On the Automorphism Groups of the Z2Z4-Linear 1-Perfect and Preparata-Like Codes

TL;DR

This paper investigates the automorphism groups of $Z_2Z_4$-linear codes, showing that for lengths over 16, these groups only contain symmetries preserving the code's algebraic structure, and it computes their orders.

Contribution

It proves that for sufficiently large code lengths, the automorphism groups are restricted to structure-preserving symmetries and calculates their sizes.

Findings

Automorphism groups are structure-preserving for code length > 16.

Orders of symmetry groups for $Z_2Z_4$-linear 1-perfect codes are determined.

Symmetry groups are characterized for extended 1-perfect and Preparata-like codes.

Abstract

We consider the symmetry group of a $Z_2Z_4$-linear code with parameters of a $1$-perfect, extended $1$-perfect, or Preparata-like code. We show that, provided the code length is greater than $16$, this group consists only of symmetries that preserve the $Z_2Z_4$ structure. We find the orders of the symmetry groups of the $Z_2Z_4$-linear (extended) $1$-perfect codes. Keywords: additive codes, $Z_2Z_4$-linear codes, $1$-perfect codes, Preparata-like codes, automorphism group, symmetry group.