A Note on the Weight Distribution of Minimal Constacyclic Codes

Abstract

In this note, we determine weight distributions of minimal constacyclic codes of length $p^n$ over the finite field $F_l$, where $p$ is a prime which is coprime to $l$.

1 Introduction

Let $F_l$ be a finite field of order $l$. A linear code $\mathcal {C}$ of length $n$ and dimension $k$ is a linear subspace of vector space $(F_l)^n$. A linear code $\mathcal {C}$ is called a cyclic code if for every word $v=(v_0,~v_1,\ldots ,v_{n-1}),$ the vector $(v_{n-1},~v_0,\ldots ,v_{n-2}),$ obtain from $v$ by the cyclic shift of co-ordinates $i\mapsto i+1,$ taken modulo $n,$ in also in $\mathcal {C}.$ A group ring code is an ideal in the group ring $RG$, where $G$ is a group and $R$ is a suitable commutative ring. Cyclic codes of length $n$ over $F_l$ can be identified with ideals of the group algebra $F_l C_n$ of the cyclic group $C_n$ of order $n.$

Using cocycles which arise from cohomology and the theory of projective representations of groups, this concept can be extended by taking ideals of a twisted group ring. In the simplest case of a cyclic group $G$, these ideals coincide with the constacyclic codes introduced by Berlekamp [1]. For abelian groups, the ideals in the twisted group rings are known as constabelian codes as defined by Lim [9]. Trivial cocycles (i.e., coboundaries) correspond to constacyclic codes that are scalar equivalent to cyclic codes. Throughout this note we take the ring $R$ to be a finite field $F_l$ of order $l$. In case the characteristic of the field does not divide the order of the group, the twisted group ring turns out to be semisimple (see [11]), so that every ideal (code) of the twisted group ring is a finite direct sum of minimal ideals (codes). The minimal ideals of twisted group ring of cyclic (respectively abelian) groups are defined as minimal constacyclic (respectively constabelian) codes. The weight of a codeword is the number of its elements that are non-zero, and the distance between two codewords (also known as Hamming distance) is the number of elements in which they differ. The distance $d$ of a linear code is minimum weight of its non-zero codewords, or equivalently the minimum distance between distinct codewords. A linear code of length $n$, dimension $k$, and distance $d$ will be called a $[n,k,d]_l$ code.

In this note, we take length of the code $n$ to be coprime with the characteristic of the finite field $F_l$. We determine weight distribution of minimal constacyclic codes. We also compare the minimum distances of constacylic and constabelian codes and it turns out that constabelian codes may be better than cyclic or constacyclic codes in the sense that larger minimum distances can be found in these codes.

2 Some Preliminaries

We need a few definitions and results from cohomology (see, for example, [6]). Let $G$ be a group. A function $\psi : G\times G\rightarrow F_l^{*}$ is called a cocycle if $\psi (1,1)=1$ and

$$\begin{aligned} \psi (xy,z) \psi (x,y)=\psi (x,yz)\psi (y,z) \quad \forall x,y,z \in G. \end{aligned}$$

A consequence of the above equation is that $\psi (1,x)=\psi (x,1)=1~ \forall x\in G$. We call a cocycle $\alpha$ on $G$ to be a coboundary if for some $\tau : G\rightarrow F_l^{*}$

$$\begin{aligned} \alpha (x,y)=\tau (x)\tau (y)\big (\tau (xy)\big )^{-1} \quad \forall x,y \in G, \end{aligned}$$

and write $\alpha =\partial \tau$. Two cocycles $\psi$ and $\psi '$ on $G$ are said to be cohomologous if $\psi =\psi '\partial \tau$ for some $\tau : G\rightarrow F_l^{*}.$ The set of cocycles from $G$ to $F_l$ forms a group under multiplication denoted by $Z^2(G, F_l)$. The set of coboundaries forms a normal subgroup of $Z^2(G, F_l)$ denoted by $B^2(G, F_l)$. The quotient group $Z^2(G, F_l)/B^2(G, F_l)$ is known as second cohomology group and is denoted by $H^2(G,F_l).$

For a cocycle $\psi$ on $G$, the twisted group algebra $F_l^{\psi }G$ over $F_l$ is defined as an algebra which has the same module structure as that of the group algebra $F_lG$ but the multiplication is defined by linearly extending the multiplication

$$\begin{aligned} \bar{g}\bar{h}=\psi (g,h)\overline{gh} \quad \forall g,\;h\in G. \end{aligned}$$

Let $a\in F_{l}^{*}=F_l\setminus \{0\}$. A linear code $\mathcal {C}$ of length $n$ over $F_l$ is said to be $a$-constacyclic if for $(c_0,c_1,\ldots ,c_{n-1})\in \mathcal {C},$ the constacyclically shifted vector $(ac_{n-1},c_0,\ldots ,c_{n-2})\in \mathcal {C}.$ In the special case when $a=-1,$ such codes are called negacyclic. Constacyclic codes can be viewed as ideals in the ring $\frac{F_l[X]}{\langle X^n-a\rangle }.$

Let $C_n=\,\langle x\rangle$ be the cyclic group of order $n$. For $a\in F_{l}^{*},$ let $\psi : C_n\times C_n\rightarrow F_{l}^{*}$ be a cocycle of $C_n$ given by

$$\begin{aligned} \quad \psi ({x^i,x^j})~= \left\{ \begin{array}{ll} 1,&{}\quad \text{ if }\,~i+j<n~~\\ ~a,&{}\quad \text{ if }\,~i+j~\ge n. \end{array} \right. \end{aligned}$$

In fact, every cocycle of $C_n$ is of above type (Theorem 3.1 in [6]). Let $F_l^{\psi }C_n$ be the twisted group algebra of $C_n$ over $F_l$ corresponding to the cocycle $\psi$. It follows that $\frac{F_l[X]}{\langle X^n-a\rangle }$ is isomorphic to the twisted group algebra $F_l^{\psi }C_n$ resulting in a one to one correspondence between the $a$-constacyclic codes of length $n$ over $F_l$ and the ideals of the twisted group algebra $F_l^{\psi }C_n$.

By the Fundamental Theorem of finite abelian groups $G=C_{n_1}\times \cdots \times C_{n_t}$, where $n_1|n_2 \ldots |n_t$ and $C_{n_i}=\,\langle x_i\rangle$. Let $\psi :G\times G\rightarrow F_l^{*}$ be a cocycle of $G$, which is the product of cocycles of $C_{n_1},\ldots ,C_{n_t}$ given by

$$\begin{aligned} \quad \psi _{i}\big ({x_i^{k_1},x_i^{k_2}}\big )~= \left\{ \begin{array}{ll} 1,&{}\quad \text{ if }\,~k_1+k_2<n_i~~~\\ ~a_i,&{}\quad \text{ if }\,~k_1+k_2~\ge n_i,\\ \end{array} \right. \end{aligned}$$

where, for $1\le i\le t,~a_i\in F_{l}^{*}.$ In [9], a code $\mathcal {C}$ is called a constabelian code associated with abelian group $G$ if there exist $a_1,\ldots ,a_t\in F_{l}^{*}$ such that $\mathcal {C}$ is an ideal of the ring $\mathcal {A}=\frac{F_l[X_1,\ldots ,X_t]}{\langle X_1^{n_1}-a_1, \ldots ,X_t^{n_t}-a_t\rangle }.$ It follows easily that $\mathcal {A}\cong F_l^{\psi }G$, where $\psi$ is the product cocycle defined above and as stated earlier, minimal (irreducible) ideals of the twisted group algebra $F_l^{\psi }G$ for abelian group $G$ are known as minimal (irreducible) constabelian codes. In [4], minimal constabelian (and constacyclic) codes have been explicitly determined for several families of abelian and cyclic groups.

3 Weight Distribution of Constacyclic Codes of Length $p^n$

Let $A_i^{(n)}$ denote the number of codewords of weight $i$ in a code $\mathcal {C}$ of length $n$. The list $A_0^{(n)},~A_1^{(n)},\ldots ,~A_n^{(n)}$ is called the weight distribution (or weight spectrum) of the code $\mathcal {C}$, and the corresponding homogeneous polynomial

$$\begin{aligned} W_\mathcal {C}(x,y)=\sum \limits _{i=0}^{n}A_{i}^{(n)}x^{n-i}y^i \end{aligned}$$

is called the Hamming weight enumerator of $\mathcal {C}$. The knowledge of the weight distribution of a code enables one to calculate the probability of undetected errors when the code is used purely for error detection. The least positive value of $i$ for which $A_i$ is non-zero is the minimum Hamming weight of the code which gives a measure that how good a code is at error correcting. The problem of the determination of weight distribution of a code is of considerable interest.

It is known that if $\mathcal {C}$ and $\mathcal {D}$ are two codes of length $n$, then the weight enumerator of the direct sum $\mathcal {C}\bigoplus \mathcal {D}$ is $W_{\mathcal {C}\bigoplus \mathcal {D}}(x,y)= W_{\mathcal {C}}(x,y)W_{\mathcal {D}}(x,y)$ (for example, see [5]). As every constacyclic code is a direct sum of irreducible constacyclic codes, it is enough to compute weight distributions of irreducible constacyclic codes.

Let $G=\,\langle C_{p^n}\rangle \,=\,\langle x\rangle$ be a cyclic group of order $p^n$, where $p$ is an odd prime. Let $F_l$ be a finite field such that $l$ is coprime to $p$. Let $a\in F_{l}^{*}$ and let, as usual, $\psi$ be the cocycle given by

$$\begin{aligned} \quad \psi \big ({x^i,x^j}\big )~= \left\{ \begin{array}{ll} 1,&{}\quad \text{ if }\,~i+j<p^n~~\\ a,&{}\quad \text{ if }\,~i+j~\ge p^n.\\ \end{array} \right. \end{aligned}$$

The following well-known result characterizes 2-cocycles of cyclic groups (See, for example [6], p. 52).

Proposition 3.1

Let $G=C_n$ be a cyclic group of order $n$ and let $K$ be a field. Then the second cohomology group $H^2(G,K^{*})$ is isomorphic to $\frac{K^{*}}{(K^{*})^n}.$

Corollary 3.2

Let $C_n$ be the cyclic group of order $n$ and let $K$ be a finite field such that $n$ is coprime to $o(K^{*})$, the order of the multiplicative group of $K$. Then $H^2(C_n, K^{*})$ is trivial.

Proof

As $n$ is coprime to $o(K^{*})$, $(K^{*})^n= K^{*}$. $\square$

From the above corollary, it follows that proper constacyclic codes of length $n$ over $F_l$ exist if and only if gcd$(n, o(K^{*}))> 1$; otherwise, they are equivalent to cyclic codes, as the corresponding twisted group algebras are isomorphic to group algebras in these cases, (see [6], Page 17]). Let $K$ be the splitting field of $x^{p^n}-a$ over $F_l$. For $1\le i\le n$, let $\varepsilon _{p^i}$ be the primitive $p^i$th root of unity. Depending on the divisibility of $l-1$ by $p$, the following cases arise:

Suppose that $p\not \mid l-1$. Then gcd$(p^n, l-1)=1$ implies that there exist integers $x'$ and $y'$ such that $p^nx'+(l-1)y'=1$, so that $a=~a^{p^nx'+(l-1)y'}= a^{p^nx'}a^{(l-1)y'}=a^{p^nx'}$ and hence $a\in (F_l)^{p^n}.$ So, $F_l^{\psi }C_{p^n}\cong F_lC_{p^n}$ ([6], p. 53), yielding that constacyclic codes in this case are equivalent to cyclic codes. Weight distributions of cyclic codes have been studied by many authors, for instance in [2, 3, 10, 12–14].

Suppose that $p$ divides $l-1$. Again, there are two possibilities; either $p\not \mid o(a)$ or $p|o(a)$. In the first case, as gcd$(p^n, o(a))=1$, it follows that $a\in (F_l)^{p^n}$, yielding $F_l^{\psi }C_{p^n}\cong F_lC_{p^n}$.

Thus, it only remains to consider the case when $p$ divides both $l-1$ and $o(a)$. Proper constacyclic codes, i.e., constacyclic codes which are not equivalent to cyclic codes exist in this case only. We shall determine the weight distributions of such constacyclic codes of length $p^n$.

The main result about the weight distribution of the irreducible constacyclic codes of length $p^n$ over the finite field $F_l$ is given in the following:

Theorem 3.3

Let $F_l$ be a finite field with $l$ elements. Let $p$ be an odd prime such that $p\mid l-1$. Let $a \in F_l^{*}=F_l\setminus \{0\}$ such that $p\mid o(a)$. The weight distribution of each $a$-constacyclic code of length $p^n$ over $F_l$ is given by

$$\begin{aligned} \quad A_i^{(p^n)}~= \left\{ \begin{array}{ll} 0, &{}\quad \text{ if }~p^s\not \mid i\\ \left( {\begin{array}{c}p^{n-s}\\ w'\end{array}}\right) (l-1)^{w'}, &{}\quad \text{ if }~i=p^sw'.\\ \end{array} \right. \end{aligned}$$

where $s$ is the largest integer such that $a\in (F_l)^{p^s}.$

Proof

Let $s$ be the largest integer such that $a\in (F_l)^{p^s},$ i.e., there exists an element $b\in F_l\setminus (F_l)^p$ such that $a=b^{p^s}$ and let $\alpha$ be an element in some extension field of $F_l$ such that $\alpha ^{p^{n}}=a$. As $\alpha ^{p^{n-s}}=b,$

$$\begin{aligned} x^{p^{n}}-a= & {} x^{p^{n}}-b^{p^{s}} \\= & {} \big (x^{p^{n-s}}-b\big )\Bigg (x^{p^{n-s}\big (p^{s}-1\big )}+ b~x^{p^{n-s}\big (p^s-2\big )}+\cdots +b^{p^s-1}\Bigg ) \\= & {} \big (x^{p^{n-s}}-b\big ) b^{p^s-1}\prod \limits _{i=1}^{s}\Phi _{p^i}\Big (\big (\alpha ^{-1}x\big )^{p^{n-s}}\Big ), \end{aligned}$$

where $\Phi _{p^i}(x)$ is cyclotomic polynomial of order $p^i.$ Now $p|o(a)$ and $p|l-1.$ Let $m$ be the largest integer such that $p^m|l-1$. Supposing $F_l^{*}= C_{p^m}\times C_{q_1^{k_1}}\times \cdots \times C_{q_t^{k_t}},$ where $q_i$’s are distinct primes, we get $(F_l^{*})^{p^m}\cong C_{q_1^{k_1}}\times \cdots \times C_{q_t^{k_t}}$ so that $a\notin (F_l^{*})^{p^m}$. It follows that $s<m$ and for $1\le i\le s,~\text{ ord }_{p^i}(l)=1,$ where $\text{ ord }_{p^i}(l)$ denotes the multiplicative order of $l$ modulo $p^i.$ Consequently, by Theorem 2.47 of [8], $\Phi _{p^i}(x)$ factors into linear polynomials over $F_l$ so that $\Phi _{p^i}\big ((\alpha ^{-1}x)^{p^{n-s}}\big )$ factors into polynomials of degree $p^{n-s}$ over $F_l$.

Let $\phi$ denote the Euler phi function. For a prime $p,$ $\phi (p^i)=p^i-p^{i-1}.$ Thus,

$$\begin{aligned}&\Phi _{p^i}\bigg (\big (\alpha ^{-1}x\big )^{n-s}\bigg )\\&\quad = \bigg (\big (\alpha ^{-1}x\big )^{p^{n-s}}-\varepsilon _{p^i}\bigg )\bigg (\big (\alpha ^{-1}x\big )^{p^{n-s}} -\varepsilon _{p^i}^2\bigg )\ldots \bigg (\big (\alpha ^{-1}x\big )^{p^{n-s}}-\varepsilon _{p^i}^{\phi \big (p^i\big )}\bigg )\\&\quad =\frac{1}{b^{\phi (p^i)}}\Bigg [\bigg (x^{p^{n-s}}-b\varepsilon _{p^i}\bigg ) \bigg (x^{p^{n-s}}-b\varepsilon _{p^i}^2\bigg )\ldots \bigg (x^{p^{n-s}} -b\varepsilon _{p^i}^{\phi \big (p^i\big )}\bigg )\Bigg ]\\&\quad = \frac{1}{b^{p^i-p^{i-1}}}\Bigg [\bigg (x^{p^{n-s}}-b\varepsilon _{p^i}\bigg ) \bigg (x^{p^{n-s}}-b\varepsilon _{p^i}^2\bigg )\ldots \bigg (x^{p^{n-s}} -b\varepsilon _{p^i}^{p^i-p^{i-1}}\bigg )\Bigg ]. \end{aligned}$$

Since $b\notin (F_l)^p$, therefore, $b\varepsilon _{p^i}\notin (F_l)^p,$ so that by ([7], Lemma 16.5) each of the polynomial $x^{p^{n-s}}-b\varepsilon _{p^i}^{j}$ is irreducible over $F_l$ and

$$\begin{aligned} x^{p^{n}}-a=\bigg (x^{p^{n-s}}-b\bigg )\prod \limits _{i=1}^{s}\prod \limits _{j=1}^{p^i- p^{i-1}}\bigg (x^{p^{n-s}}-b\varepsilon _{p^i}^{j}\bigg ). \end{aligned}$$

We observe that $\varepsilon _{p^i}^{h^j}, ~1\le i\le s$, $0\le j\le p^i-p^{i-1}-1$, are contained in $F_l$. Thus, replacing $b$ by $b\varepsilon _{p^i}^{h^j}$ will yield the same factorization of $x^{p^n}-a,$ as $a=b^{p^s}=(b\varepsilon _{p^i}^{h^j})^{p^s}.$

The ring $\frac{F_l[x]}{\langle x^{p^n}-a\rangle }$ is a principal ideal ring and any ideal in it is generated by (the image of) a divisor of $x^{p^n}-a.$ Consider the ideal $I$ generated by the polynomial

$$\begin{aligned} g(x)=x^{p^{n-s}\big (p^s-1\big )}+b x^{p^{n-s}\big (p^s-2\big )}+\cdots + b^{p^s-1}, \end{aligned}$$

so that, $I=\frac{F_l[x]}{\langle x^{p^{n-s}}-b\rangle }$ is an irreducible constacyclic code. Dimension of the ideal $I$ over $F_l$ is $p^{n-s}.$ Note that $g(x),~ x^{p^{s}}g(x),~x^{2p^{s}}g(x), \ldots ,x^{(p^{n-s}-1)p^s}g(x)$ form a basis of the ideal $I$ over $F_l$, as this is a linearly independent set and the number of elements in this set is equal to the dimension of $I$.

Let $e_i, ~1\le i\le p^n,$ be the standard basis of $(F_l)^{p^n}$ as a vector space over $F_l$. Identifying the polynomial $g(x)$ with the tuple $\sum \nolimits _{j=1}^{p^s}b^{j-1}e_{p^n-jp^{n-s}}$, any codeword $c$ in $I$ can be written as

$$\begin{aligned} c=\sum \limits _{i=0}^{p^{n-s}-1}\sum \limits _{j=1}^{p^s}\alpha _i b^{j-1} e_{p^s\big (p^{n-s}+i\big )-jp^{n-s}}, \end{aligned}$$

where $\alpha _i\in F_l$ [as $x^{ip^s}g(x)$ is the touple $\sum \nolimits _{j=1}^{p^s}b^{j-1}e_{{p^n}-jp^{n-s}+ip^s}$ for $0\le i\le p^{n-s}-1$]. Clearly, wt$(c)=p^sw'$, where wt$(c)$ denotes the weight of the codeword $c$ and $w'$ is the number of non-zeros $\alpha _i$’s. Therefore,

$$\begin{aligned} A_w^{p^n}=0, \quad \text{ when }~ p^s\not \mid w, \end{aligned}$$

and a codeword in $I$ has weight $w=p^sw'$ if and only if it is a linear combination of any $w'$ basis vectors over $F_l$ out of total $p^{n-s}$ basis vectors of $I$. Therefore, there are $\left( {\begin{array}{c}p^{n-s}\\ w'\end{array}}\right) (l-1)^{w'}$ codewords of weight $w=p^sw'$ in $I$.

If we take another ideal of the ring $\frac{F_l[x]}{\langle x^{p^n}-a\rangle }$, for instance $I'=\frac{F_l[x]}{\langle x^{p^{n-s}}-b\varepsilon _{p^i}^{h^j}\rangle },$ for some $1\le i\le s$ and $1\le j\le \phi (p^i)$, on replacing $b$ by $b\varepsilon _{p^i}^{h^j}$, we get the same result as $a=b^{p^s}=\big (b\varepsilon _{p^i}^{h^j}\big )^{p^s}$ and it follows that the weight enumerator of every irreducible constacyclic code of length $p^n$ is the same, which is given by

$$\begin{aligned} \quad A_i^{(p^n)}~= \left\{ \begin{array}{ll} 0, &{}\quad \text{ if }~p^s\not \mid i\\ \left( {\begin{array}{c}p^{n-s}\\ w'\end{array}}\right) (l-1)^{w'}, &{}\quad \text{ if }~i=p^sw'\\ \end{array} \right. \end{aligned}$$

where $s$ is the largest integer such that $a\in (F_l)^{p^s}.$ $\square$

Corollary 3.4

It follows that the minimum distance of an irreducible constacyclic codes of length $p^n$ and dimension $p^{n-s}$ over $F_l$ is $p^s$, as $A_i^{(p^n)}=0$ for $p^s\not \mid i$, so that the least value of $i$ for which $A_i^{(p^n)}$ is non-zero is $p^s$.

Example 3.5

Let $C_{27}=\langle x\rangle$ be cyclic group of order $27$. Let $l=19$ and let the cocycle $\psi :C_{27}\times C_{27}\rightarrow F_{19}^{*}$ be given by

$$\begin{aligned} \quad \psi \big ({x^i,x^j}\big )~= \left\{ \begin{array}{ll} 1, &{}\quad \text{ if }~i+j<27~~\\ 7, &{}\quad \text{ if }~i+j~\ge 27,\\ \end{array} \right. \end{aligned}$$

for $0\le i,j\le 27.$ The order of $7$ in $F_{19}$ is $3$. In the above notations, $s=1,~m=2$, and $x^{27}-7=(x^9-4)(x^9-6)(x^9-9)$ is the irreducible factorization of $x^{27}-7$ over $F_{19}.$ Consider the ideal $I=\frac{F_{19}[x]}{\langle x^9-4\rangle }=\langle g(x)\rangle$, where $g(x)= (x^9-6)(x^9-9)=x^{18}-15x^9+16.$ We note that $g(x),~ x^3g(x),~x^6g(x),\ldots , x^{24}g(x)$ form a basis of $I$ and the weight enumerators are given by

$$\begin{aligned} \quad A_i^{(27)}~= \left\{ \begin{array}{ll} 0,&{}\quad \text{ if }~3\not \mid i\\ \left( {\begin{array}{c}9\\ w'\end{array}}\right) (18)^{w'}, &{}\quad \text{ if }~i=3w'.\\ \end{array} \right. \end{aligned}$$

It would be of considerable interest to determine weight distribution of other minimal constacyclic and in general, of constabelian codes, which at present seems to be difficult.

We now compare the minimum distances of constacyclic codes with the minimum distances of constabelian codes.

Example 3.6

Let $C_{729}=\langle z|z^{729}=1\rangle$ be the cyclic group of order $729.$ Let $l=19$ and let the cocycle $\psi$ be given by

$$\begin{aligned} \quad \psi \big ({z^i,z^j}\big )~= \left\{ \begin{array}{ll} 1, &{}\quad \text{ if }~i+j<729 \\ 7, &{}\quad \text{ if }~i+j~\ge 729.\\ \end{array} \right. \end{aligned}$$

The order of $7$ in $F_{19}$ is $3.$ In the notations described earlier, $m=2$ (as $3^2 \mid 19-1$, but $3^3\not \mid 19-1$), $s=1$ [as $7\in F_{19}$, but $7\notin (F_{19})^2$] and over $F_{19}$, $x^{729}-7=(x^{243}-4)(x^{243}-6)(x^{243}-9).$ By Corollary 3.4, we get that the minimum distance of the minimal constacyclic code of length 729 is $3^s=3^1=3$. Thus, we get a $[729, 243,3]_{19}$ constacyclic code.

Let $G=C_9\times C_{81}=\langle x,y|x^{9}=1=y^{81}\rangle$ be an abelian group of order $729$. Let $\psi =\psi _1\psi _2$ be given by

$$\begin{aligned} \quad \psi _1\big ({x^i,x^j}\big )~= \left\{ \begin{array}{ll} 1, &{}\quad \text{ if }~i+j<9~\\ 7, &{}\quad \text{ if }~i+j~\ge 9\\ \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \quad \psi _2\big ({y^{i'},y^{j'}}\big )~= \left\{ \begin{array}{ll} 1, &{}\quad \text{ if }~i'+j'<81\\ 7^{9}=1\,(\text{ mod } \,\,19), &{}\quad \text{ if }~i'+j'~\ge 81.\\ \end{array} \right. \end{aligned}$$

Note that $F_{19}^{\psi _2}C_{81}\cong F_{19}C_{81}.$ As ord$(7)=3$ and ord$(7^9)=1,$ $3|\text{ ord }(7)$ but $3\not \mid \text{ ord }(7^9).$

One checks that constabelian code generated by the idempotent $e=\frac{1}{243}\big (1+4x^3+16 x^6\big )\big (\sum \nolimits _{v=0}^{80}y^v\big )\in F_{19}^{\psi }\big (C_9\times C_{81}\big )$ has minimum distance $3\times 81=243,$ as any element in this ideal will be of the type $\alpha =\sum _{u,v}\alpha _{u,v}x^uy^v. e=\sum \nolimits _{u,v}\alpha _{u,v}x^u. e$. Since the length of the element $\sum \nolimits _{v=0}^{80}y^v$ is $81$, the minimum distance of this ideal will be $81$ times the minimum distance of the constacyclic code of length $9$ and dimension $3$, which is $3$ and we get a $[729,3,243]_{19}$ constabelian code. Thus, minimum distance of the constabelian code of length 729 is 81 times more than the constacyclic code of length 729.

We have Table1.

Table 1 Comparison of constacyclic and constablelian codes

It turns out that constabelian codes may have larger minimum distances than constacyclic codes of same length.