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On metacyclic p-group codes

Seema Chahal, Sugandha Maheshwary

TL;DR

The paper advances the theory of group codes by providing a comprehensive framework to construct and analyze metacyclic codes from semisimple group algebras using strong Shoda pairs and primitive central idempotents. It extends prior results to metacyclic $p$-groups for odd primes and to $2$-groups with maximal cyclic subgroups, delivering explicit Wedderburn decompositions, bases, and distance bounds for the associated codes. It also develops non-central codes via Bass and bicyclic units, producing codes with improved parameters and several best-known examples, and extends the construction to arbitrary lengths through direct products. The work unifies and generalizes existing dihedral and quaternion code results, providing practical guidance for code construction with potential cryptographic and coding-theory applications.

Abstract

In this article, we study the metacyclic p-group codes arising from finite semisimple group algebras. In [CM25], we studied group codes arising from metacyclic groups with order divisible by two distinct odd primes. In the current work, we focus on metacyclic p-group codes, as a result of which we are also able to extend the results of [CM25] for metacyclic groups with order divisible by any two primes, not necessarily odd or distinct. Consequently, existing results on group algebras of some important classes of groups, including dihedral and quaternion groups, have been extended. Additionally, we provide left codes for the undertaken group algebras. Finally, we construct non-central codes using units motivated by Bass and bicyclic units, which are inequivalent to any abelian group codes and yield best known parameters.

On metacyclic p-group codes

TL;DR

The paper advances the theory of group codes by providing a comprehensive framework to construct and analyze metacyclic codes from semisimple group algebras using strong Shoda pairs and primitive central idempotents. It extends prior results to metacyclic -groups for odd primes and to -groups with maximal cyclic subgroups, delivering explicit Wedderburn decompositions, bases, and distance bounds for the associated codes. It also develops non-central codes via Bass and bicyclic units, producing codes with improved parameters and several best-known examples, and extends the construction to arbitrary lengths through direct products. The work unifies and generalizes existing dihedral and quaternion code results, providing practical guidance for code construction with potential cryptographic and coding-theory applications.

Abstract

In this article, we study the metacyclic p-group codes arising from finite semisimple group algebras. In [CM25], we studied group codes arising from metacyclic groups with order divisible by two distinct odd primes. In the current work, we focus on metacyclic p-group codes, as a result of which we are also able to extend the results of [CM25] for metacyclic groups with order divisible by any two primes, not necessarily odd or distinct. Consequently, existing results on group algebras of some important classes of groups, including dihedral and quaternion groups, have been extended. Additionally, we provide left codes for the undertaken group algebras. Finally, we construct non-central codes using units motivated by Bass and bicyclic units, which are inequivalent to any abelian group codes and yield best known parameters.

Paper Structure

This paper contains 11 sections, 17 theorems, 49 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Metacyclic 2-group codes
  4. $\mathbf{D_{2^{n+1}}}$
  5. $\mathbf{G_{2^{n+1}}}$
  6. Metacyclic $p$-group codes, $p\neq 2$
  7. Metacyclic $p$-group codes having maximal cyclic subgroup, $p \neq 2$
  8. Metacyclic $p$-group codes, $p\neq 2$
  9. Good $p$-group codes
  10. Metacyclic group codes of arbitrary length
  11. Some Non-Central Good Codes

Key Result

Theorem 2.1

Let $\mathbb{F}_q G$ be a finite semisimple group algebra. If $K$ is a normal subgroup of $G$ such that $G/K=\langle gK\rangle$, for some $g \in G$, then the code corresponding to the pci(s) $e:=e_C(G, G, K)$, $C \in \mathcal{R}(G/K)$, satisfy the following: In particular, if $|G/K|=p^j$ where $j \in \mathbb{N}$ and $p$ is an odd prime such that $o_{p^j}(q)=\phi(p^j)$, then the minimum distance

Theorems & Definitions (39)

  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Corollary 3.4
  • ...and 29 more