Consta-dihedral Codes over Finite Fields

TL;DR

This paper studies the algebraic and asymptotic properties of consta-dihedral codes over finite fields, showing conditions under which self-dual or LCD codes are asymptotically good, and corrects previous errors in the literature.

Contribution

It extends prior work by analyzing consta-dihedral codes, establishing new conditions for their asymptotic goodness, and correcting earlier inaccuracies.

Findings

Self-dual consta-dihedral codes are asymptotically good if q is even or divisible by 4 minus 1.

LCD consta-dihedral codes are asymptotically good if q is odd and not divisible by 4 minus 1.

Provides a new technique to correct errors in previous literature.

Abstract

It is proved in a reference (Fan, Lin, IEEE TIT, vol.67, pp.5016-5025) that the self-dual (LCD respectively) dihedral codes over a finite field~$F$ with ${|F|=q}$ are asymptotically good if $q$ is even (odd respectively). In this paper, we investigate the algebraic property and the asymptotic property of conta-dihedral codes over $F$, and show that: if $q$ is even or $4\,|\,(q-1)$, then the self-dual consta-dihedral codes are asymptotically good; otherwise, the LCD consta-dihedral codes are asymptotically good. And, with the help of a technique developed in this paper, some errors in the reference mentioned above are corrected.

Equations

$$FG⟶FG,\sum _{g\in G}{a}_{g}g⟼\sum _{g\in G}{a}_g{g}^{-1}.$$

$$\displaystyle\sigma:~{}FG\longrightarrow F,~{}~{}\sum\limits_{g\in G}a_{g}g\longmapsto a_{1_{G}}~{}~{}~{}(\mbox{$1_{G}$ is the identity of $G$}).$$

$$FH=FH{e}_0\oplus FH{e}_1\oplus \cdots \oplus FH{e}_{\ell },$$

$$\displaystyle\big{\langle}FHe_{i},FHe_{j}\big{\rangle}=\begin{cases}0,&\mbox{if~{} }e_{i}\neq\overline{e}_{j};\\ F,&\mbox{if~{} }e_{i}=\overline{e}_{j}.\end{cases}$$

$$\displaystyle\lambda(n)=\min\big{\{}\dim_{F}(FHe_{1}),\cdots,\dim_{F}(FHe_{\ell})\big{\}}.$$

$$X^{2}+gXY+Y^{2}=\big{(}X+\frac{g}{2}Y\big{)}^{2}+\big{(}1-\frac{g^{2}}{4}\big{)}Y^{2}=X^{\prime 2}+bY^{2},$$

$$Mf,f=\left(\begin{array}{cc}a& 1\\ 0& 0\end{array}\right),\forall a\in F;\text{mbox}orf=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right).$$

$$\displaystyle G=\langle u,v\,|\,u^{n}=1=v^{2},vuv^{-1}=u^{-1}\rangle,\quad\mbox{$n>1$ is odd},\quad\gcd(n,q)=1;$$

$$H=⟨u⟩,T=⟨v⟩,\text{mbox}thenG=H⋊T.$$

$$\tilde{G}=⟨u,\dot{v}|u^n=1={\dot{v}}^4,\dot{v}u{\dot{v}}^{-1}=u^{-1}⟩=H⋊\tilde{T},$$

$$\displaystyle\big{\{}u^{i}\dot{v}^{j}\,\big{|}\,0\leq i<n,\,0\leq j<2\big{\}}=\{1,u,\cdots,u^{n-1},\dot{v},\;u\dot{v},\cdots,u^{n-1}\dot{v}\}$$

$$u^n=1,{\dot{v}}^2=-1,\dot{v}u=u^{-1}\dot{v}.$$

$${F\kern-1.0pt{*}G}\kern 1.0pt=F[X,Y]\big{/}\langle X^{n}-1,\,Y^{2}+1,\,XYX-Y\rangle,$$

$$\displaystyle{F\kern-1.0pt{*}G}\kern 1.0pt=F\tilde{G}\big{/}\langle\dot{v}^{2}+1\rangle.$$

$$\alpha (g_1,g_2{g}_3)\alpha (g_2,g_3)=\alpha (g_1{g}_2,g_3)\alpha (g_1,g_2),\forall g_1,g_2,g_3\in G.$$

$$g_1\cdot g_2=\alpha (g_1,g_2)(g_1{g}_2),\forall g_1,g_2\in G;$$

$$\gamma (u^i{v}^s,u^j{v}^t)=\{\begin{array}{ll}-1,& s=t=1;\\ 1,& \text{mbox}otherwise;\end{array}0\le i,j<n,0\le s,t<2;$$

$$F\tilde{G}\to F\tilde{G},\sum _{g\in \tilde{G}}{a}_{g}g\mapsto \overline{\sum _{g\in \tilde{G}}{a}_{g}g}=\sum _{g\in \tilde{G}}{a}_g{g}^{-1}.$$

$$\overline{\sum _{h\in H}{a}_{h}h+\sum _{h\in H}{a}_{h\dot{v}}h\dot{v}}=\sum _{h\in H}{a}_h{h}^{-1}+\sum _{h\in H}{a}_{h\dot{v}}(h\dot{v}{)}^{-1}\in F\ast G.$$

$$\overline{\sum _{h\in H}{a}_{h}h+\sum _{h\in H}{a}_{h\dot{v}}h\dot{v}}=\sum _{h\in H}{a}_h{h}^{-1}-\sum _{h\in H}{a}_{h\dot{v}}h\dot{v}\in F\ast G.$$

$$\overline{\overline{a}}=a,\overline{ab}=\bar{b}\bar{a},\forall a,b\in F\ast G.$$

$$\sigma :F\ast G\to F,\sum _{0\le i<n,0\le j<2}{a}_{ij}{u}^i{\dot{v}}^j\mapsto a_{00},$$

$$a\bar{b}$$

$$=\sum _{0\le i,i^{\prime }<n,0\le j,j^{\prime }<2}{a}_{ij}{b}_{i^{\prime }{j}^{\prime }}{u}^i{\dot{v}}^j{\dot{v}}^{-j^{\prime }}{u}^{-i^{\prime }}.$$

$$\sigma (a\bar{b})=\sum _{(ij{i}^{\prime }{j}^{\prime })}{a}_{ij}{b}_{i^{\prime }{j}^{\prime }}{s}_{ij{i}^{\prime }{j}^{\prime }},$$

$$\displaystyle{F\kern-1.0pt{*}G}\kern 1.0pt=FH\oplus FH\dot{v}=\big{\{}a+a^{\prime}\dot{v}\,\big{|}\,a,a^{\prime}\in FH\big{\}}.$$

$$\dot{v}(r{e}_0+e_0\dot{v})=r{e}_0\dot{v}+\dot{v}{e}_0\dot{v}=r{e}_0\dot{v}-e_0=r{e}_0\dot{v}+r^2{e}_0=r(r{e}_0+e_0\dot{v}).$$

$$(r{e}_0+e_0\dot{v})\overline{(r{e}_0+e_0\dot{v})}=(r{e}_0+e_0\dot{v})(r{e}_0+\bar{\dot{v}}{e}_0)$$

$$=(r{e}_0+e_0\dot{v})(r{e}_0-e_0\dot{v})=(r{e}_0{)}^2-(e_0\dot{v}{)}^2=-e_0+e_0=0.$$

$$\displaystyle\begin{array}[]{ccc}{\rm M}_{2}(\tilde{F})&\mathop{\longrightarrow}\limits^{\cong}&FHe\oplus FHe\dot{v}\oplus FH\bar{e}\oplus FH\bar{e}\dot{v},\\[3.0pt] \begin{pmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{pmatrix}&\longmapsto&a_{11}e-a_{12}\,e\,\dot{v}+\overline{a_{21}}\,\bar{e}\,\dot{v}+\overline{a_{22}}\,\bar{e},\end{array}$$

$$\displaystyle\big{(}a_{11}e-a_{12}e\dot{v}+\overline{a_{21}}\,\bar{e}\dot{v}+\overline{a_{22}}~{}\overline{e}\big{)}\big{(}b_{11}e-b_{12}e\dot{v}+\overline{b_{21}}\,\bar{e}\dot{v}+\overline{b_{22}}~{}\overline{e})$$

Full text

Key words: Finite fields; dihedral codes; consta-dihedral codes; self-dual codes; LCD codes.

*Email address: yfan@mail.ccnu.edu.cn (Yun Fan); *

1 Introduction

Let $F$ be a finite field with cardinality $|F|=q$, where $q$ is a power of a prime. Any $a=(a_{1},\cdots,a_{n})\in{F}^{n}$, $a_{i}\in{F}$, is called a word. The Hamming weight ${\rm w}(a)$ is defined to be the number of such indexes $i$ that $a_{i}\neq 0$. The Hamming distance between two words $a,a^{\prime}\in F^{n}$ is defined as ${\rm d}(a,a^{\prime})={\rm w}(a-a^{\prime})$. Any $\emptyset\neq C\subseteq F^{n}$ is called a code of length $n$ over $F$; the words in the code are called code words. The minimal Hamming distance ${\rm d}(C)$ is the minimum distance between distinct codewords of $C$. If $C$ is a linear subspace of $F^{n}$, then $C$ is called a linear code, the minimal Hamming weight ${\rm w}(C)$ is defined to be the minimal weight of the nonzero code words of $C$, and it is known that ${\rm w}(C)={\rm d}(C)$. The fraction $\Delta(C)=\frac{{\rm d}(C)}{n}=\frac{{\rm w}(C)}{n}$ is called the relative minimum distance of $C$, and ${\rm R}(C)=\frac{\dim_{F}C}{n}$ is called the rate of $C$. A code sequence $C_{1},C_{2},\cdots$ is said to be asymptotically good if the length $n_{i}$ of $C_{i}$ goes to infinity and both ${\rm R}(C_{i})$ and $\Delta(C_{i})$ are positively bounded from below. A class of codes is said to be asymptotically good if there is an asymptotically good sequence of codes within the class. The inner product of words $a=(a_{1},\cdots,a_{n})$ and $b=(b_{1},\cdots,b_{n})$ is defined to be $\langle a,b\rangle=\sum_{i=1}^{n}a_{i}b_{i}$. Then the self-orthogonal codes, self-dual codes, LCD codes etc. are defined as usual, e.g., cf. [14].

Let $G$ be a finite group of order $n$. The group algebra $FG$ is the $F$-vector space with basis $G$ and equipped with the multiplication induced by the multiplication of the group $G$. Any element $\sum_{x\in{G}}a_{x}x\in FG$ is identified with a word $(a_{x})_{x\in G}\in F^{n}$. Then any left ideal of $FG$ is called an $FG$-code. If $G$ is a cyclic (abelian, dihedral, resp.) group, the $FG$-codes are called cyclic (abelian, dihedral, resp.) codes. Any $FG$-submodule of $FG\times FG$ is called a quasi-$FG$ code of index $2$. If $G$ is cyclic (abelian, resp.), the quasi-$FG$ codes of index $2$ are also called quasi-cyclic (quasi-abelian, resp.) codes of index $2$.

If $G\!=\!\langle x\,|\,x^{n}\!=\!1\rangle$ is a cyclic group, then $FG$ is an $F$-algebra generated by $x$ with a relation $x^{n}\!=\!1$. For $0\!\neq\!\lambda\in F$, the $F$-algebra generated by $x$ with the relation $x^{n}\!=\!\lambda$ is called a constacyclic group algebra, and its ideals are called constacyclic codes. Next, let $G=\langle u,v\,|\,u^{n}\!=\!1,v^{2}\!=\!1,vuv^{-1}\!=\!u^{-1}\rangle$ be a dihedral group. Then $FG$ is the $F$-algebra (non-commutative) generated by $u,v$ with three relations $u^{n}\!=\!1$, $v^{2}\!=\!1$ and $vu\!=\!u^{-1}v$. If we replace the relation “$v^{2}\!=\!1$” by the relation “$v^{2}\!=\!-1$” and keep the other two relations invariant, then the obtained $F$-algebra is called a consta-dihedral group algebra, and its left ideals are called consta-dihedral codes (cf. Section 3 for details).

It is a long-standing open question (cf. [20]): are the cyclic codes over a finite field asymptotically good? However, it is well-known long ago that, if the characteristic ${\rm char}(F)=2$, quasi-cyclic codes of index $2$ over $F$ are asymptotically good, see [6, 7, 15]. Finite dihedral groups are near to finite cyclic groups, because a dihedral group of order $2n$ has a normal cyclic subgroup of order $n$. Bazzi and Mitter [4] proved that the binary dihedral codes are asymptotic good. Afterwords, Martínez-Pérez and Willems [21] proved the asymptotic goodness of binary self-dual quasi-cyclic codes of index $2$. Borello and Willems [5] proved the asymptotic goodness of such $FG$-codes that $|F|=p$ is an odd prime and $G$ is a semidirect product of the cyclic group of order $p$ by a finite cyclic group.

For any finite field $F$, i.e., for any prime power $q$, in the dissertation [17] it has been shown that the quasi-cyclic codes of index $2$ over $F$ are asymptotically good; and, if $q$ is even or $4\,|\,(q-1)$ (i.e., $q\,{\not\equiv}\,3~{}({\rm mod}\;4)$), the self-dual quasi-cyclic codes of index $2$ over $F$ are asymptotically good. Note that self-dual quasi-cyclic codes over $F$ of index $2$ exist if and only if $q\,{\not\equiv}\,3~{}({\rm mod}\;4)$, cf. [19] or [18, Corollary IV.5]. Based on Artin’s primitive root conjecture, with the same assumption on $q$, Alahmadi, Özdemir and Solé [1] also proved the asymptotic goodness of the self-dual quasi-cyclic codes of index $2$. Lin and Fan [18] exhibited further that, if $q\,{\not\equiv}\,3~{}({\rm mod}\;4)$, the self-dual quasi-abelian codes (including the quasi-cyclic case) of index $2$ are asymptotically good. Recently, Fan and Liu [12] discussed the quasi-constacyclic codes of index $2$ and showed that such codes are asymptotically good.

On the other hand, Fan and Lin [10] extend the asymptotic goodness of dihedral codes to any q-ary case; more precisely, they proved that the self-dual dihedral codes (if $q$ is even) and the LCD dihedral codes (if $q$ is odd) are asymptotically good. As consequences, the asymptotic goodness of the self-dual (if $q$ is even) and the LCD (if $q$ is odd) quasi-cyclic codes of index $2$ are also obtained. By the way, we observed some errors in the proofs of two theorems of the reference [10], as a result of the errors, [10, Theorem IV.5(1)] is false; see Section 6 below for details. Fortunately the issue does not affect the correctness of the results stated above.

About the relationship between the quasi-cyclic codes of index $2$ and the dihedral codes, any dihedral (consta-dihedral) code is a quasi-cyclic code of index $2$ in a natural way. Alahmadi, Özdemir and Solé [1] showed that, if $q$ is even, the self-dual double circulant codes (a family of self-dual quasi-cyclic codes of index $2$) are dihedral codes. Fan and Zhang [13] extended it and showed a necessary and sufficient condition for a self-dual quasi-cyclic code of index $2$ being a dihedral code (if $q$ is even) or a consta-dihedral code (if $q$ is odd).

The research outlines lead us to focus on the consta-dihedral codes. In this paper we investigate the algebraic property and the asymptotic property of the consta-dihedral codes, and address the issue in the reference [10]. To study the algebraic property of consta-dihedral codes, we develop a technique to evaluate the orthogonality of (consta-)dihedral codes by matrix computation; and construct a class of consta-dihedral codes which possess good algebraic properties (self-orthogonal, LCD etc.). With the help of this technique we address the issue in [10] mentioned above and recover the correct version of the false theorem of [10] (Theorem 6.7 below). To study the asymptotic property of consta-dihedral codes, in the class of consta-dihedral codes we constructed, we count the number of such codes which have bad asymptotic property; the number is much less than the total quantity of the class, so that we obtain the following results.

Theorem 1.1** (Theorem 5.12 below).**

Assume that $q$ is even or $4\,|\,(q-1)$. Then the self-dual consta-dihedral codes over $F$ are asymptotically good. In particular, self-dual quasi-cyclic codes of index $2$ over $F$ are asymptotically good.

Theorem 1.2** (Theorem 5.8 below).**

Assume that $q$ is odd and ${4\!\nmid\!(q-1)}$. Then the LCD consta-dihedral codes over $F$ are asymptotically good. In particular, LCD quasi-cyclic codes of index $2$ over $F$ are asymptotically good.

In Section 2, some preliminaries are sketched.

In Section 3, we describe the consta-dihedral group algebras in three ways, exhibit properties of them; and characterize the structures of the minimal ideals of consta-dihedral group algebras.

In Section 4, with the structures of the consta-dihedral group algebras characterized in the last section, we construct a class of consta-dihedral codes, and characterize their algebraic property precisely (self-orthogonal, or LCD, etc.).

Section 5 is devoted to the study on the asymptotic property of consta-dihedral codes. The two theorems listed above are proved in this section.

In Section 6, we analyze the cause of the issue in [10] mentioned above, with the technique developed in this paper we address the issue and recover the correct version of [10, Theorem IV.5(1)].

Finally, conclusion is made in Section 7.

2 Preliminaries

In this paper $F$ is always a finite field with $|F|=q$ which is a power of a prime, where $|S|$ denotes the cardinality of any set $S$. And $n>1$ is an integer.

Let $G$ be a finite group, the group algebra $FG=\{\sum_{x\in{G}}a_{x}x\,|\,a_{x}\in{F}\}$ is the $F$-vector space with basis $G$ and equipped with the multiplication induced by the multiplication of the group. So $FG$ is an $F$-algebra, called the group algebra of $G$ over $F$. Any $\sum_{x\in{G}}a_{x}x\in{FG}$ is viewed as a word $(a_{x})_{x\in{G}}$ over $F$ with coordinates indexed by $G$. Any left ideal $C$ of $FG$ is called a group code of $G$ over $F$. We also say that $C$ is an $FG$-code for short.

It is an anti-automorphism of the group: $G\to G$, $g\mapsto g^{-1}$, which induces an anti-automorphism of the group algebra:

[TABLE]

We denote $\sum_{g\in G}a_{g}g^{-1}=\overline{\sum_{g\in G}a_{g}g}$, and call Eq.(2.1) the “bar” map of $FG$ for convenience. So, $\overline{\overline{a}}=a$, $\overline{ab}=\overline{b}\overline{a}$, for $a,b\in FG$. It is an automorphism of $FG$ once $G$ is abelian. The following is a linear form of $FG$:

[TABLE]

For $a=\sum_{g\in G}a_{g}g,b=\sum_{g\in G}b_{g}g\in{FG}$, the inner product $\langle a,b\rangle=\sum_{g\in G}a_{g}b_{g}$. The following is just [10, Lemma II.4].

Lemma 2.1**.**

(1)* $\sigma(ab)=\sigma(ba)$, $\forall~{}a,b\in FG$.*

(2)* $\langle a,b\rangle=\sigma(a\overline{b})=\sigma(\overline{a}b)$, $\forall~{}a,b\in FG$.*

(3)* $\langle da,\,b\rangle=\langle a,\,\overline{d}b\rangle$, $\forall~{}a,b,d\in FG$.*

(4)* If $C$ is an $FG$-code, then so is $C^{\bot}$.*

(5)* For $FG$-codes $C$ and $D$, $\langle C,D\rangle=0$ if and only if $C\overline{D}=0$.*

Let $H$ be a cyclic group of odd order $n>1$ with $\gcd(n,q)=1$. Let $e_{0},e_{1},\cdots,e_{\ell}$ be all primitive idempotents of $FH$, where $e_{0}=\frac{1}{n}\sum_{x\in{H}}x$. Thus, $FH$ is a semisimple algebra, i.e., $FH$ is the direct sum of simple ideals as follows:

[TABLE]

where $FHe_{i}$ being a field with identity $e_{i}$.

Since the bar map Eq.(2.1) is an automorphism of $FH$ of order $2$, it permutes the primitive idempotents $e_{0},e_{1},\cdots,e_{\ell}$ in Eq.(2.3) (note that $\overline{e}_{0}=e_{0}$). By Lemma 2.1(5) we have:

[TABLE]

For any ring $R$ (with identity $1_{R}$), by $R^{\times}$ we denote the unit group, i.e., the multiplicative group of all the units (invertible elements) of $R$. For the field $F$, $F^{\times}=F\backslash\{0\}$. By $\mathbb{Z}_{n}$ we denote the integer residue ring modulo $n$, hence $\mathbb{Z}_{n}^{\times}$ is the multiplicative group consisting of the reduced residue classes. Then $q\in{\mathbb{Z}_{n}^{\times}}$ (since $\gcd(n,q)=1$). In the multiplicative group $\mathbb{Z}_{n}^{\times}$, ${\rm ord}_{{\mathbb{Z}}_{n}^{\times}}(q)$ denotes the order of $q$, and ${\langle q\rangle}_{\mathbb{Z}_{n}^{\times}}$ denotes the cyclic subgroup generated by $q$. The following facts are well-known.

Lemma 2.2**.**

Keep the notation be as above in Eq.(2.3).

(1)* ([16, Theorem 1])  $\overline{e_{j}}=e_{j}$, $\forall j\geq 0$, if and only if $-1\in{\langle q\rangle}_{\mathbb{Z}_{n}^{\times}}$.*

(2)* ([2, Theorem 6])  $\overline{e_{j}}\neq e_{j}$, $\forall j>0$, if and only if ${\rm ord}_{{\mathbb{Z}}_{n}^{\times}}(q)$ is odd.*

Let $e_{0},e_{1},\cdots,e_{\ell}$, where $e_{0}=\frac{1}{n}\sum_{x\in H}x$, be the all primitive idempotents of $FH$ as in Eq.(2.3). Denote

[TABLE]

It is known (cf. [18, Lemma II.2]) that:

Lemma 2.3**.**

\lambda(n)=\min\big{\{}{\rm ord}_{{\mathbb{Z}}_{p}^{\times}}(q)\,\big{|}\;\mbox{p$runs over the prime divisors of$n}\big{\}}.

For an equation of $X,Y$ over $F$, we’ll need the following result.

Lemma 2.4**.**

Let $g\in F$ and $g\neq\pm 2$. Then the equation $X^{2}+gXY+Y^{2}=-1$ has a solution $(s,s^{\prime})\in F\times F$; and, there is a solution $(s,s^{\prime})$ such that $s^{\prime}=0$ if and only if either $q$ is even or $4\,|\,(q-1)$.

Proof.

Let ${\cal Q}=\{a^{2}\,|\,a\in F\}$ be a subset of $F$. If $q$ is even, then ${\cal Q}=F$ and the lemma holds obviously. Assume that $q$ is odd. Then $|{\cal Q}|=\frac{q+1}{2}$, and

[TABLE]

where $X^{\prime}=X+\frac{g}{2}Y$ and $b=1-\frac{g^{2}}{4}\neq 0$ (as $g\neq\pm 2$). Then \big{|}(-1-b{\cal Q})\big{|}=\big{|}{\cal Q}\big{|}, and {\big{|}{\cal Q}\big{|}+\big{|}(-1-b{\cal Q})\big{|}}=q+1>|F|. So ${\cal Q}\cap(-1-b{\cal Q})\neq\emptyset$, and there are $s^{\prime},t^{\prime}\in F$ such that $X^{\prime}=t^{\prime}$ and $Y=s^{\prime}$ satisfying $t^{\prime 2}=-1-bs^{\prime 2}$. Thus $X=s=t^{\prime}-\frac{gs^{\prime}}{2}$ and $Y=s^{\prime}$ are a solution of the equation. There is a solution $(s,s^{\prime})$ such that $s^{\prime}=0$ if and only if $-1$ is a square of $F$, so the second conclusion is obvious. ∎

By ${\rm M}_{2}(F)$ we denote the $F$-algebra consisting of all $F$-matrices of degree $2$.

Lemma 2.5**.**

Let $M={\rm M}_{2}(F)$, and $\varphi(X)=X^{2}+gX+1$ be an irreducible polynomial over $F$. Then there is a subalgebra $E$ of $M$ such that $E\cong F[X]/\langle\varphi(X)\rangle$, hence $E$ is an extension field over $F$ of degree $2$; and the following hold.

(1)* For any $f\in M$ with ${\rm rank}(f)=1$, $Ef=Mf=:L$ is a simple left ideal of $M$ and, for $0\neq c\in L$, $a,b\in E^{\times}$, $ac=cb$ if and only if $a=b\in F^{\times}$.*

(2)* There are altogether $q+1$ simple left ideals of $M$ as follows:*

[TABLE]

(3)* Let $L$ be a simple left ideal of $M$. When $\beta$ runs over $E^{\times}$, $L\beta$ runs over the simple left ideals of $M$, with each of them appears exactly $q-1$ times.*

Proof.

(1) and (3) have been proved in [10, Lemma III.6]. We prove (2). For $f\in M$ with ${\rm rank}(f)=1$ and $\alpha\in M$, each row of $\alpha f$ is a linear combination of the rows of $f$. Then the rows of $\alpha f$ with $\alpha$ running on $M$ form exactly a $1$-dimensional subspace of $F\times F$. For the $q+1$ vectors listed in (2), i.e., $(a,1)$, $a\in F$, and $(1,0)$, any two of them are linearly independent. Thus each of the $q+1$ vectors generates a $1$-dimensional subspace, and any two of the $q+1$ obtained $1$-dimensional subspaces are distinct. There are altogether $q+1$ $1$-dimensional subspaces of $F\times F$. So the $q+1$ simple left ideals listed in (2) are the all simple left ideals of $M$. ∎

3 Consta-dihedral group algebras

Keep the notation in Section 2. In this section we characterize consta-dihedral group algebras. From now on to the end of the paper we assume that (except for other explicit specified):

[TABLE]

i.e., $G$ is the dihedral group of order $2n$; and denote

[TABLE]

We further consider the group

[TABLE]

which is a semidirect product of the cyclic group $H=\langle u\rangle$ of order $n$ by the cyclic group $\tilde{T}=\langle\dot{v}\rangle$ of order $4$ with relation $\dot{v}u\dot{v}^{-1}=u^{-1}$. Obviously, $Z=\{1,\dot{v}^{2}\}$ is a central subgroup of $\tilde{G}$, and the quotient group $\tilde{G}/Z\cong G$ is the usual dihedral group of order $2n$. The $\tilde{G}$ is called a dicyclic group in literature, e.g., [3].

Definition 3.1**.**

Let ${F\kern-1.0pt{*}G}\kern 1.0pt$ be the $F$-vector space with basis

[TABLE]

and endowed with the $F$-linear multiplication induced by the multiplication of $\tilde{G}$ with identifying that $\dot{v}^{2}=-1\in F$, i.e., subject to the following relations:

[TABLE]

The $F$-algebra {F\kern-1.0pt{*}G}\kern 1.0pt=\big{\{}\sum_{h\in H}a_{h}h+\!\sum_{h\in H}a_{h\dot{v}}h\dot{v}\,\big{|}\,a_{h},a_{h\dot{v}}\in F\big{\}}, i.e., {F\kern-1.0pt{*}G}\kern 1.0pt=\big{\{}\sum_{j=0}^{1}\sum_{i=0}^{n-1}a_{ij}u^{i}\dot{v}^{j}\,\big{|}\,a_{ij}\in F\big{\}}, is called the consta-dihedral group algebra, and any left ideal of ${F\kern-1.0pt{*}G}\kern 1.0pt$ is called a consta-dihedral code (cf. [24]).

In another notation, the consta-dihedral group algebra

[TABLE]

where $F[X,Y]$ is the non-commutative $F$-polynomial algebra of $X$ and $Y$, and $\langle X^{n}\!-\!1,\,Y^{2}+1,\,XYX-Y\rangle$ is the ideal generated by $X^{n}\!-\!1,\,Y^{2}+1,\,XYX-Y$. Therefore, ${F\kern-1.0pt{*}G}\kern 1.0pt$ is identified with the quotient algebra of the group algebra $F\tilde{G}$ over the ideal $\langle\dot{v}^{2}+1\rangle$ generated by $\dot{v}^{2}+1$:

[TABLE]

Remark 3.2**.**

For any finite group $G$, by a general theory ([8, p.268]), a function $\alpha:G\times G\rightarrow F^{\times}$ is called a $2$-cocycle of G if

[TABLE]

For a $2$-cocycle $\alpha$, the twisted group algebra of $G$ by $\alpha$, denoted by $F^{\alpha}G$, is the $F$-vector space with basis $G$ and endowed with the $F$-bilinear product $FG\times FG\rightarrow FG$ defined by

[TABLE]

(the associativity of the multiplication follows from Eq.(3.5)). Turn back to the dihedral group $G=\langle u,v\,|\,u^{n}=1=v^{2},uvu^{-1}=u^{-1}\rangle$. It is easy to check that the following $\gamma$ is a $2$-cocycle of $G$:

[TABLE]

and the consta-dihedral group algebra ${F\kern-1.0pt{*}G}\kern 1.0pt$ defined above (Definition 3.1) is just the twisted group algebra $F^{\gamma}G$ by the above $2$-cocycle $\gamma$.

Remark 3.3**.**

By Eq.(2.1), we have the bar map on the group algebra $F\tilde{G}$:

[TABLE]

Since $\overline{\dot{v}^{2}}=\dot{v}^{2}$ (as $\dot{v}^{4}=1$), the ideal $\langle\dot{v}^{2}+1\rangle$ of $F\tilde{G}$ is invariant by the bar map (i.e., $\overline{\langle\dot{v}^{2}+1\rangle}=\langle\dot{v}^{2}+1\rangle$), so the bar map of $F\tilde{G}$ induces a transformation (called and denoted by “bar map” again) of the quotient algebra ${F\kern-1.0pt{*}G}\kern 1.0pt=F\tilde{G}/\langle\dot{v}^{2}+1\rangle$ (see Eq.(3.4)) as follows: for $\sum_{h\in H}a_{h}h+\!\sum_{h\in H}a_{h\dot{v}}h\dot{v}\in{F\kern-1.0pt{*}G}\kern 1.0pt$,

[TABLE]

Obviously, $\overline{\dot{v}}=\dot{v}^{-1}=-\dot{v}$; and $\dot{v}\,a=\overline{a}\,\dot{v}$, $\forall\,a\in FH$. Eq.(3.7) can be rewritten in a linear combination of the standard basis of ${F\kern-1.0pt{*}G}\kern 1.0pt$ in Eq.(3.2):

[TABLE]

Because the bar map on ${F\kern-1.0pt{*}G}\kern 1.0pt$ is induced by the bar map on $F\tilde{G}$ which is an anti-automorphism, the bar map on ${F\kern-1.0pt{*}G}\kern 1.0pt$ is again an anti-automorphism:

[TABLE]

Remark 3.4**.**

Similarly to Eq.(2.2), with the basis Eq.(3.2) we get the map

[TABLE]

which is a linear form of ${F\kern-1.0pt{*}G}\kern 1.0pt$. And we have that:

• •

Lemma 2.1 is still valid for the consta-dihedral group algebra ${F\kern-1.0pt{*}G}\kern 1.0pt$.

The proof is similar to [10, Lemma II.4]. For any elements $a,b\in{{F\kern-1.0pt{*}G}\kern 1.0pt}$,

[TABLE]

Rewriting it as a linear combination of the basis Eq.(3.2) and picking up the coefficient of $u^{0}\dot{v}^{0}=1_{\tilde{G}}$, we get

[TABLE]

where the sum is over the indexes $(iji^{\prime}j^{\prime})$ satisfying the following two:

(i)  $0\leq i,i^{\prime}<n$ and $0\leq j,j^{\prime}<2$;

(ii)  $u^{i}\dot{v}^{j}\dot{v}^{-j^{\prime}}u^{-i^{\prime}}=s_{iji^{\prime}j^{\prime}}\cdot 1_{\tilde{G}}$ for an $s_{iji^{\prime}j^{\prime}}\in F$.

By (i), it is easy to see that (ii) holds if and only if $i=i^{\prime}$ and $j=j^{\prime}$; and at that case $s_{iji^{\prime}j^{\prime}}=1$ (note that (i) is necessary for the conclusion; e.g, $\dot{v}^{3}\cdot\dot{v}^{-1}=-1\cdot 1_{\tilde{G}}$ in ${F\kern-1.0pt{*}G}\kern 1.0pt$ but $3\neq 1$). So $\sigma(a\bar{b})=\sum_{j=0}^{1}\sum_{i=0}^{n-1}a_{ij}b_{ij}$; i.e., (1) of Lemma 2.1 holds for ${F\kern-1.0pt{*}G}\kern 1.0pt$. In a similar way, (5) of Lemma 2.1 for ${F\kern-1.0pt{*}G}\kern 1.0pt$ holds. And, (2), (3) and (4) of Lemma 2.1 for ${F\kern-1.0pt{*}G}\kern 1.0pt$ can be checked by (1) directly. In particular, if $C$ is a consta-dihedral code, then so is the orthogonal code $C^{\bot}$.

By Definition 3.1, the cyclic group algebra $FH$ is a commutative subalgebra of the consta-dihedral group algebra ${F\kern-1.0pt{*}G}\kern 1.0pt$; and, as $FH$-modules, we have

[TABLE]

Lemma 3.5**.**

Let $FH=FHe_{0}\oplus FHe_{1}\oplus\cdots\oplus FHe_{\ell}$ as in Eq.(2.3). Then the idempotent $e_{0}$ is central in ${F\kern-1.0pt{*}G}\kern 1.0pt$ and the ideal ${F\kern-1.0pt{*}G}\kern 1.0pte_{0}$ is a commutative $F$-algebra of dimension $2$, and the following hold.

(1)* If $q$ is odd and $4\nmid(q-1)$, then ${F\kern-1.0pt{*}G}\kern 1.0pte_{0}$ is a field extension over $F$ with degree $|{F\kern-1.0pt{*}G}\kern 1.0pte_{0}:F|=2$.*

(2)* If either $q$ is even or $4\,|\,(q-1)$, then there is an element $r\in F$ such that $r^{2}=-1$ and $C_{0}={F\kern-1.0pt{*}G}\kern 1.0pt(re_{0}+e_{0}\dot{v})$ is an $1$-dimensional ideal of ${F\kern-1.0pt{*}G}\kern 1.0pte_{0}$, and $\langle C_{0},C_{0}\rangle=0$.*

Proof.

Since $\overline{e_{0}}=e_{0}$, $\dot{v}e_{0}=\overline{e_{0}}\,\dot{v}=e_{0}\dot{v}$. So $e_{0}$ is a central element of ${F\kern-1.0pt{*}G}\kern 1.0pt$. It is known that ${F\kern-1.0pt{*}G}\kern 1.0pte_{0}=FHe_{0}\oplus FHe_{0}\dot{v}$ and $FHe_{0}=\{ae_{0}\,|\,a\in F\}\cong F$. Thus ${F\kern-1.0pt{*}G}\kern 1.0pte_{0}$ is a commutative $F$-algebra with $e_{0}$, $e_{0}\dot{v}$ being a basis.

(1) Since the group $F^{\times}$ is a cyclic group having no element of order $4$, the polynomial $X^{2}+1$ is irreducible over $F$. Because $(e_{0}\dot{v})^{2}=-e_{0}$, we have an isomorphism ${F\kern-1.0pt{*}G}\kern 1.0pte_{0}\cong F[X]/\langle X^{2}+1\rangle$ which is a field extension over $F$ of degree $2$.

(2) If $q$ is even, then $-1=1$ and $r=1$ satisfies that $r^{2}=-1$. If $4\,|(q-1)$, then $F^{\times}$ has an element $r$ of order $4$, and so $r^{2}=-1$. Thus

[TABLE]

So $\dim_{F}({F\kern-1.0pt{*}G}\kern 1.0pt(re_{0}+e_{0}\dot{v}))=1$. And

[TABLE]

By Remark 3.4 and Lemma 2.1(5), $\langle C_{0},C_{0}\rangle=0$. ∎

Lemma 3.6**.**

Keep the notation in Eq.(3.8) and Eq.(2.3). Let e be a primitive idempotent of $FH$ other than $e_{0}$ with $\overline{e}\neq e$. Then $\tilde{F}:=FHe$ is a field extension over F, $e+\overline{e}$ is a primitive central idempotent of ${F\kern-1.0pt{*}G}\kern 1.0pt$ and:

(1)* The ideal ${F\kern-1.0pt{*}G}\kern 1.0pt(e+\overline{e})=FHe\oplus FH\bar{e}\oplus FHe\dot{v}\oplus FH\bar{e}\dot{v}\cong{\rm M}_{2}(\tilde{F}).$*

(2)* With the isomorphism in (1), if $f\in{F\kern-1.0pt{*}G}\kern 1.0pt(e+\overline{e})$ corresponds to the matrix $\begin{pmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{pmatrix}\in{\rm M}_{2}(\tilde{F})$, then $\overline{f}$ corresponds to the matrix $\begin{pmatrix}a_{22}&-a_{12}\\ -a_{21}&a_{11}\end{pmatrix}$.*

Proof.

Since $e\bar{e}=0$, $e+\bar{e}$ is an idempotent of ${F\kern-1.0pt{*}G}\kern 1.0pt$. For $\dot{v}\in{{F\kern-1.0pt{*}G}\kern 1.0pt}$, we have $\dot{v}(e+\bar{e})=\dot{v}e+\dot{v}\bar{e}=\bar{e}\dot{v}+e\dot{v}=(\bar{e}+e)\dot{v}$ . Thus, $e+\bar{e}$ is a central element of ${F\kern-1.0pt{*}G}\kern 1.0pt$. So, ${F\kern-1.0pt{*}G}\kern 1.0pt(e+\bar{e})=FHe\oplus FHe\dot{v}\oplus FH\bar{e}\oplus FH\bar{e}\dot{v}$ is an ideal of ${F\kern-1.0pt{*}G}\kern 1.0pt$. We first show an $\tilde{F}$-algebra isomorphism. Define a map:

[TABLE]

which is a linear isomorphism. For $a_{ij},b_{ij}\in\widetilde{F}$, $1\leq i,j\leq 2$, noting that $\dot{v}a_{ij}=\overline{a_{ij}}\,\dot{v}$ and $\dot{v}\dot{v}=-1$, we have

[TABLE]

So, Eq.(3.11) is an $\tilde{F}$-algebra isomorphism, and (1) holds.

Next, we have the bar map image of $a_{11}e-a_{12}e\dot{v}+\overline{a_{21}}\,\bar{e}\dot{v}+\overline{a_{22}}\,\bar{e}$ (note that $\overline{\dot{v}}={\dot{v}}^{-1}=-\dot{v}$) as follows:

[TABLE]

Thus, this image corresponds the matrix:

[TABLE]

We are done. ∎

Lemma 3.7**.**

Let e be a primitive idempotent of FH with $\bar{e}=e\neq e_{0}$. Then $FHe$ is a field extension over F, $e$ is a primitive central idempotent of ${F\kern-1.0pt{*}G}\kern 1.0pt$, $\tilde{F}:=\{a\,|\,a\in{FHe},\,a=\bar{a}\}$ is a subfield of $FHe$ with degree $|FHe:\tilde{F}|=2$, the ideal ${F\kern-1.0pt{*}G}\kern 1.0pte=FHe\oplus FHev\cong{\rm M}_{2}(\tilde{F})$, and the center ${\rm Z}({F\kern-1.0pt{*}G}\kern 1.0pte)=\tilde{F}$.

Proof.

Since $\dot{v}e=\bar{e}\dot{v}=e\dot{v}$ (as $e=\bar{e}$), $e$ is a primitive central idempotent of ${F\kern-1.0pt{*}G}\kern 1.0pt$ . The $FHe$ is a field with identity e. Since $n>1$ is odd, $\tilde{F}$ is a subfield of $FHe$ and $|FHe:\tilde{F}|=2$ (cf. [18, Lemma II.3]). Since $FHe=\sum_{i=0}^{n-1}Fu^{i}e=\sum_{i=0}^{n-1}F(ue)^{i}$, $FHe=\tilde{F}\oplus\tilde{F}(ue)$ is an extension over $\tilde{F}$ by the element $ue$. And, the minimal polynomial of $ue$ over $\tilde{F}$ is $\varphi_{ue}(X)=X^{2}+gX+1$, where $\pm 2\neq g\in\tilde{F}$ such that $g$ and $2$ cannot be both zero in $\tilde{F}$ (because $\varphi_{ue}(X)$ is irreducible); cf. [10, Lemma III.3]. By Lemma 2.4, we take $s,s^{\prime}\in\tilde{F}$ such that $s^{2}+gss^{\prime}+s^{\prime 2}=-1$, and set

[TABLE]

Then the characteristic polynomial of $\eta$ is $\varphi_{\eta}(X)=X^{2}+gX+1=\varphi_{ue}(X)$, and $\nu^{2}=-\varepsilon$ and $\nu\eta\nu^{-1}=\eta^{-1}$. Mapping $e\mapsto\varepsilon$, $ue\mapsto\eta$ and $\dot{v}e\mapsto\nu$, we get

[TABLE]

and ${\rm M}_{2}(\tilde{F})=\tilde{F}\varepsilon\oplus\tilde{F}\eta\oplus\tilde{F}\nu\oplus\tilde{F}\eta\nu$, and the following (where $a,b,c,d\in{\tilde{F}}$)

[TABLE]

is an algebra isomorphism. Any $ae\in\tilde{F}$ is mapped to $a\varepsilon$; so $\tilde{F}={\rm Z}({F\kern-1.0pt{*}G}\kern 1.0pte)$. ∎

Theorem 3.8**.**

The consta-dihedral group algebra ${F\kern-1.0pt{*}G}\kern 1.0pt$ is an orthogonal direct sum of ideals $A_{t}$:

[TABLE]

where $A_{0}={F\kern-1.0pt{*}G}\kern 1.0pte_{0}$ is described in Lemma 3.5 and, for $1\leq t\leq m$, the ideal $A_{t}\cong M_{2}(F_{t})$ with $F_{t}$ is a field extension over F with $\dim_{F}F_{t}=k_{t}$ and one of the following two holds:

(1)* The identity $1_{A_{t}}=e+\bar{e}$ for a primitive idempotent $e$ of $FH$ with $\bar{e}\neq e$, and $A_{t}={F\kern-1.0pt{*}G}\kern 1.0pt(e+\bar{e})\cong{\rm M}_{2}(F_{t})$ as in Eq.(3.11), where $F_{t}=FHe$.*

(2)* The identity $1_{A_{t}}=e$ for a primitive idempotent $e$ of $FH$ with $\bar{e}=e\neq e_{0}$, and $A_{t}={F\kern-1.0pt{*}G}\kern 1.0pte\cong{\rm M}_{2}(F_{t})$ as in Eq.(3.14), where $F_{t}=\{a\,|\,a\in{FHe},\,a=\bar{a}\}$ is the subfield of the field $FHe$ with degree $|FHe:F_{t}|=2$.*

Proof.

By Lemma 3.5, Lemma 3.6 and Lemma 3.7, the direct sum in Eq.(3.15) follows at once; and for $1\leq t\leq m$, either (1) or (2) holds. If $0\leq s\neq t\leq m$,

[TABLE]

By Remark 3.4 and Lemma 2.1(5), $\langle A_{s},A_{t}\rangle=0$, $0\leq s\neq t\leq m$. That is, Eq.(3.15) is an orthogonal decomposition. ∎

Corollary 3.9**.**

Keep the notation in Theorem 3.8. Let $C$, $D$ be any left ideals of ${F\kern-1.0pt{*}G}\kern 1.0pt$. Then:

(1)* $C=C_{0}\oplus C_{1}\oplus\cdots\oplus C_{m}$, where $C_{t}=C\cap A_{t}=1_{A_{t}}\!\cdot\!C$, $t=0,1,\cdots,m$. (We call $C_{t}$ the $A_{t}$-component of $C$.)*

(2)* Let $D=D_{0}\oplus D_{1}\oplus\cdots\oplus D_{m}$ be as in (1). Then for $c=c_{0}+c_{1}+\cdots+c_{m}\in C$ and $d=d_{0}+d_{1}+\cdots+d_{m}\in D$, the inner product $\langle c,d\rangle=\sum_{t=0}^{m}\langle c_{t},d_{t}\rangle$.*

(3)* $C^{\bot}=C_{0}^{\bot_{\!A_{0}}}\oplus C_{1}^{\bot_{\!A_{1}}}\oplus\cdots\oplus C_{m}^{\bot_{\!A_{m}}}$, where $C_{t}^{\bot_{\!A_{t}}}$, $0\leq t\leq m$, denotes the orthogonal subspace of $C_{t}$ in $A_{t}$. Both $C_{t}$ and $C_{t}^{\bot_{\!A_{t}}}$ are left ideals of $A_{t}$.*

(4)* $C$ is self-orthogonal if and only if every $C_{t}$, $0\leq t\leq m$, is self-orthogonal in $A_{t}$.*

(5)* $C$ is LCD if and only if every $C_{t}$, $0\leq t\leq m$, is LCD in $A_{t}$.*

Proof.

(1). By Eq.(3.15), $1=1_{A_{0}}+1_{A_{1}}+\cdots+1_{A_{m}}$. Since $C_{t}\subseteq C$, we have $C_{0}\oplus C_{1}\oplus\cdots\oplus C_{m}\subseteq C$. On the other hand, for $c\in C$,

[TABLE]

so $C\subseteq C_{0}\oplus C_{1}\oplus\cdots\oplus C_{m}$. We call $c_{t}=1_{\!A_{t}}\!\cdot\!c$ the $A_{t}$-component of $c$.

(2). By Remark 3.4 and Lemma 2.1(1), we have

[TABLE]

(3). By (1), $C^{\bot}=C_{0}^{\prime}\oplus C_{1}^{\prime}\oplus\cdots\oplus C_{m}^{\prime}$, where $C_{t}^{\prime}=C^{\bot}\!\cap\!A_{t}$. Since $C_{t}=C\cap A_{t}\subseteq C$ and $C^{\bot}\cap A_{t}\subseteq C^{\bot}$, $\langle C_{t},\,C_{t}^{\prime}\rangle\subseteq\langle C,\,C^{\bot}\rangle=0$. Thus $C_{t}^{\prime}\subseteq C_{t}^{\bot_{A_{t}}}$. Conversely, assume that $a_{t}\in C_{t}^{\bot_{A_{t}}}$, then the $A_{j}$-component of $a_{t}$ is zero provided $j\neq t$; for any $c=c_{0}+c_{1}+\cdots+c_{m}\in C$, by the above (2), $\langle c,a_{t}\rangle=\langle c_{t},a_{t}\rangle=0$; so $a_{t}\in C^{\bot}\cap A_{t}=C^{\prime}_{t}$. We get that $C_{t}^{\bot_{A_{t}}}\subseteq C^{\prime}_{t}$.

(4). If $\langle C,C\rangle=0$, since $C_{t}\subseteq C$, we have $\langle C_{t},C_{t}\rangle\subseteq\langle C,C\rangle=0$; i.e., $C_{t}$ is self-orthogonal in $A_{t}$. Conversely, if $\langle C_{t},C_{t}\rangle=0$ for all $t=0,1,\cdots,m$, then, by the above (2), $\langle C,C\rangle=\sum_{t=0}^{m}\langle C_{t},C_{t}\rangle=0$.

(5). As $(C\cap D)\cap A_{t}=(C\cap A_{t})\cap(D\cap A_{t})$, with notation in (1) we have

[TABLE]

By (3), $C^{\bot}\cap A_{t}=C_{t}^{\bot_{A_{t}}}$. Applying Eq.(3.16) to $D=C^{\bot}$, we get

[TABLE]

Therefore, $C\cap C^{\bot}=\{0\}$ if and only if $C_{t}\cap C_{t}^{\bot_{A_{t}}}=\{0\}$ for $t=0,1,\cdots,m$. ∎

Corollary 3.10**.**

Keep the notation in Theorem 3.8.

(1)* $k_{1}+k_{2}+\cdots+k_{m}=\frac{n-1}{2}$.*

(2)* $2k_{t}\geq\lambda(n)$, $t=1,\ldots,m$, where $\lambda(n)$ is defined in Eq.(2.5).*

Proof.

By Eq. (3.15), we have that $2n=\dim_{F}{F\kern-1.0pt{*}G}\kern 1.0pt=\sum_{t=0}^{m}\dim_{F}A_{t}$, where $\dim_{F}A_{0}=\dim_{F}{F\kern-1.0pt{*}G}\kern 1.0pte_{0}=2$, see Lemma 3.5(1). Since $A_{t}\cong M_{2}(F_{t})$ for $t=1,\cdots,m$, $\dim_{F}A_{t}=4k_{t}$, and so $2(k_{1}+k_{2}+\cdots+k_{m})=n-1$. The second conclusion is obvious. ∎

4 Consta-dihedral codes

Any left ideal of the consta-dihedral algebra ${F\kern-1.0pt{*}G}\kern 1.0pt$ is called a consta-dihedral code over $F$ of length $2n$, cf. Definition 3.1. In this section we construct some consta-dihedral codes and investigate their algebraic properties.

Remark 4.1**.**

Keep the notation in Theorem 3.8: ${F\kern-1.0pt{*}G}\kern 1.0pt=A_{0}\oplus A_{1}\oplus\cdots\oplus A_{m}$, where $A_{0}={F\kern-1.0pt{*}G}\kern 1.0pte_{0}$ and, for $t=1,\cdots,m$, the ideal $A_{t}\cong{\rm M}_{2}(F_{t})$ as described in Theorem 3.8(1) and (2), and $\dim_{F}F_{t}=k_{t}$. By the isomorphism $A_{t}\cong{\rm M}_{2}(F_{t})$, applying Lemma 2.5 to ${\rm M}_{2}(F_{t})$, we get a field $K_{t}\subseteq A_{t}$ corresponding the subfield (denoted by $E$ in Lemma 2.5) of ${\rm M}_{2}(F_{t})$ of dimension $2$ over $F_{t}$. So $\dim_{F}K_{t}=2k_{t}$ because $\dim_{F}F_{t}=k_{t}$. And the following hold.

• (1)

The following is a subgroup of the multiplicative unit group $({F\kern-1.0pt{*}G}\kern 1.0pt)^{\times}$:

[TABLE]

where $K_{t}^{\times}=K_{t}\backslash\{0\}$ is the multiplicative unit group of the field $K_{t}$. If $C$ is a left ideal of $A_{t}$ and $\beta\in K_{t}^{\times}$, then $C\beta$ is a left ideal of $A_{t}$ which is isomorphic to $C$ (cf. Lemma 2.5).

• (2)

If $1_{A_{t}}=e$ for a primitive idempotent $e$ of $FH$ with $\bar{e}=e\neq e_{0}$, then we choose $K_{t}=FHe$, hence $F_{t}=\tilde{F}=\{a\in{K_{t}}\,|\,\bar{a}=a\}={\rm Z}(A_{t})$ as described in Lemma 3.7, where ${\rm Z}(A_{t})$ denotes the center of $A_{t}$.

Lemma 4.2**.**

Assume that $1_{A_{t}}\!=e+\bar{e}$ for a primitive idempotent $e$ of $FH$ with $\bar{e}\neq e$. Let $C_{t}=A_{t}e$ and $\beta_{t}\in K_{t}^{\times}$. Then $C_{t}\beta_{t}$ is a simple left ideal of $A_{t}$ and $\langle C_{t}\beta_{t},C_{t}\beta_{t}\rangle=0$.

Proof.

Let $M:={\rm M}_{2}(F_{t})$, where $F_{t}=FHe$, see Theorem 3.8(1). By Eq.(3.11), $C_{t}=A_{t}e$ corresponds to $M\!\cdot\!\begin{pmatrix}1&0\\ 0&0\end{pmatrix}$, which is a simple left ideal of $M$. Hence $C_{t}$ and $C_{t}\beta_{t}$ are simple left ideal of $A_{t}$. If $C_{t}\beta_{t}=C_{t}$, then $C_{t}\beta_{t}\cdot\overline{C_{t}\beta_{t}}=A_{t}e\overline{A_{t}e}=A_{t}e\overline{e}A_{t}=0$. By Remark 3.4 and Lemma 2.1(5), $\langle C_{t}\beta,C_{t}\beta\rangle=0$. In the following, we assume that $C_{t}\beta_{t}\neq C_{t}$. By Lemma 3.6 and Lemma 2.5, the simple left ideal $C_{t}\beta_{t}$ corresponds the simple left ideal $Mf$ of $M$ as follows

[TABLE]

Let $f^{\prime}\in A_{t}$ correspond $f\in M$ by the isomorphism Eq.(3.11). Then $C_{t}\beta_{t}=A_{t}f^{\prime}$. By Lemma 3.6(2) (cf. Eq.(3.12)), $\overline{f^{\prime}}$ corresponds the matrix $\begin{pmatrix}0&-1\\ 0&a\end{pmatrix}$. Because

[TABLE]

So $C_{t}\beta_{t}\cdot\overline{C_{t}\beta_{t}}=A_{t}f^{\prime}\!\cdot\!\overline{f^{\prime}}A_{t}=0$, i.e., $\langle C_{t}\beta_{t},C_{t}\beta_{t}\rangle=0$. ∎

Lemma 4.3**.**

Assume that $1_{\!A_{t}}\!=e$ for a primitive idempotent $e$ of $FH$ with $\bar{e}=e\neq e_{0}$, and set $C_{t}=A_{t}f$ where $f=se-s^{\prime}ue+\dot{v}e\in A_{t}$ with $s,s^{\prime}\in F_{t}$ as in Eq.(3.13). Then $C_{t}$ is a simple left ideal of $A_{t}$, and for any $\beta_{t}\in K_{t}^{\times}$,

[TABLE]

Proof.

Note that in this case we choose $K_{t}=FHe$, $F_{t}=\{a\,|\,a\in K_{t},\overline{a}=a\}={\rm Z}(A_{t})$ with $|K_{t}:F_{t}|=2$, see Theorem 3.8(2) and Remark 4.1(2). By Eq.(3.13) and its notation, $g,s,s^{\prime}\in F_{t}$ satisfy that $g$ and $2$ are not both zero, $g\neq\pm 2$ which implies that $\det\begin{pmatrix}2&g\\ g&2\end{pmatrix}\neq 0$, and $s^{2}+gss^{\prime}+s^{\prime 2}=-1$. So the matrix

[TABLE]

has non-zero first column, hence ${\rm rank}(s\varepsilon-s^{\prime}\eta+\nu)=1$. By Lemma 2.5(1), ${\rm M}_{2}(F_{t})(s\varepsilon-s^{\prime}\eta+\nu)$ is a simple left ideal of ${\rm M}_{2}(F_{t})$. By the isomorphism Eq.(3.14), $s\varepsilon-s^{\prime}\eta+\nu$ corresponds the element $f=se-s^{\prime}ue+\dot{v}e\in A_{t}$, and $C_{t}=A_{t}f$ is a simple left ideal of $A_{t}$. For $\beta_{t}\in K_{t}^{\times}$, obviously, $\overline{\beta_{t}\bar{\beta}_{t}}=\beta_{t}\bar{\beta}_{t}$; so $\beta_{t}\bar{\beta}_{t}\in F_{t}={\rm Z}(A_{t})$. Then we have

[TABLE]

Thus $C_{t}\beta\cdot\overline{C_{t}\beta}=0$ if and only if $f\,\overline{f}=0$. Since $\overline{u}=u^{-1}$ and $\overline{\dot{v}}=-\dot{v}$, we get

[TABLE]

By Eq.(3.14), $u^{-1}e$ corresponds the matrix

[TABLE]

So we have the following correspondence:

[TABLE]

Thus $ue-u^{-1}e$ is invertible because $\det\begin{pmatrix}g&-2\\ 2&-g\end{pmatrix}=4-g^{2}\neq 0$. And $(s^{2}+s^{\prime 2}+1)e-ss^{\prime}(u^{-1}e+ue)$ corresponds to the matrix

[TABLE]

Because $s^{2}+s^{\prime 2}+1+gss^{\prime}=0$, we get $(s^{2}+s^{\prime 2}+1)e-ss^{\prime}(u^{-1}e+ue)=0$. Therefore

[TABLE]

It follows that $f\,\bar{f}=0$ if and only if $s^{\prime}=0$. As $s,s^{\prime}\in F_{t}$ and $|F_{t}|=q^{k_{t}}$, Eq.(4.2) follows from Lemma 2.4. ∎

Remark 4.4**.**

In the following we always fix:

— $C_{t}=A_{t}e$ as in Lemma 4.2 if it is the case of Theorem 3.8(1);

— $C_{t}=A_{t}f$ as in Lemma 4.3 if it is the case of Theorem 3.8(2);

and consider the following consta-dihedral code

[TABLE]

We have the following:

(1)  For $C$ in Eq.(4.4) the rate ${\rm R}(C)=\frac{1}{2}-\frac{1}{2n}$, because

[TABLE]

(2)  By $C_{t}^{\bot_{\!A_{t}}}$ we denote the orthogonal submodule (left ideal) of $C_{t}$ in $A_{t}$ (see Corollary 3.9(3)), and so $C_{t}\cap C_{t}^{\bot_{\!A_{t}}}$ is still a left ideal of $A_{t}$. Since $C_{t}$ is simple, $C_{t}\cap C_{t}^{\bot_{\!A_{t}}}$ is either $C_{t}$ or [math]. Hence,

[TABLE]

Lemma 4.5**.**

Assume that $q$ is odd and $4\nmid(q-1)$. Assume that there are $m^{\prime}$ indexes $1\leq i_{1}<\cdots<i_{m^{\prime}}\leq m$ such that, for $t=1,\cdots,m^{\prime}$, $e_{i_{t}}=\overline{e}_{i_{t}}$ and $k_{i_{t}}$ is odd. Let $C^{\prime}=C_{i_{1}}\oplus\cdots\oplus C_{i_{m^{\prime}}}$ and $\widehat{C}^{\prime}=A_{0}\oplus C^{\prime}$. Then for any $\beta\in K^{*}$ both $C^{\prime}\beta$ and $\widehat{C}^{\prime}\beta$ are LCD consta-dihedral codes.

Proof.

Note that $q^{k_{i_{t}}}-1=(q-1)(q^{k_{i_{t}}-1}+q^{k_{i_{t}}-2}+\cdots+q+1)$. Since both $k_{i_{t}}$ and $q$ are odd and $4\nmid(q-1)$, we have $4\nmid(q^{k_{i_{t}}}-1)$. Write $\beta=e_{0}+\beta_{1}+\cdots+\beta_{m}$, where $\beta_{t}\in K_{t}^{\times}$ for $t=1,\cdots,m$. Then

[TABLE]

By Lemma 4.3, $\langle C_{i_{t}}\beta_{i_{t}},C_{i_{t}}\beta_{i_{t}}\rangle\neq 0$ for $t=1,\cdots,m^{\prime}$. By Eq(4.6), we have $C_{i_{t}}\beta_{i_{t}}\cap(C_{i_{t}}\beta_{i_{t}})^{\bot_{A_{i_{t}}}}=0$. Then $C_{i_{t}}\beta_{i_{t}}$ is an LCD consta-dihedral code in $A_{i_{t}}$. By Lemma 3.9 (5), $C^{\prime}\beta$ is an LCD consta-dihedral code. Moreover, by Lemma 3.5(1), $\langle A_{0},A_{0}\rangle=A_{0}\overline{A_{0}}=A_{0}\neq 0$; so $\widehat{C}^{\prime}\beta$ is an LCD consta-dihedral code. ∎

For integers $s,t$ and a prime $p$, $p^{s}\,\|\,t$ means that $p^{s}\,|\,t$ but $p^{s+1}\nmid t$.

Theorem 4.6**.**

Assume that $q$ is odd and $4\nmid(q-1)$. Assume that $-1\in{\langle q\rangle}_{\mathbb{Z}_{n}^{\times}}$ and $2\,\|\,{\rm ord}_{{\mathbb{Z}}_{n}^{\times}}(q)$. Then for any $\beta\in K^{*}$, both $C\beta$ and $A_{0}\oplus C\beta$ are LCD consta-dihedral codes.

Proof.

Since $-1\in{\langle q\rangle}_{\mathbb{Z}_{n}^{\times}}$, by Lemma 2.2(1), $\overline{e}_{t}=e_{t}$, $t=1,\cdots,m$. By Lemma 4.5, it is enough to show that any $k_{t}$ is odd for $1\leq t\leq m$. By Theorem 3.8(2), $k_{t}=\frac{1}{2}\dim_{F}FHe_{t}$, i.e., $FHe_{t}$ is a field extension over $F$ with degree $2k_{t}$. There exists a $q$-coset $Q\subseteq{\mathbb{Z}}_{n}$ such that $\dim_{F}FHe_{t}=|Q|$. However, $|Q|$ is a divisor of ${\rm ord}_{{\mathbb{Z}}_{n}^{\times}}(q)$. Hence, by the assumption that $2\,\|\,{\rm ord}_{{\mathbb{Z}}_{n}^{\times}}(q)$, $k_{t}=\frac{1}{2}|Q|$ is odd. ∎

Theorem 4.7**.**

Assume that $q$ is even or 4\,\big{|}\,(q-1), Assume that $r\in F$ satisfies that $r^{2}=-1$. Set $C_{0}=A_{0}(re_{0}+e_{0}\dot{v})$ as in Lemma 3.5(2), and

[TABLE]

Then, for any $\beta\in K^{*}$, $\widehat{C}\beta$ is a self-dual consta-dihedral code.

Proof.

By Eq.(4.1) we write $\beta=e_{0}+\beta_{1}+\cdots+\beta_{m}$ with $\beta_{t}\in K_{t}^{\times}$ for $t=1,\cdots,m$. Then

[TABLE]

and by Corollary 3.9(2),

[TABLE]

By Lemma 4.2 and Lemma 4.3, $\langle C_{t}\beta_{t},C_{t}\beta_{t}\rangle=0$, $t=1,\cdots,m$. By Lemma 3.5, $\langle C_{0},C_{0}\rangle=0$. Hence $\widehat{C}\beta$ is self-orthogonal. Finally, by Lemma 3.5 and Eq.(4.5), $\dim_{F}(\widehat{C}\beta)=\dim_{F}(C_{0})+\dim_{F}(C)=n$. So $\widehat{C}\beta$ is self-dual. ∎

Remark 4.8**.**

Recall that, for any subset $I_{*}=\{i_{1},\cdots,i_{k}\}\subseteq I=\{1,2,\cdots,n\}$ ($1\leq i_{1}<\cdots<i_{k}\leq n$), there is the projection $\rho_{I_{*}}:$ ​$F^{I}\to F^{I_{*}}$,​ $(a_{1},a_{2}\cdots,a_{n})\mapsto(a_{i_{1}},\cdots,a_{i_{k}})$. For $B\subseteq F^{n}$ with $|B|=q^{k}$, if there are subsets (repetition is allowed) $I_{1},\cdots,I_{s}$ of $\{1,2,\cdots,n\}$ such that: (1) for $1\leq j\leq s$, the projection $\rho_{I_{j}}:$ $F^{n}\to F^{I_{j}}$ maps $B$ bijectively onto $F^{I_{j}}$ (such $I_{j}$ is called an information index set of the code $B$); (2) there is an integer $t$ such that for any $1\leq i\leq n$ the number of the subsets $I_{j}$ which contains $i$ (i.e., $i\in I_{j}$) equals $t$; then $B$ is called a balanced code. An important result (cf. [9, Corollary 3.4]) is that, if $B$ is balanced, then the cardinality $|B^{\leq\delta}|\leq q^{kh_{q}(\delta)}$ for $0\leq\delta\leq 1-q^{-1}$, where

[TABLE]

and

[TABLE]

is the $q$-entropy function. It is easy to see that group codes are balanced, cf. [10, Remark II.3]. And it is also easy to prove that constacyclic codes are balanced, see [12, Lemma II.8].

Lemma 4.9**.**

If $B$ is a consta-dihedral code over $F$, then $B$ is balanced; in particular, for $0\leq\delta\leq 1-q^{-1}$, the cardinality $|B^{\leq\delta}|\leq q^{\dim_{F}\!B\cdot h_{q}(\delta)}$.

Proof.

For any $a=\sum_{i=0}^{n-1}a_{i0}u^{i}\dot{v}^{0}+\sum_{i=0}^{n-1}a_{i1}u^{i}\dot{v}^{1}\in{F\kern-1.0pt{*}G}\kern 1.0pt$, as a word of $F^{2n}$,

[TABLE]

the coordinates of the word $a$ indexed by the standard basis Eq.(3.2) of ${F\kern-1.0pt{*}G}\kern 1.0pt$:

[TABLE]

Let ${\rm Sym}(I)$ be the symmetric group of the set $I$. The dihedral group $G$ (as in Eq.(3.1)) acts on the set $I$ through the homomorphism $\theta:G\to{\rm Sym}(I)$, $g\mapsto\theta_{g}$, as follows: $\theta_{u}$ is the permutation: $\theta_{u}(u^{i}\dot{v}^{j})=u^{i+1}\dot{v}^{j}$ (since $u\cdot u^{i}\dot{v}^{j}=u^{i+1}\dot{v}^{j}$), i.e.,

[TABLE]

is a double circulant permutation of $I$; and $\theta_{v}$ is the permutation:

[TABLE]

i.e.,

[TABLE]

is a product of $n$ transpositions of $I$. In fact, there is a bijection $I\to G$ (by dropping the dot from $\dot{v}$) such that the action of $G$ on $I$ defined by $\theta$ as above is equivalent to the left regular action of $G$ on $G$ itself. In particular, $G$ acts on $I$ transitively. Next, let $\Theta_{u}$ be the permutation matrix of the permutation $\theta_{u}$, i.e.,

[TABLE]

As shown in Eq.(4.11), we should take

[TABLE]

By Eq.(4.9), any $a\in{F\kern-1.0pt{*}G}\kern 1.0pt$ is identified with a word in $F^{2n}$, i.e., a $1\times 2n$ matrix; in this way $a\Theta_{u}$ and $a\Theta_{\dot{v}}$ make sense. By Eq.(4.10) and Eq.(4.11) we have

[TABLE]

Let $J=\{u^{i_{1}}\dot{v}^{0},u^{i_{2}}\dot{v}^{0},\cdots,u^{i_{s}}\dot{v}^{0},u^{j_{1}}\dot{v}^{1},u^{j_{2}}\dot{v}^{1},\cdots,u^{j_{t}}\dot{v}^{1}\}\subseteq I$, $s+t=k$, $0\leq i_{1}<i_{2}<\cdots<i_{s}\leq n-1$ and $0\leq j_{1}<j_{2}<\cdots<j_{t}\leq n-1$. For $a=\sum_{i=0}^{n-1}a_{i0}u^{i}\dot{v}^{0}+\sum_{i=0}^{n-1}a_{i1}u^{i}\dot{v}^{1}\in{F\kern-1.0pt{*}G}\kern 1.0pt$, by Remark 4.8 we can write

[TABLE]

by Eq.(4.9), we can identify $\rho_{J}(a)$ with such a word in $F^{2n}$ whose coordinates outside $J$ are zero. Then $\rho_{J}(a)\Theta_{u}$ ($\rho_{J}(a)\Theta_{\dot{v}}$, resp.) makes sense and it is a word in $F^{2n}$ whose coordinates outside $\theta_{u}(J)$ (outside $\theta_{v}(J)$, resp.) are zero. By Eq.(4.12), $\rho_{J}(a)\Theta_{u}=\rho_{\theta_{u}\!(\!J\!)}(ua)$ and $\rho_{J}(a)\Theta_{\dot{v}}=\rho_{\theta_{v}\!(\!J\!)}(\dot{v}a)$. Replacing $a$ by $u^{-1}a$ (and replacing $a$ by $\dot{v}^{-1}a$) we have

[TABLE]

Assume that $\dim_{F}B=k$, and $I_{*}\subseteq I$ is an information index set of $B$, i.e., $|I_{*}|=k$ and $\rho_{I_{*}}(B)=F^{I_{*}}$. Because $B$ is a left ideal of ${F\kern-1.0pt{*}G}\kern 1.0pt$, $u^{-1}B=B=\dot{v}^{-1}B$. Note that both $\Theta_{u}$ and $\Theta_{\dot{v}}$ are invertible. By Eq.(4.13),

[TABLE]

In other words, both $\theta_{u}(I_{*})$ and $\theta_{v}(I_{*})$ are information index sets of $B$. For any $g\in G$, because $g$ can be written as a product of several $u$ and $v$, $\theta_{g}$ is a product of several $\theta_{u}$ and $\theta_{v}$; so $\theta_{g}(I_{*})$ is an information index set of $B$ too.

Finally, fix an information index set $I_{*}\subseteq I$ of $B$. Then the $2n$ subsets (repetition allowed): $\theta_{g}(I_{*})$, $g\in G$, are all information index sets of $B$. And, since $G$ acts on $I$ transitively, by [12, Lemma II.9], there is an integer $t$ such that for any $u^{i}\dot{v}^{j}\in I$ the number of such $g\in G$ that $u^{i}\dot{v}^{j}\in\theta_{g}(I)$ equals $t$. In conclusion, $B$ is a balance code. ∎

5 Asymptotic property of consta-dihedral codes

Keep the notation in Section 3 and Section 4. In this section we always denote

[TABLE]

Then ${F\kern-1.0pt{*}G}\kern 1.0pt=A_{0}\oplus A$. By Corollary 3.10, we have that

[TABLE]

We further assume that

[TABLE]

From now on, let $\delta$ be a real number satisfying that ($h_{q}(\delta)$ is defined in Eq.(4.8))

[TABLE]

In this section, we prove that the consta-dihedral codes constructed in the last section are asymptotically good.

5.1 Consta-dihedral codes of rate $\frac{1}{2}-\frac{1}{2n}$

In this subsection, we consider the consta-dihedral code $C=C_{1}\oplus\cdots\oplus C_{m}$ defined in Eq.(4.4). Recall that $K^{*}=\{e_{0}\}\times K_{1}^{\times}\times\cdots\times K_{m}^{\times}$ with each field $K_{t}\subseteq A_{t}$ of dimension $\dim_{F}K_{t}=2k_{t}$, see Remark 4.1. For any $\beta\in K^{*}$, $\beta=e_{0}+\beta_{1}+\cdots+\beta_{m}$, we have a consta-dihedral code $C\beta=C_{1}\beta_{1}\oplus\cdots\oplus C_{m}\beta_{m}$.

For any $0\neq d\in A$, there is a unique subset $\omega_{d}=\{t_{1},\cdots,t_{r}\}\subseteq\{1,2,\cdots,m\}$ such that $d=d_{t_{1}}+\cdots+d_{t_{r}}$, where $d_{t_{i}}\in A_{t_{i}}\backslash\{0\}$ for $i=1,\cdots,r$; we denote

[TABLE]

Lemma 5.1**.**

Let $0\neq d\in A$. Set ${\cal K}(C)_{d}=\{\beta\in K^{*}\;|\;d\in C\beta\}$. Then

[TABLE]

Proof.

Assume that $\omega_{d}=\{t_{1},\cdots,t_{r}\}\subseteq\{1,2,\cdots,m\}$, and $d=d_{t_{1}}+\cdots+d_{t_{r}}$ for $d_{t_{i}}\in A_{t_{i}}\backslash\{0\}$. Then $d\in C\beta$ if and only if $d_{t_{i}}\in C_{t_{i}}\beta_{t_{i}}$, $i=1,\cdots,r$. The $C_{t_{i}}\beta_{t_{i}}$ is a simple left ideal of $A_{t_{i}}$. In $A_{t_{i}}$, the intersection of any two distinct simple left ideals is [math]; so there is at most one simple left ideal $C^{\prime}_{t_{i}}$ containing $d_{t_{i}}$. By Lemma 2.5(3), there are exactly $q^{k_{t_{i}}}-1$ elements $\beta_{t_{i}}$ in $K_{t_{i}}^{\times}$ such that $C_{t_{i}}\beta_{t_{i}}=C^{\prime}_{t_{i}}$. Thus

[TABLE]

Since $\dim_{F}K_{t_{i}}=2k_{t_{i}}$, see Remark 4.1, we get $q^{k_{t_{i}}}-1=\frac{|K_{t_{i}}^{\times}|}{(q^{k_{t_{i}}}+1)}$. Set $\omega_{d}^{\prime}=\{1,2,\cdots,m\}\backslash\omega_{d}$. Then

[TABLE]

i.e., \big{|}{\cal K}(C)_{d}\big{|}\leq|K^{*}|\big{/}q^{k_{t_{1}}+\cdots+k_{t_{r}}}=|K^{*}|\big{/}q^{\ell_{d}}. ∎

Denote

[TABLE]

which is the set of all ideals of $A$.

Lemma 5.2**.**

Set ${\cal K}(C)^{\!\leq\delta}=\{\,\beta\in K^{*}\,|\,\Delta(C\beta)\leq\delta\,\}$. If $\frac{1}{4}-h_{q}(\delta)-\frac{\log_{q}n}{\lambda(n)}>0$, then

[TABLE]

Proof.

For any subset $\omega\subseteq\{1,\cdots,m\}$, we denote $A_{\omega}=\bigoplus_{t\in\omega}A_{t}$, so $A_{\omega}\in{\Omega}$. For $k_{1}\leq\ell\leq\frac{n-1}{2}$, we set

[TABLE]

For $A_{\omega}\in{\cal A}_{\ell}$, $\dim_{F}A_{\omega}=4\sum_{t\in\omega}k_{t}=4\ell$ and $k_{t}\geq k_{1}$, by the assumption Eq.(5.2), we have that $|\omega|\leq\ell/k_{1}$. Thus,

[TABLE]

It is obvious that (where ${A_{\omega}}^{\!\leq\delta}$ is defined in Eq.(4.7))

[TABLE]

By Lemma 4.9, for $A_{\omega}\in{\cal A}_{\ell}$, we have that $|{{A}_{\omega}}^{\!\leq\delta}|\leq q^{4\ell h_{q}(\delta)}$ since $\dim_{F}A_{\omega}=4\ell$. By Eq.(5.11), we get

[TABLE]

By Lemma 5.1, |{\cal K}(C)_{d}|\leq|K^{*}|\big{/}q^{\ell}. From Eq.(5.11) we obtain

[TABLE]

Because $\frac{1}{4}-h_{q}(\delta)-\frac{\log_{q}n}{4k_{1}}>0$ and $\ell\geq k_{1}$, we further get

[TABLE]

The last inequality holds since $\frac{n-1}{2}-k_{1}+1\leq n=q^{\log_{q}n}$. Further, $\frac{1}{4}-h_{q}(\delta)>0$ and $2k_{1}\geq\lambda(n)$. So

[TABLE]

That is, \big{|}{\cal K}(C)^{\leq\delta}\big{|}\leq|K^{\!*}|\!\cdot\!q^{-2\lambda(n)\big{(}\frac{1}{4}-h_{q}(\delta)-\frac{\log_{q}n}{\lambda(n)}\big{)}}. ∎

Remark 5.3**.**

Let ${\cal P}$ be the set of all primes, and ${\cal P}_{t}$ the set of primes less than or equal to $t$. We denote ${\cal G}_{t}=\{p\in{\cal P}\,|\,q<p\leq t,\,{\rm ord}_{{\mathbb{Z}}_{p}^{\times}}(q)\geq(\log_{q}t)^{2}\}$, and denote ${\cal G}=\bigcup_{t=1}^{\infty}{\cal G}_{t}$. Then the density of ${\cal G}$ is \lim\limits_{t\to\infty}|{\cal G}_{t}|\big{/}|{\cal P}_{t}|=1, see [10, Lemma II.6]. Hence, by Lemma 2.3, there are positive odd integers $n_{1},n_{2},\cdots$ with every $n_{i}$ coprime to $q$ and $n_{i}\to\infty$ such that

[TABLE]

Theorem 5.4**.**

Let $\delta$ be as in Eq.(5.3), and $n_{1},n_{2},\cdots$ as in Eq.(5.12). Then there are consta-dihedral code $C^{(i)}$ of length $2n_{i}$, for $i=1,2,\cdots$, such that

(1)* the length $2n_{i}$ of $C^{(i)}$ is going to infinity;*

(2)* ${\rm R}(C^{(i)})=\frac{1}{2}-\frac{1}{2n_{i}}$ for $i=1,2,\cdots$;*

(3)* the relative minimum distance $\Delta(C^{(i)})>\delta$ for $i=1,2,\cdots$;*

hence the code sequence $C^{(1)},C^{(2)},\cdots$ is asymptotically good.

Proof.

Since $\frac{1}{4}-h_{q}(\delta)>0$, by dropping finitely many terms (if necessary), we can further assume that the sequence $n_{1},n_{2},\cdots$ satisfy Eq.(5.12) and that $\frac{1}{4}-h_{q}(\delta)-\frac{\log_{q}n_{i}}{\lambda(n_{i})}>0$, for  $i=1,2,\cdots$. In Lemma 5.2, take $n=n_{i}$, we get

[TABLE]

Thus we can take $\beta^{(i)}\in K^{*}\backslash\mathcal{K}(C)^{\leq\delta}$ for $i=1,2,\cdots$. Set $C^{(i)}=C\beta^{(i)}$. Then $C^{(i)}$ is a consta-dihedral code of length $2n_{i}$ and the obtained code sequence

[TABLE]

satisfy the statements (1), (2) and (3). ∎

The density of the set ${\cal G}$ of primes in Remark 5.3 equals $1$, so that we can take some subsets of ${\cal G}$ which satisfy more conditions.

Lemma 5.5** ([10, Corollary II.8]).**

There are positive odd integers $n_{1},n_{2},\cdots$ with every $n_{i}$ coprime to $q$ and $n_{i}\to\infty$ such that

[TABLE]

Theorem 5.6**.**

The self-orthogonal consta-dihedral codes over any finite field $F$ are asymptotically good.

Proof.

Taking $n_{1},n_{2},\cdots$ as in Eq.(5.14). By Lemma 2.2(2), for the cyclic group $H$ of order $n_{i}$, any primitive idempotent $e$ of $FH$ other than $e_{0}$ satisfies that $\overline{e}\neq e$. By Lemma 4.2 and Corollary 3.9(4), the consta-dihedral code $C^{(i)}$ with length $2n_{i}$ in Eq.(5.13) is self-orthogonal. ∎

Lemma 5.7**.**

Assume that $q$ is odd and $4\nmid(q-1)$. Then there are positive odd integers $n_{1},n_{2},\cdots$ with every $n_{i}$ coprime to $q$ and $n_{i}\to\infty$ such that

[TABLE]

where “ $2\,\|\,t$” means that $2\,|\,t$ but $2^{2}\nmid t$.

Proof.

Let {\mathcal{O}}=\{\,p\in{\cal P}\,|\,\mbox{{\rm ord}{{\mathbb{Z}}{p}^{\times}}(q) is odd}\,\} and $\overline{\mathcal{O}}={\cal P}\backslash{\mathcal{O}}$. By the assumption, we can write $q=r^{s}$ for an odd prime $r$ and an odd positive integer $s$. By [22, Theorem 1], the density of ${\mathcal{O}}$ in ${\cal P}$ equals $\frac{1}{3}$, hence the density of $\overline{\mathcal{O}}$ equals $\frac{2}{3}$. On the other hand, we consider

[TABLE]

By a Dirichlet’s theorem on density (cf. [23, Ch.6 §4 Theorem 2]), the density of ${\cal T}$ in ${\cal P}$ equals $\frac{1}{2}$. Thus the density of $\overline{\mathcal{O}}\cap{\cal T}$ is at least $\frac{2}{3}+\frac{1}{2}-1=\frac{1}{6}$. Hence the density of $\overline{\mathcal{O}}\cap{\cal T}\cap{\cal G}$ is at least $\frac{1}{6}$, where ${\cal G}$ is defined in Remark 5.3. For any $p\in\overline{\mathcal{O}_{t}}\cap{\cal T}\cap{\cal G}$, we have that $-1\in\langle q\rangle_{{\mathbb{Z}}_{p}^{\times}}$ (because: ${\rm ord}_{{\mathbb{Z}}_{p}^{\times}}(q)$ is even and $-1$ is the unique element of order $2$ in ${\mathbb{Z}}_{p}^{\times}$), and $2\,\|\,{\rm ord}_{{\mathbb{Z}}_{p}^{\times}}(q)$ (because: ${\rm ord}_{{\mathbb{Z}}_{p}^{\times}}(q)\,|\,(p-1)$ but $4\nmid(p-1)$). Thus, there are positive odd integers $n_{1},n_{2},\cdots$ with every $n_{i}$ coprime to $q$ and $n_{i}\to\infty$ such that Eq.(5.15) holds. ∎

Theorem 5.8**.**

Assume that $q$ is odd and $4\nmid(q-1)$ (i.e. $q\equiv 3~{}({\rm mod}~{}4)$). Then the LCD consta-dihedral codes over $F$ are asymptotically good. In particular, LCD quasi-cyclic codes of index $2$ over $F$ are asymptotically good.

Proof.

Take $n_{1},n_{2},\cdots$ as in Eq.(5.15). By Theorem 4.6, the $C^{(i)}$ with length $2n_{i}$ in Eq.(5.13) is an LCD consta-dihedral code. Thus the LCD consta-dihedral code sequence Eq.(5.13) is asymptotically good. By Eq.(3.8), any consta-cyclic code is a quasi-cyclic code of index $2$. So the “In particular” part holds. ∎

5.2 Self-dual consta-dihedral codes

In this subsection we always assume that $q$ is even or $4\,|\,(q-1)$, i.e., $q\,{\not\equiv}\,3~{}({\rm mod}~{}4)$. Keep the notation in Theorem 4.7:

• •

$\widehat{C}=C_{0}\oplus C=C_{0}\oplus C_{1}\oplus\cdots\oplus C_{m}$ where $C_{0}=A_{0}(re_{0}+e_{0}\dot{v})$ is defined in Lemma 3.5(2);

• •

For any $\beta=e_{0}+\beta_{1}+\cdots+\beta_{m}\in K^{*}$, the consta-dihedral code

[TABLE]

is self-dual; in particular, the rate ${\rm R}(\widehat{C}\beta)=\frac{1}{2}$.

We will find the $\beta$ such that the relative minimal distance $\Delta(\widehat{C}\beta)>\delta$.

Note that ${F\kern-1.0pt{*}G}\kern 1.0pt=A_{0}\oplus A$. For any $\widehat{d}=d_{0}+d\in{F\kern-1.0pt{*}G}\kern 1.0pt$ with $d_{0}\in A_{0}$ and $d\in A$, if $d_{0}\notin C_{0}$, then $\widehat{d}\notin\widehat{C}\beta$ for any $\beta\in K^{*}$.

Lemma 5.9**.**

Assume that $0\neq\widehat{d}=d_{0}+d\in C_{0}\oplus A$ with $d_{0}\in C_{0}$ and $d\in A$. Set $\mathcal{K}(\widehat{C})_{\widehat{d}}=\{\beta\in K^{*}\;|\;\widehat{d}\in\widehat{C}\beta\}$. Then

[TABLE]

where $\ell_{d}$ is defined in Eq.(5.4).

Proof.

It is clear that $\widehat{d}\in\widehat{C}\beta$ if and only if $d\in C\beta$. So this lemma follows from Lemma 5.1 immediately. ∎

Lemma 5.10**.**

Let ${\mathcal{K}(\widehat{C})}^{\!\leq\delta}=\{\,\beta\in K^{*}\,|\,\Delta(\widehat{C}\beta)\leq\delta\,\}$. If $\frac{1}{4}-h_{q}(\delta)-\frac{\log_{q}n}{\lambda(n)}>0$, then

[TABLE]

Proof.

For $k_{1}\leq\ell\leq\frac{n-1}{2}$, we extend the notation ${\cal D}_{\ell}$ in Eq.(5.9) and set

[TABLE]

With ${\cal A}_{\ell}$ defined in Eq.(5.9), we have that

[TABLE]

For $A_{\omega}\in{\cal A}_{\ell}$, we have that $|(C_{0}\oplus{\cal A}_{\omega})^{\!\leq\delta}|\leq q^{(4\ell+1)h_{q}(\delta)}$ since $\dim_{F}(C_{0}\oplus A_{\omega})=4\ell+1$; see Lemma 4.9. By Eq.(5.17) and Eq.(5.10), we have

[TABLE]

For $\widehat{d}=d_{0}+d\in\widehat{D}_{\ell}$, we have $\ell_{d}=\ell$. By Eq.(5.17) and Lemma 5.9,

[TABLE]

Therefore,

[TABLE]

By the same argument for Lemma 5.2, we can get

[TABLE]

We are done. ∎

Theorem 5.11**.**

Assume that $q$ is even or $4\,|\,(q-1)$. Let $\delta$ be as in Eq.(5.3), and $n_{1},n_{2},\cdots$ as in Eq.(5.12). Then there are self-dual consta-dihedral codes $\widehat{C}^{(i)}$ of length $2n_{i}$ such that $\Delta(\widehat{C}^{(i)})>\delta$ for all $i=1,2,\cdots$; hence the code sequence $\widehat{C}^{(1)},\widehat{C}^{(2)},\cdots$ is asymptotically good.

Proof.

In Lemma 5.10, we set $n=n_{i}$, so

[TABLE]

We can take $\beta^{(i)}\in K^{*}\backslash\mathcal{K}(\widehat{C})^{\leq\delta}$ for $i=1,2,\cdots$. Set $\widehat{C}^{(i)}=\widehat{C}\beta^{(i)}$. Then $\widehat{C}^{(i)}$ is a self-dual consta-dihedral code of length $2n_{i}$ and the code sequence

[TABLE]

satisfies that $\Delta(\widehat{C}^{(i)})>\delta$ for $i=1,2,\cdots$. ∎

We get the following immediately (cf. the proof of Theorem 5.8).

Theorem 5.12**.**

Assume that $q$ is even or $4\,|\,(q-1)$ (i.e. $q\,{\not\equiv}\,3~{}({\rm mod}~{}4)$). Then the self-dual consta-dihedral codes over $F$ are asymptotically good. In particular, self-dual quasi-cyclic codes of index $2$ over $F$ are asymptotically good.

It is known that if $q$ is odd then LCD dihedral codes of rate $\frac{1}{2}$ are asymptotically good, see [10, Theorem 1.2]. We get the following consequence.

Corollary 5.13**.**

If $q\equiv 1~{}({\rm mod}~{}4)$, then the self-dual quasi-cyclic codes of index $2$ over $F$ and the LCD quasi-cyclic codes of index $2$ over $F$ of rate $\frac{1}{2}$ are both asymptotically good.

6 Remarks on dihedral codes

The purpose of this section is to correct some mistakes in [10]. We begin with a subtle remark.

Remark 6.1**.**

Lemma 2.1(5) (i.e., [10, Lemma II.4(5)]) provides an efficient technique to evaluate the orthogonality of group codes, i.e., for $FG$-codes $C,D$,

[TABLE]

However, a subtle point is that the following is incorrect:

[TABLE]

Here is a counterexample for Eq.(6.2).

Example. Take $|F|=7$, $n=3$, $G=\langle u,v\,|\,u^{3}=1=v^{2},vuv^{-1}=u^{-1}\rangle=H\rtimes\langle v\rangle$ be the dihedral group of order $6$, where $H=\{1,u,u^{2}\}$ is the cyclic group of order $3$. Then $\frac{1}{3}$ equals 5 in $F$ and 2, 4 are primitive $3$’th roots of unity. $e_{0},e,\bar{e}$ are all primitive idempotents of $FH$, where

[TABLE]

Denote $FG=A_{0}\oplus A_{1}$, where

[TABLE]

are minimal ideals of $FG$. Take $C=D=A_{1}e=FHe\oplus FH\bar{e}v$. Then

[TABLE]

By Eq.(6.1), $\langle C,D\rangle=0$. However, $\overline{D}C=\bar{e}\overline{A}_{1}\cdot A_{1}e=\bar{e}A_{1}e\neq 0$, because we can choose an element $\bar{e}(\bar{e}v)e\in\bar{e}A_{1}e$ and $\bar{e}(\bar{e}v)e=\bar{e}ve=\bar{e}\bar{e}v=\bar{e}v\neq 0$.

Remark 6.2**.**

Turn to the mistakes of [10]. The main issue in [10] is that

• (I)

In the proofs of [10, Theorem IV.3] and [10, Theorem IV.5], some citations of [10, Lemma II.4(5)] (i.e., Eq.(6.1)) are in fact misuses of the incorrect version Eq.(6.2).

We first show the effects of the issue, then explain how to address it.

[10, Theorem IV.3] considers the case that ${\rm char}(F)=2$. Though the incorrect Eq.(6.2) was misused in its proof, [10, Theorem IV.3] is itself correct. Because: if ${\rm char}(F)=2$, then $-1=1$ and the consta-dihedral group algebra is identified with the dihedral group algebra, i.e., ${F\kern-1.0pt{*}G}\kern 1.0pt=FG$, cf. Eq.(3.3); hence all the results in this paper are applied to $FG$ and to dihedral codes provided ${\rm char}(F)$ is even. Thus, [10, Theorem IV.3] is a consequence of Theorem 4.7 (by taking even $q$) of this paper.

[10, Theorem IV.5] considers the case that ${\rm char}(F)$ is odd, and consists of two parts: (1) the case that ${\rm ord}_{{\mathbb{Z}}_{n}^{\times}}(q)$ is odd; (2) the case that $-1\in\langle q\rangle_{{\mathbb{Z}}_{n}^{\times}}$. For [10, Theorem IV.5(2)], though there were gaps in the proof, its conclusion is still correct and proved in [11, Lemma 8.6(2)]. For [10, Theorem IV.5(1)] (the case that ${\rm char}(F)$ is odd and ${\rm ord}_{{\mathbb{Z}}_{n}^{\times}}(q)$ is odd), however, it is unlucky that the misuse of the incorrect Eq.(6.2) results in an incorrect conclusion.

Remark 6.3**.**

The issue described in Remark 6.2 implies that on some occasions Eq.(6.1) is not enough to recognize orthogonality of group codes. In this paper we developed a technique to recognize the orthogonality of group codes by matrix computations, e.g., see the proofs of Lemma 4.2 and Lemma 4.3. That is one of the contributions of this paper. By this technique, to look for a correct version of [10, Theorem IV.5(1)], we begin with the dihedral group algebra version of Lemma 3.6.

Keep the assumption Eq.(3.1) and Eq.(2.3).

Lemma 6.4**.**

Let e be a primitive idempotent of $FH$ with $\overline{e}\neq e$. Then $\tilde{F}:=FHe$ is a field extension over F, $e+\overline{e}$ is a primitive central idempotent of $FG$ and:

(1)* The ideal $FG(e+\overline{e})=FHe\oplus FH\bar{e}\oplus FHev\oplus FH\bar{e}v\cong{\rm M}_{2}(\tilde{F})$.*

(2)* With the isomorphism in (1), if $f\in FG(e+\overline{e})$ corresponds to the matrix $\begin{pmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{pmatrix}\in{\rm M}_{2}(\tilde{F})$, then $\overline{f}$ corresponds to the matrix $\begin{pmatrix}a_{22}&a_{12}\\ a_{21}&a_{11}\end{pmatrix}$.*

Proof.

Similarly to the proof of Lemma 3.6, $\tilde{F}:=FHe$ is a field extension over F and $e+\overline{e}$ is a primitive central idempotent of $FG$. So $FG(e+\bar{e})=FHe\oplus FHev\oplus FH\bar{e}\oplus FH\bar{e}v$ is an ideal of $FG$. Define a map:

[TABLE]

which is obviously a linear isomorphism. For $a_{ij},b_{ij}\in\widetilde{F}$, $1\leq i,j\leq 2$, noting that $va_{ij}=\overline{a_{ij}}\,v$ and $v^{2}=1$, we have

[TABLE]

So, Eq.(6.5) is an $\tilde{F}$-algebra isomorphism.

For ${a_{11}e+a_{12}\,e\,v+\overline{a_{21}}\,\bar{e}v+\overline{a_{22}}~{}\overline{e}}\in{FHe\oplus FHev\oplus FH\bar{e}\oplus FH\bar{e}v}$,

we have (note that $\overline{v}=v$):

[TABLE]

Thus, the bar image of ${a_{11}e+a_{12}\,e\,v+\overline{a_{21}}\,\bar{e}v+\overline{a_{22}}~{}\overline{e}}$ corresponds the matrix:

[TABLE]

We are done. ∎

Recall that (Lemma 2.5(2)), there are altogether $|\tilde{F}|+1$ simple left ideals of ${\rm M}_{2}(\tilde{F})$ with generators:

[TABLE]

Lemma 6.5**.**

Let notation be as above in Lemma 6.4. Denote $A=FG(e+\bar{e})$ for short. Let $f_{ab}\in A$ be the element corresponding to $\begin{pmatrix}a&b\\ 0&0\end{pmatrix}\in{\rm M}_{2}(\tilde{F})$. Then

(1)* If ${\rm char}(F)=2$, then $\langle Af_{ab},Af_{ab}\rangle=0$, for $\,a,b\in\tilde{F}$.*

(2)* If ${\rm char}(F)$ is odd, then $\langle Af_{ab},Af_{ab}\rangle=0$ if and only if $ab=0$; in particular, there are exactly two simple left ideals of $A$ which is self-dual in $A$, and the other $|\tilde{F}|-1$ simple left ideals of $A$ are LCD in $A$.*

Proof.

By Lemma 2.1(5), $\langle Af_{ab},Af_{ab}\rangle=0$ if and only if $f_{ab}\overline{f_{ab}}=0$. By Lemma 6.4 (Eq.(6.5) and Eq.(6.6)), $f_{ab}\overline{f_{ab}}=0$ if and only if

[TABLE]

If ${\rm char}(F)=2$, it is always true that $2ab=0$, hence (1) holds. Next assume that ${\rm char}(F)\neq 2$. Then $2ab=0$ if and only if $ab=0$. In Eq.(6.7), there exactly two cases such that $ab=0$, i.e., $\begin{pmatrix}0&1\\ 0&0\end{pmatrix}$ and $\begin{pmatrix}1&0\\ 0&0\end{pmatrix}$. Hence (2) follows. ∎

Remark 6.6**.**

Assume that $q$ is odd and ${\rm ord}_{{\mathbb{Z}}_{n}^{\times}}(q)$ is odd. For all primitive idempotents $e_{0}=\frac{1}{n}\sum_{i=0}^{n-1}u^{i},\;e_{1},\cdots,e_{\ell}$ of $FH$, except for $e_{0}$, the other primitive idempotents are pairwise partitioned (see Lemma 2.2(2)):

[TABLE]

(1) Set $A_{0}=FGe_{0}$ and $A_{t}=FG(e_{t}+\overline{e_{t}})$ for $t=1,\cdots,m$. By Lemma 6.4,

[TABLE]

where $F_{t}=FHe_{t}$ is a field extension over $F$, denote $k_{t}=|F_{t}:F|$. By Eq.(6.1), $\langle A_{i},A_{j}\rangle=0$ for $0\leq i\neq j\leq m$; hence Corollary 3.9 is still valid for $FG=A_{0}\oplus A_{1}\oplus\cdots\oplus A_{m}$.

(2) Let $\widehat{e}_{0}=e_{0}+e_{0}v\in A_{0}$, then $\langle A_{0}\widehat{e}_{0},A_{0}\widehat{e}_{0}\rangle=A_{0}\widehat{e}_{0}\overline{\widehat{e}_{0}}\overline{\widehat{A}_{0}}\neq 0$ since $\widehat{e}_{0}\overline{\widehat{e}_{0}}=2\widehat{e}_{0}\neq 0$, hence $A_{0}\widehat{e}_{0}$ is LCD in $A_{0}$.

Thus the following is the correct version of [10, Theorem IV.5(1)].

Theorem 6.7**.**

Let notation be as in Remark 6.6. We consider the following dihedral codes:

[TABLE]

Then $\dim_{F}(C_{ab})=n$, and

(1)* The number of the dihedral codes in Eq.(6.8) equals $\prod_{t=1}^{m}(q^{k_{t}}+1)$.*

(2)* The number of the dihedral codes in Eq.(6.8) which are LCD equals $\prod_{t=1}^{m}(q^{k_{t}}-1)$.*

Proof.

By [10, Lemma III.2(3)], $A_{0}\widehat{e}_{0}$ is an $1$-dimensional ideal of $A_{0}$, hence $\dim_{F}(C_{ab})=1+\dim_{F}A_{1}f_{a_{1}b_{1}}+\cdots+\dim_{F}A_{m}f_{a_{m}b_{m}}=n$, where $\dim_{F}A_{t}f_{a_{t}b_{t}}=2k_{t}$, $t=1,\cdots,m$. By Eq.(6.7) and Lemma 6.5(2), we get (1) and (2) at once. ∎

7 Conclusion

We studied the consta-dihedral codes, and addressed an issue of the reference [10] which is a research on dihedral codes.

To investigate the algebraic property of the consta-dihedral codes, the existing methods, cf. in [10], are not enough to recognize the orthogonality of group codes; so we developed a technique to evaluate the orthogonality of group codes by matrix computation. We characterized the algebraic structure of consta-dihedral group algebras. By the algebraic structure and with the technique mentioned just now, we constructed a class of consta-dihedral codes which possess good algebraic property (self-orthogonal, or LCD).

Next, we showed the existence of asymptotic good sequences of the consta-dihedral codes in the class we constructed. Instead of probabilistic methods, in the class we counted directly the number of the consta-dihedral codes with bad asymptotic property. This number is much less than the total quantity of the class. In this way we obtained (recall that $F$ is a finite field with $|F|=q$):

— If $q$ is even or $4\,|\,(q-1)$, then the self-dual consta-dihedral codes over $F$ are asymptotically good.

— If $q$ is odd and $4\nmid(q-1)$, then the LCD consta-dihedral codes over $F$ are asymptotically good.

Finally, with the help of the technique developed in this paper, we addressed the issue in [10] and obtained the correct version of the false theorem [10, Theorem IV.5(1)].

Acknowledgements