D}=0$	3
$\mathsf{D}=1$	4
$\mathsf{D}=2$	5
$3\le \mathsf{D}\le 7$	6
$\mathsf{D}=8$	8
$9\le \mathsf{D}\le 14$	9
$15\le \mathsf{D}\le 21$	12
$22\le \mathsf{D}\le 28$	15
$29\le \mathsf{D}\le 35$	18
$36\le \mathsf{D}\le 42$	21
$\mathsf{D}=43$	28
$\mathsf{D}=44$	35
$\mathsf{D}=45$	42
$46\le \mathsf{D}\le 47$	49

▹

$\mathtt{Type}\mathtt{8}$$({\mathfrak{C}}_8)$: ${\mathfrak{C}}_8=⟨{(x-\psi )}^{\alpha }+u{(x-\psi )}^{t_1}{f}_1(x)+u^2{(x-\psi )}^{t_2}{f}_2(x),u{(x-\psi )}^{\beta }+u^2{(x-\psi )}^{t_3}{f}_3(x),u^2{(x-\psi )}^{\gamma }⟩$, where $f_1(x),f_2(x),f_3(x)$ are 0 or are units in $\mathfrak{R}\left[x\right]/⟨x^{49}-\psi ⟩$, $0\le \gamma <\beta <\alpha \le 48$, $0\le t_1<\beta$, $0\le t_2<\gamma$, $0\le t_3<\gamma$, then, the symbol-triple distance, $d_{\mathit{st}}({\mathfrak{C}}_8)={\mathbf{\Delta }}_{\gamma },\mathit{where}\mathbf{0}\le \gamma \le 46$, is presented in Table 7.

Table 7. Symbol-triple distance of $\mathbf{\psi }$-constacyclic codes over ${\mathbb{F}}_{\mathbf{7}}+\mathit{u}{\mathbb{F}}_{\mathbf{7}}+{\mathit{u}}^{\mathbf{2}}{\mathbb{F}}_{\mathbf{7}}$ of $\mathtt{Type}\mathtt{8}$.

Range of $\mathit{\gamma }$	${\mathtt{d}}_{\mathbf{st}}({\mathfrak{C}}_8)$
$\gamma =0$	3
$\gamma =1$	4
$\gamma =2$	5
$3\le \gamma \le 7$	6
$\gamma =8$	8
$9\le \gamma \le 14$	9
$15\le \gamma \le 21$	12
$22\le \gamma \le 28$	15
$29\le \gamma \le 35$	18
$36\le \gamma \le 42$	21
$\gamma =43$	28
$\gamma =44$	35
$\gamma =45$	42
$\gamma =46$	49

Example 2.

Table 8 presents all ψ-constacyclic codes of length 3 over ${\mathbb{F}}_3+u{\mathbb{F}}_3+u^2{\mathbb{F}}_3$, where $\psi \in \{1,2\}$, with their symbol-triple distances $d_{\mathit{st}}$ and their number of codewords $\left|\mathfrak{C}\right|$. In all codes, we have ${\nu }_0,{\theta }_0\in \{1,2\}$ and ${\theta }_1\in \{0,1,3\}$.

Table 8. $\mathbf{\psi }$-constacyclic codes of length 3 over ${\mathbb{F}}_{\mathbf{3}}+\mathit{u}{\mathbb{F}}_{\mathbf{3}}+{\mathit{u}}^{\mathbf{2}}{\mathbb{F}}_{\mathbf{3}}$.

Ideal $(\mathfrak{C})$	${\mathtt{d}}_{\mathbf{st}}(\mathfrak{C})$	$\left|\mathfrak{C}\right|$
$\to \mathtt{Type}\mathtt{1}$ :
$\ast ⟨0⟩$	0	1
$\ast ⟨1⟩$	3	19,683
$\to \mathtt{Type}\mathtt{2}$ :
$\ast ⟨u^2⟩$	3	27
$\ast ⟨u^2(x-\psi )⟩$	3	9
$\ast ⟨u^2{(x-\psi )}^2⟩$	3	3
$\to \mathtt{Type}\mathtt{3}$ :
$\ast ⟨u⟩$	3	729
$\ast ⟨u(x-\psi )⟩$	3	81
$\ast ⟨u{(x-\psi )}^2⟩$	3	9
$\ast ⟨u(x-\psi )+{\nu }_0{u}^2⟩$	3	81
$\ast ⟨u{(x-\psi )}^2+{\nu }_0{u}^2⟩$	3	27
$\ast ⟨u{(x-\psi )}^2+{\nu }_0{u}^2(x-\psi )⟩$	3	9
$\to \mathtt{Type}\mathtt{4}$ :
$\ast ⟨u(x-\psi ),u^2⟩$	3	243
$\ast ⟨u{(x-\psi )}^2,u^2⟩$	3	81
$\ast ⟨u{(x-\psi )}^2,u^2(x-\psi )⟩$	3	27
$\to \mathtt{Type}\mathtt{5}$ :
$\ast ⟨(x-\psi )⟩$	3	729
$\ast ⟨{(x-\psi )}^2⟩$	3	27
$\ast ⟨(x-\psi )+{\nu }_0{u}^2⟩$	3	729
$\ast ⟨{(x-\psi )}^2+{\nu }_0{u}^2⟩$	3	81
$\ast ⟨{(x-\psi )}^2+{\nu }_0{u}^2(x-\psi )⟩$	3	27
$\ast ⟨(x-\psi )+{\nu }_{0}u⟩$	3	729
$\ast ⟨{(x-\psi )}^2+{\nu }_{0}u⟩$	3	243
$\ast ⟨{(x-\psi )}^2+{\nu }_{0}u(x-\psi )⟩$	3	27
$\ast ⟨(x-\psi )+{\nu }_{0}u+{\theta }_0{u}^2⟩$	3	729
$\ast ⟨{(x-\psi )}^2+{\nu }_{0}u+{\theta }_0{u}^2⟩$	3	243
$\ast ⟨{(x-\psi )}^2+{\nu }_{0}u(x-\psi )+{\theta }_0{u}^2+{\theta }_1{u}^2(x-\psi )⟩$	3	27
$\ast ⟨{(x-\psi )}^2+{\nu }_{0}u(x-\psi )+{\theta }_0{u}^2(x-\psi )⟩$	3	27
$\to \mathtt{Type}\mathtt{6}$ :
$\ast ⟨(x-\psi ),u^2⟩$	3	2187
$\ast ⟨{(x-\psi )}^2,u^2⟩$	3	243
$\ast ⟨{(x-\psi )}^2,u^2(x-\psi )⟩$	3	81
$\ast ⟨(x-\psi )+{\nu }_{0}u,u^2⟩$	3	2187
$\ast ⟨{(x-\psi )}^2+{\nu }_{0}u,u^2⟩$	3	729
$\ast ⟨{(x-\psi )}^2+{\nu }_{0}u(x-\psi ),u^2⟩$	3	243
$\ast ⟨{(x-\psi )}^2+{\nu }_{0}u(x-\psi ),u^2(x-\psi )⟩$	3	81
$\ast ⟨{(x-\psi )}^2+{\nu }_{0}u(x-\psi )+{\theta }_0{u}^2,u^2(x-\psi )⟩$	3	81
$\to \mathtt{Type}\mathtt{7}$ :
$\ast ⟨(x-\psi ),u⟩$	3	6561
$\ast ⟨{(x-\psi )}^2,u⟩$	3	2187
$\ast ⟨{(x-\psi )}^2,u(x-\psi )⟩$	3	243
$\ast ⟨{(x-\psi )}^2,u(x-\psi )+{\theta }_0{u}^2⟩$	3	243
$\ast ⟨{(x-\psi )}^2+{\theta }_0{u}^2,u(x-\psi )⟩$	3	243
$\ast ⟨{(x-\psi )}^2+{\nu }_0{u}^2,u(x-\psi )+{\theta }_0{u}^2⟩$	3	243
$\to \mathtt{Type}\mathtt{8}$ :
$\ast ⟨{(x-\psi )}^2,u(x-\psi ),u^2⟩$	3	729

5. Conclusions

In this research article, we determined all symbol-triple distances of $\psi$-constacyclic codes of length $p^{\vartheta }$ over $\mathfrak{R}={\mathbb{F}}_q+u{\mathbb{F}}_q+u^2{\mathbb{F}}_q$, where $\psi \in {\mathbb{F}}_q^{*}$, $u^3=0$, $q=p^m$ and $p\ge 3$. For a given length and size, MDS symbol-triple codes have the highest capability in error detecting as well as in error correcting. Then, it would be interesting to investigate MDS symbol-triple codes in this constacyclic codes class.