Group codes over fields are asymptotically good

TL;DR

This paper proves that group codes over finite fields of any characteristic are asymptotically good, extending previous results from binary fields to all finite fields.

Contribution

It establishes the asymptotic goodness of group codes over arbitrary finite fields, generalizing prior work limited to binary fields.

Findings

Group codes over finite fields are asymptotically good.

The result extends to fields of any characteristic.

Supports the potential of group codes in coding theory applications.

Abstract

Group codes are right or left ideals in a group algebra of a finite group over a finite field. Following ideas of Bazzi and Mitter on group codes over the binary field, we prove that group codes over finite fields of any characteristic are asymptotically good.

Equations

$$G_{p,q,m}:=⟨\alpha ,\beta |{\alpha }^p={\beta }^q=1,\alpha \beta {\alpha }^{-1}={\beta }^m⟩=⟨\beta ⟩⋊⟨\alpha ⟩.$$

$$r=r_0+\alpha r_1+\cdots +{\alpha }^{p-1}{r}_{p-1}$$

$$\hat{a}:=\sum _{i=0}^{q-1}{a}_i{\beta }^{i\cdot m}$$

$$\alpha a=\hat{a}\alpha$$

$$Q=\underset{i=0}{\overset{s}{⨁}}{Q}_i$$

$$x^q-1=\prod _{i=0}^s{f}_i$$

$$Q_i=⟨\frac{x^q-1}{f_i}⟩\cong {\mathbb{F}}_p[x]/⟨f_i⟩\cong {\mathbb{F}}_{p^{\mathrm{deg}{f}_i}}.$$

$$f_i=\prod _{a\in A_i}(x-{\zeta }_q^a)$$

$$\dim Q_i:=l_i=s_p(q)$$

$$\mathcal{P}:=\{q\mid q\text{a prime},p\mid s_p(q)\}.$$

$$R_i=Q_i\oplus \alpha Q_i\oplus \dots \oplus {\alpha }^{p-1}{Q}_i$$

$${\mathbb{F}}_{p}G=R_0\oplus \dots \oplus R_s.$$

$$\dim Z_i=\frac{q-1}{sp}=\frac{s_p(q)}{p}=\frac{l_i}{p}.$$

$$x(1+\alpha +\dots {\alpha }^{p-1})y=(1+\alpha +\dots {\alpha }^{p-1})$$

$$x(1+\alpha +\dots {\alpha }^{p-1})y=xy+x\hat{y}\alpha +\hat{\hat{y}}{\alpha }^2+\dots$$

$$y=b{b}^{\prime -1}\in Z_i^{\times },$$

$$P(r)=\{t\text{ prime}|s_p(t)\ge r(t)\}.$$

$$0<a<r(t)\text{ and }{p}^a-1=kt.$$

$$\#B_n\le \#\{(a,k)|0<a<r(t)\text{ and }(p^a-1)/k\text{ is prime}\}\le r(t)\cdot \underset{0<a<r(t)}{\max }\#\{\text{prime factors of }{p}^a-1\}$$

$$\le r(t)\cdot {\log }_2(p^{r(t)}-1)\le r(t{)}^2\cdot {\log }_2(p)<<\frac{t}{\log t}.$$

$$U=Q^{\ast \times }f(1+\alpha +\dots +{\alpha }^{p-1})Q^{\ast \times },$$

$$U=\sum _{i\in S}{Q}_i^{\times }(1+\alpha +\dots +{\alpha }^{p-1})Q_i^{\times }.$$

$$Q_i^{\times }(1+\alpha +\dots +{\alpha }^{p-1})Q_i^{\times }=\underset{[b]\in Q_i^{\times }/Z_i^{\times }}{⨆}{I}_{[b]}^i\setminus \{0\},$$

$$\#(Q_i^{\times }(1+\alpha +\dots +{\alpha }^{p-1})Q_i^{\times })=\frac{p^{l_i}-1}{p^{l_i/p}-1}\cdot (p^{l_i}-1)\ge p^{(p-1)l_i/p}\cdot p^{l_i}.$$

$$\#U\le \sum _{i\in S}{p}^{(p-1)l_i/p}\cdot p^{l_i}=p^{\frac{2p-1}{p}l},$$

$$A_w(C):=\#\{c\mid c\in C,\mathrm{wt}(c)=w\}\le p^{k\cdot h_p(w/n)}$$

$$h_p(x):=-(1-x){\log }_p(1-x)-x{\log }_p\left(\frac{x}{p-1}\right)$$

$$I=\underset{i=1}{\overset{s}{⨁}}{I}_i$$

$$p^{-p\cdot s_p(q)\cdot \left(\frac{p-1}{p^2}-h_p(\delta )\right)+(2p+1){\log }_p(q)}.$$

$$I_{[b]}=Q(1+\alpha +\dots +{\alpha }^{p-1})b=Q^{\ast }(1+\alpha +\dots +{\alpha }^{p-1})b$$

Full text

M. Borello is with Université Paris 8, Laboratoire de Géométrie, Analyse et Applications, LAGA, Université Sorbonne Paris Nord, CNRS, UMR 7539, F-93526, Saint-Denis, France

W. Willems is with Otto-von-Guericke Universität, Magdeburg, Germany, and Universidad del Norte, Barranquilla, Colombia

Keywords. Group algebra, group code, asymptotically good

MSC classification. 94A17, 94B05, 20C05

1. Introduction

Let $\mathbb{F}$ be a finite field of characteristic $p$ and let $G$ be a finite group. By a group code or, more precisely, a $G$-code we denote a right or left ideal in the group algebra $\mathbb{F}G$. Many interesting linear codes are group codes. For example, cyclic codes of length $n$ are group codes for a cyclic group $C_{n}$; Reed-Muller codes are group codes for an elementary abelian $p$-group [4, 7]; the binary extended self-dual $[24,12,8]$ Golay code is a group code for the symmetric group ${\rm S}_{4}$ on $4$ letters [5] and the dihedral group $D_{24}$ of order $24$ [14]. Many best known codes are group codes as well. For instance, $\mathbb{F}_{5}(C_{6}\times C_{6})$ contains a $[32,28,6]$ and $\mathbb{F}_{5}(C_{12}\times C_{6})$ a $[72,62,6]$ group code [13]. Both codes improved earlier examples in Grassl’s list [11].

Already in 1965, Assmus, Mattson and Tyrun [2] asked the question whether the class of cyclic codes, i.e., the class of group codes over cyclic groups, is asymptotically good. The answer is still open. In [3], Bazzi and Mitter proved that the class of group codes over the binary field is asymptotically good. Using the trivial fact that by field extensions neither the dimension nor the minimum distance changes, group codes are asymptotically good in characteristic $2$. In this note we use the ideas of Bazzi and Mitter to prove our main result.

Theorem. Group codes over fields are asymptotically good in any characteristic.

The proof mainly follows the lines of [3] and does not distinguish between the prime $p=2$ and $p$ odd for the characteristic of the underlying field.

For different primes $p\not=q$ let $s_{p}(q)$ denote the order of $p$ modulo $q$. In order to construct a sequence of particular binary group algebras over dihedral groups, in [3] the authors need a set of primes $q$ with $2\mid s_{2}(q)$ which has positive density in the set of all primes. Such a set is obviously given by all primes $q\equiv\pm 5\bmod 8$. For odd primes $p$ the analog is far less obvious, but has already been proved by Wiertelak in 1977 (see [15]). In the following unified proof (i.e., $p$ any prime) we heavily use results from modular representation theory.

2. The structure of the group algebra $\mathbb{F}_{p}G_{p,q,m}$

Let $p$ be a fixed prime and let $q$ be a prime such that $p$ divides $q-1$ (there are infinitely many such $q$, by Dirichlet’s Theorem). For $m\in\mathbb{N}$ such that $m\not\equiv 1\bmod q$ and $m^{p}\equiv 1\bmod q$, we define the group $G_{p,q,m}$ by

[TABLE]

Note that $G_{p,q,m}$ is a nonabelian metacyclic group. In the case $p=2$ and $m=q-1$ the group $G_{2,q,q-1}$ is a dihedral group which has been considered in [3] to prove the Theorem over the binary field $\mathbb{F}_{2}$.

Next we put $N:=\langle\beta\rangle$ and $Q:=\mathbb{F}_{p}N$. Any element $r$ of $\mathbb{F}_{p}G_{p,q,m}$ can uniquely be written as

[TABLE]

with $r_{0},\ldots,r_{p-1}\in Q$. If $a=\sum_{i=0}^{q-1}a_{i}\beta^{i}$ (with $a_{i}\in\mathbb{F}_{p}$) is an element of $Q$, we define $\hat{a}$ by

[TABLE]

Clearly, the map $\hat{}:Q\to Q$ is an $\mathbb{F}_{p}$-algebra automorphism. From the relation $\alpha\beta=\beta^{m}\alpha$ we get $\alpha\beta^{i}=\beta^{i\cdot m}\alpha$ for all $i\in\{0,\ldots,q-1\}$, so that

[TABLE]

for all $a\in Q$.

Now we realize $Q$ as $\mathbb{F}_{p}[x]/\langle x^{q}-1\rangle$. Since $Q$ is a semisimple algebra by Maschke’s Theorem ([1], p. 116), we have, due to Wedderburn’s Theorem ([1], Chap. 5, Sect. 13, Theorem 16), a unique decomposition

[TABLE]

into $2$-sided ideals $Q_{i}$, where each $Q_{i}$ is a simple algebra over $\mathbb{F}_{p}$. If

[TABLE]

is a factorization of $x^{q}-1$ into irreducible polynomials $f_{i}\in\mathbb{F}_{p}[x]$, then

[TABLE]

We may suppose that $f_{0}=x-1$, so that $Q_{0}=\langle 1+\ldots+x^{q-1}\rangle\cong\mathbb{F}_{p}$.

Now let $\zeta_{q}$ be a primitive $q$-th root of unity in an extension field of $\mathbb{F}_{p}$. It is well-known by basic Galois theory that, for every $i\in\{1,\ldots,s\}$, there exists exactly one coset $A_{i}$ in $\mathbb{F}_{q}^{\times}/\langle p\rangle$ such that

[TABLE]

and the map $f_{i}\mapsto A_{i}$ is one-to-one. Furthermore, $\deg f_{i}=s_{p}(q)$, which is the multiplicative order of $p$ in $\mathbb{F}_{q}^{\times}$. In particular,

[TABLE]

for $i\in\{1,\ldots,s\}$. The automorphism  $\hat{}$   maps each $Q_{i}$ to some $Q_{j}$. More precisely, $\hat{Q_{i}}$ corresponds to the coset $mA_{i}$. In particular, $\hat{Q_{i}}=Q_{i}$ iff $mA_{i}=A_{i}$.

In what follows we need to understand which conditions on $q$ and $m$ imply $\hat{Q}_{i}=Q_{i}$ for all $i\in\{1,\ldots,s\}$. Note that obviously $\hat{Q}_{0}=Q_{0}$.

Lemma 2.1**.**

The following conditions are equivalent.

1. (1)

$\hat{Q_{i}}=Q_{i}$* for all $i\in\{0,1,\ldots,s\}$.* 2. (2)

There exists $i\in\{1,\ldots,s\}$ such that $\hat{Q_{i}}=Q_{i}$. 3. (3)

$m\in\langle p\rangle\leq\mathbb{F}_{q}^{\times}$.

Proof.

Clearly $(1)$ implies $(2)$. By the discussion above, $\hat{Q_{i}}=Q_{i}$ for some $i\geq 1$ iff $mA_{i}=A_{i}$, which happens iff $m\in\langle p\rangle\leq\mathbb{F}_{q}^{\times}$. So $(2)$ implies $(3)$. Obviously $(1)$ follows from $(3)$. ∎

Let $s_{p}(q)$ denote the order of $p$ modulo $q$ and suppose that $p\mid s_{p}(q)$. Thus $s_{p}(q)=pu$ for some $u\in\mathbb{N}$. We may take $m:=p^{u}$ in the definition of $G_{p,q,m}$, since $m\not\equiv 1\bmod q$ and $m^{p}\equiv 1\bmod q$. In this case we have $\hat{Q_{i}}=Q_{i}$ for $i\in\{0,1,\ldots,s\}$, by Lemma 2.1.

Now let

[TABLE]

The set ${\mathcal{P}}$ of primes is infinite and it has positive density (see for instance [15]).

From now on, we assume that $q\in\mathcal{P}$.

Let $G:=G_{p,q,p^{s_{p}(q)/p}}$ and recall that $Q=\mathbb{F}_{p}N=Q_{0}\oplus\ldots\oplus Q_{s}$ with $Q_{0}=(\sum_{i=0}^{q-1}\beta^{i})\mathbb{F}_{p}$. If we put

[TABLE]

for $i\in\{0,\ldots,s\}$, then obviously

[TABLE]

Theorem 2.2**.**

The structure of $R_{i}$ is as follows.

• a)

All $R_{i}$ are $2$-sided ideals of $\mathbb{F}_{p}G$.

• b)

As a left $\mathbb{F}_{p}G$-module we have $R_{0}\cong\mathbb{F}_{p}G/N$. In particular, $R_{0}$ is uniserial of dimension $p$ and all composition factors are isomorphic to the trivial $\mathbb{F}_{p}G$-module.

• c)

For $i>0$ all minimal left ideals in $R_{i}$ are projective $\mathbb{F}_{p}G$-modules. Thus $R_{i}$ is a completely reducible left $\mathbb{F}_{p}G$-module for $i>0$.

• d)

$R_{i}$* is indecomposable as a $2$-sided ideal, hence a $p$-block of $\mathbb{F}_{p}G$. In particular, $R_{i}$ contains up to isomorphism exactly one irreducible left $\mathbb{F}_{p}G$-module which is of dimension $l_{i}=s_{p}(q)$.*

• e)

$R_{i}\cong{\rm Mat}_{p}(\mathbb{F}_{{p}^{{l_{i}}/{p}}})$* for $i>0$ and $R_{i}$ contains up to isomorphism exactly one irreducible left $\mathbb{F}_{p}G$-module, say $M_{i}$, of dimension $l_{i}=s_{p}(q)$.*

Proof.

a) Clearly, $R_{i}$ is a left ideal. It is also a right ideal since $Q_{i}=\hat{Q}_{i}$ by Lemma 2.1, and $\alpha a=\hat{a}\alpha$ for $a\in Q$.

b) This follows immediately from representation theory (see for instance ([12], Chap. VII, Example 14.10)).

c) Let $\bar{\mathbb{F}}_{p}\supseteq\mathbb{F}_{p}$ be a finite splitting field for $N$ ([12], Chap. VII, Theorem 2.6). Thus every irreducible character $\chi$ of $\bar{\mathbb{F}}_{p}N$ is of degree $1$. If $\chi$ is not the trivial character, then, according to the action of $\alpha$ on $\beta$, the induced character $\chi^{G}$ is an irreducible character for $G$, by Clifford’s Theorem. Furthermore $\chi^{G}$ is afforded by an irreducible projective $\bar{\mathbb{F}}_{p}G$-module ([12], Chap. VII, Theorem 7.17). Thus all non-trivial irreducible $\bar{\mathbb{F}}_{p}G$-modules are projective. Now, let $M$ be an irreducible non-trivial $\bar{\mathbb{F}}_{p}G$-module and denote by $M_{0}$ the space $M$ regarded as an $\mathbb{F}_{p}G$ module. Then, by ([12], Chap. VII, Theorem 1.16 a)), $M_{0}\otimes_{\mathbb{F}_{p}}\bar{\mathbb{F}}_{p}$ is a direct sum of Galois conjugates of $M$, which are all projective since no one is the trivial module. Finally, by ([12], Chap. VII, Ex. 19 in Sec. 7), the module $M_{0}$ is a projective $\mathbb{F}_{p}G$-module, and by ([12], Chap. VII, Theorem 1.16 d)), $M_{0}\cong W\oplus\ldots\oplus W$ for some irreducible $\mathbb{F}_{p}G$-module $W$. Thus $W$ is projective. Since obviously all irreducible non-trivial $\mathbb{F}_{p}G$-modules can be described this way we are done.

d) Note that $R_{i}$ is not irreducible as a left module since $M_{i}:=Q_{i}(1+\alpha+\ldots+\alpha^{p-1})$ is a minimal ideal in $R_{i}$. Clearly, $Q_{i}\cong M_{i}$ as a left $\mathbb{F}_{p}N$-module. Thus $Q_{i}$ has an extension to the irreducible $\mathbb{F}_{p}G$-module $M_{i}$. But all extensions are isomorphic since $G/N$ is a $p$-group. Thus $R_{i}$ has up to isomorphism exactly one irreducible $F_{p}G$-module and $\mathbb{F}_{p}G$ has exactly $s+1$ non-isomorphic $\mathbb{F}_{p}G$-modules. If some $R_{i}$ is a direct sum of two non-zero $2$-sided ideals, then $R_{i}$ contains at least two non-isomorphic irreducible $\mathbb{F}_{p}G$-modules, a contradiction.

e) By c) and d), we know that $R_{i}$ contains up to isomorphism exactly one irreducible left $\mathbb{F}_{p}G$-module, say $M_{i}$, which has dimension $l_{i}$. Thus $R_{i}\cong M_{i}\oplus\ldots\oplus M_{i}$ with $p$ components $M_{i}$. That $R_{i}$ has the indicated matrix algebra structure now follows by Wedderburn’s Theorem. ∎

Lemma 2.3**.**

For $i>0$ we have

• a)

$Z_{i}:=\{a\in Q_{i}\mid a=\hat{a}\}$* is a subfield of $Q_{i}$.*

• b)

$\#Z_{i}=p^{\frac{l_{i}}{p}}=p^{\frac{s_{p}(q)}{p}}$.

Proof.

a) This is obviously true.

b) Since $\alpha$ acts fixed point freely on $N\setminus\{1\}$ we get $\dim\{a\in Q^{\ast}\mid\hat{a}=a\}=\frac{q-1}{p}$. Now, it is sufficient to show that $\dim Z_{1}=\dim Z_{j}$ for $j\geq 1$, which implies

[TABLE]

Let $\bar{\mathbb{F}}_{p}$ be a splitting field for $G$. To prove that $\dim Z_{1}=\dim Z_{j}$ for $j\geq 1$ first note that $Q_{i}\otimes_{\mathbb{F}_{p}}\bar{\mathbb{F}}_{p}=V_{1}\oplus\ldots\oplus V_{l_{i}}$, where $V_{j}=(\frac{1}{|N|}\sum_{x\in N}\chi_{j}(x^{-1})x)\bar{\mathbb{F}}_{p}$ and $\chi_{j}$ is a linear non-trivial character of $\bar{\mathbb{F}}_{p}N$. Thus $\alpha$ acts regularly on the set $\{V_{1},\ldots,V_{l_{i}}\}$, which proves that the fixed point space of $\alpha$ on $V_{1}\oplus\ldots\oplus V_{l_{i}}$ has dimension $\frac{l_{i}}{p}$. This implies that the fixed point space on $W_{i}$ also has dimension $\frac{l_{i}}{p}$, i.e. $\#Z_{i}=p^{\frac{l_{i}}{p}}$. ∎

In order to determine all minimal left ideals in $R_{i}$ we need the following notation. For $b\in Q_{i}^{\times}$ we denote by $[b]$ the image of $b$ in the factor group $Q_{i}^{\times}/Z_{i}^{\times}$.

Lemma 2.4**.**

For $i>0$ we have the following.

• a)

For $b\in Q_{i}^{\times}$, the space $Q_{i}(1+\alpha+\ldots\alpha^{p-1})b$ is a minimal left ideal in $R_{i}$.

• b)

$Q_{i}(1+\alpha+\ldots\alpha^{p-1})b=Q_{i}(1+\alpha+\ldots\alpha^{p-1})b^{\prime}$* iff $[b]=[b^{\prime}]$.*

• c)

Each minimal left ideal of $R_{i}$ is of the form $I_{[b]}^{i}=Q_{i}(1+\alpha+\ldots\alpha^{p-1})b$ with $b\in Q_{i}^{\times}$.

Proof.

a) This is clear since $\alpha a=\hat{a}\alpha$ for $a\in Q$ and $\hat{Q_{i}}=Q_{i}$.

b) Suppose that $0\not=a(1+\alpha+\ldots\alpha^{p-1})b=a^{\prime}(1+\alpha+\ldots\alpha^{p-1})b^{\prime}$ with $a,a^{\prime},b,b^{\prime}\in Q_{i}^{\times}$. Thus

[TABLE]

with $x=a^{\prime-1}a$ and $y=bb^{\prime-1}$. Since

[TABLE]

we obtain $xy=1=x\hat{y}$, hence $y=\hat{y}$. It follows

[TABLE]

hence $[b]=[b^{\prime}]$. Conversely, if $[b]=[b^{\prime}]$, then obviously $Q_{i}(1+\alpha+\ldots\alpha^{p-1})b=Q_{i}(1+\alpha+\ldots\alpha^{p-1})b^{\prime}$.

c) Since $\#Z_{i}=p^{\frac{l_{i}}{p}}$ by Lemma 2.3, we have constructed so far exactly $\frac{p^{l_{i}}-1}{p^{l_{i}/p}-1}$ minimal left ideals. According to Lemma 2.2 e) we have $R_{i}\cong{\rm Mat}_{p}(\mathbb{F}_{{p}^{{l_{i}}/{p}}})$. It is well-known that there is a bijection between the set of minimal left ideals in ${\rm Mat}_{p}(\mathbb{F}_{{p}^{{l_{i}}/{p}}})$ and the set of 1-dimensional subspaces in a $p$-dimensional vector space over $\mathbb{F}_{{p}^{{l_{i}}/{p}}}$, which has cardinality $\frac{p^{l_{i}}-1}{p^{l_{i}/p}-1}$. ∎

3. Asymptotically good group codes

In this section we prove that group codes are asymptotically good in any characteristic. We set here $G:=G_{p,q,p^{s_{p}(q)/p}}$ and we consider the group algebra $\mathbb{F}_{p}G$. All the notations are as in Section 2.

Lemma 3.1** (Chepyzhov [8]).**

Let $r:\mathbb{N}\longrightarrow\mathbb{N}$ denote a non-decreasing function and let

[TABLE]

If $r(t)<<\sqrt{\gamma\cdot t/\log_{p}t}$, with $\gamma=\log_{p}(e)\cdot\log_{p}(2)$, then $P(r)$ is infinite and dense in the set of all primes. In particular, if $\log_{p}(t)<<r(t)<<\sqrt{\gamma\cdot t/\log_{p}t}$, then the set of primes $t$ such that $s_{p}(t)$ grows faster than $\log_{p}(t)$ is infinite and dense in the set of all primes.

Proof.

Let $B_{n}$ be the set of primes $t$ less than $n$ which are not in $P(r)$ (i.e., if $\pi(n)$ is the set of primes less than $n$, then $\pi(n)=B_{n}\sqcup(P(r)\cap\pi(n))$). Since $s_{p}(t)$ is the multiplicative order of $p$ modulo $t$, there exists, for every $t$ in $B_{n}$, two integers $a\in\mathbb{N}$ and $k\in\mathbb{N}$ such that

[TABLE]

Thus

[TABLE]

[TABLE]

By the Prime Number Density Theorem, we have $\pi(n)\sim n/\log n$. Thus the set $P(r)$ is infinite, even dense in the set of all primes. ∎

Remark 3.2**.**

Since $\mathcal{P}$ has positive density, there are infinitely many $q\in\mathcal{P}$ such that $s_{p}(q)$ grows faster than $\log_{p}(q)$.

Lemma 3.3**.**

If $\Omega_{l}$ be the set of left ideals in $Q$ of dimension $l$, then $\#\Omega_{l}\leq q^{l/s_{p}(q)+1}$.

Proof.

Recall that $Q_{0},Q_{1},\ldots,Q_{s}$ are the irreducible modules in $Q$ where $\dim_{\mathbb{F}_{p}}Q_{0}=1$ and $\dim_{\mathbb{F}_{p}}Q_{i}=s_{p}(q)$ for $i\in\{1,\ldots,s\}$. An ideal of dimension $l$ is a direct sum of at most $l/s_{p}(q)+1$ of these irreducible modules. There are at most $(s+1)^{l/s_{p}(q)+1}$ such sums and the assertion follows from $s+1\leq q=s_{p}(q)\cdot s+1$. ∎

Let $Q^{\ast}=\bigoplus_{i=1}^{s}Q_{i}$ and let $Q^{\ast\times}$ be the multiplicative group of units of $Q^{\ast}$.

Lemma 3.4**.**

If $f\in Q^{\ast}$ such that $\dim fQ=l$ and

[TABLE]

then $\#U\geq p^{{\frac{2p-1}{p}}l}$.

Proof.

We may decompose $f=\sum_{i=1}^{s}f_{i}$, with $f_{i}\in Q_{i}$ and put $S:=\{i\ |\ f_{i}\neq 0\}$. Since $f_{i}Q_{i}^{\times}=Q_{i}^{\times}$ for $i\in S$ (recall that $Q_{i}$ is isomorphic to a field), we get

[TABLE]

By Lemma 2.4, we have

[TABLE]

where $\#I^{i}_{[b]}=p^{l_{i}}$ and $\#Q_{i}^{\times}/Z_{i}^{\times}=\#\{\text{irreducible left ideals in }R_{i}\}=\frac{p^{l_{i}}-1}{p^{l_{i}/p}-1}.$ It follows

[TABLE]

Finally,

[TABLE]

since $l=\sum_{i\in S}l_{i}$. ∎

In order to prove Theorem 3.6 we need the following result which is a special case of ([10], Theorem 3.3). Let us recall that a group code is a balanced code, as observed in [3, Lemma 2.2.].

Lemma 3.5**.**

Let $C$ be a $[n,k]_{p}$ group code. Then

[TABLE]

for all $0\leq w\leq\frac{p-1}{p}\cdot n$, where

[TABLE]

is the $p$-ary entropy function.

Theorem 3.6**.**

Let $R:=\mathbb{F}_{p}G$ and consider the unique decomposition $R=\bigoplus_{i=0}^{s}R_{i}$ into the $p$-blocks $R_{i}$ described in Theorem 2.2.

Now we choose a left ideal $I$ of $R$ as

[TABLE]

*where each $I_{i}$ is taken uniformly at random among the $1+p^{l_{i}/p}+\ldots+p^{(p-1)l_{i}/p}$ non-zero irreducible left ideals of $R_{i}$.

If $0<\delta\leq\frac{p-1}{p}$ satisfies $h_{p}(\delta)\leq\frac{p-1}{p^{2}}-\frac{\log_{p}(q)}{p\cdot s_{p}(q)}$, then the probability that the minimum relative distance of $I$ is below $\delta$ is at most

[TABLE]

Proof.

Since every irreducible left ideal $I_{i}$ is of the form given in Lemma 2.4, the above randomized construction is equivalent to consider

[TABLE]

where $[b]$ is selected uniformly at random from $Q^{\ast\times}/Z$ with $Z:=\{a\in Q^{\ast\times}\mid\hat{a}=a\}$. Since $Q^{\ast\times}$ is a group, we have $Q^{\ast\times}=aQ^{\ast\times}$ for all $a\in Q^{\ast\times}$, hence

[TABLE]

for all $a\in Q^{\ast\times}$. Let

[TABLE]

By definition of the minimum distance, we have that

[TABLE]

We can partition $Q$ as

[TABLE]

so that

[TABLE]

Let $\Omega_{l}$ be the set of left ideals in $Q$ of dimension $l$. Then

[TABLE]

by Lemma 3.3. For any $l$ and any $f\in D_{l}^{\ast}$, we can define

[TABLE]

as in Lemma 3.4. Using this we get

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $fQ^{(w)}$ is the set of elements of weight $w$ in $fQ$.

It is easy to see that each $r\in U$ can occur with the same probability as $af(1+\alpha+\ldots+\alpha^{p-1})b$, so that the above probability is independent of $r$. Thus we have

[TABLE]

by Lemma 3.4.

Moreover, $fQ$ is a $[pq,l]_{p}$ group code, so that, by Lemma 3.5, we have

[TABLE]

for all $w\leq(p-1)\cdot q$ (which is true, since $\delta\leq\frac{p-1}{p}$). Putting together all previous inequalities we have

[TABLE]

so that, by the convexity,

[TABLE]

Finally, if $h_{p}(\delta)\leq\frac{p-1}{p^{2}}-\frac{\log_{p}(q)}{p\cdot s_{p}(q)}$, then

[TABLE]

∎

Corollary 3.7**.**

Group codes over finite fields are asymptotically good.

Proof.

We have to prove the assertion only for prime fields. The general case then follows by field extension (see ([9], Proposition 12)). According to Lemma 3.1 and Remark 3.2, we may choose a sequence of primes $q_{i}$ in ${\mathcal{P}}$ such that $q_{1}<q_{2}<\ldots$ and $\frac{s_{p}(q_{i})}{\log_{p}(q_{i})}\longrightarrow\infty$ for $i\longrightarrow\infty$. Let $0<\delta\leq\frac{p-1}{p}$ with $h_{p}(\delta)\leq\frac{p-1}{p^{2}}-\frac{\log_{p}(q_{1})}{p\cdot s_{p}(q_{1})}$. Thus the assumption in Theorem 3.6 is satisfied for all $q_{i}$ and we can find a left ideal $I_{q_{i}}$ in $\mathbb{F}_{p}G_{p,q_{i},p^{s_{p}(q_{i})/p}}$ with relative minimum distance at least $\delta$. Furthermore, $\dim I_{q_{i}}=s\cdot s_{p}(q_{i})=q_{i}-1$. Thus

[TABLE]

This shows that the sequence of the left ideals $I_{q_{i}}$ is asymptotically good. ∎

Remark 3.8**.**

Note that the groups $G_{p,q,m}$ are $p$-nilpotent with cyclic Sylow $p$-subgroups. Thus the asymptotically good sequence we constructed in Corollary 3.7 is a sequence of group codes in code-checkable group algebras [6]. In such algebras all left and right ideals are principal.**

Acknowledgement. The first author was partially supported by PEPS - Jeunes Chercheur-e-s - 2018. We are very grateful to Pieter Moree who brought to our attention his paper [15]. Thanks also goes to anonymous referees who pointed out some inconsistencies in an earlier version.