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Involutory permutation automorphisms of binary linear codes

Fatma Altunbulak Aksu, Roghayeh Hafezieh, İpek Tuvay

TL;DR

The paper investigates binary linear codes of even length with involutory permutation automorphisms, focusing on how fixed subcodes constrain the code structure and the possible automorphism groups. By bounding the dimension of fixed subcodes and analyzing fixed-point-free involutions, it proves nonexistence results for low-dimensional quasi group codes with $\mathrm{PAut}(C) \cong C_2$ and establishes tight constraints on fixed subspaces under involutory symmetries. A key methodological advancement is a generalized framework that relates fixed-point data to an involution's action, which is then applied to putative extremal self-dual codes, notably $[72,36,16]$ and $[96,48,20]$, to reveal stringent structural requirements. These results illuminate which involutory automorphisms can occur in important binary self-dual codes and guide future constructions by narrowing possible symmetry patterns.

Abstract

We investigate the properties of binary linear codes of even length whose permutation automorphism group is a cyclic group generated by an involution. Up to dimension or co-dimension $4$, we show that there is no quasi group code whose permutation automorphism group is isomorphic to $C_2$. By generalizing the method we use to prove this result, we obtain results on the structure of putative extremal self-dual $[72, 36, 16]$ and $[96, 48, 20]$ codes in the presence of an involutory permutation automorphism.

Involutory permutation automorphisms of binary linear codes

TL;DR

The paper investigates binary linear codes of even length with involutory permutation automorphisms, focusing on how fixed subcodes constrain the code structure and the possible automorphism groups. By bounding the dimension of fixed subcodes and analyzing fixed-point-free involutions, it proves nonexistence results for low-dimensional quasi group codes with and establishes tight constraints on fixed subspaces under involutory symmetries. A key methodological advancement is a generalized framework that relates fixed-point data to an involution's action, which is then applied to putative extremal self-dual codes, notably and , to reveal stringent structural requirements. These results illuminate which involutory automorphisms can occur in important binary self-dual codes and guide future constructions by narrowing possible symmetry patterns.

Abstract

We investigate the properties of binary linear codes of even length whose permutation automorphism group is a cyclic group generated by an involution. Up to dimension or co-dimension , we show that there is no quasi group code whose permutation automorphism group is isomorphic to . By generalizing the method we use to prove this result, we obtain results on the structure of putative extremal self-dual and codes in the presence of an involutory permutation automorphism.
Paper Structure (5 sections, 15 theorems, 31 equations)

This paper contains 5 sections, 15 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Background and lemmas
  3. Codes of even length with lower dimensions or co-dimensions
  4. Fixed point free automorphisms
  5. A generalization and an application to some putative extremal self-dual binary codes

Key Result

Lemma 2.1

Let $C$ be a binary linear code of length $n$ and dimension $k$. Let $\beta \in \mathrm{PAut}(C)$ be an involution and $f=\mathrm{dim}(F_{\beta}(C))$. We have that $f\geq (k-f)$ or equivalently $f \geq \lceil k/2 \rceil$.

Theorems & Definitions (33)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • ...and 23 more