Involutory permutation automorphisms of binary linear codes

TL;DR

This paper studies binary linear codes with automorphisms generated by involutions, revealing structural constraints and excluding certain automorphism groups for codes of small dimension or co-dimension.

Contribution

It proves the non-existence of quasi group codes with cyclic involutory automorphism groups up to certain dimensions and analyzes extremal self-dual codes with involutions.

Findings

No quasi group code with automorphism group isomorphic to C_2 exists up to dimension or co-dimension 4.

Structural results on extremal self-dual codes with involutory automorphisms.

Insights into the automorphism structure of high-distance self-dual codes.

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.