A note on regular subgroups of the automorphism group of the linear Hadamard code

TL;DR

This paper investigates the structure of regular subgroups within the automorphism group of the linear Hadamard code, revealing specific conditions under which certain groups are regular and analyzing cases derived from the length 15 Hamming code.

Contribution

It characterizes when the dihedral group is a regular subgroup of the automorphism group of the linear Hadamard code and explores regular subgroups related to the length 15 Hamming code.

Findings

Dihedral group D_{2^{r-1}} is regular only when r=3.

Regular subgroups of the automorphism group are linked to those of the Hamming code.

Analysis of automorphism groups for length 15 Hamming code.

Abstract

We consider the regular subgroups of the automorphism group of the linear Hadamard code. These subgroups correspond to the regular subgroups of $GA(r,2)$, w.t.r action on the vectors of $F_2^{r}$, where $n=2^r-1 $ is the length of the Hamadard code. We show that the dihedral group $D_{2^{r-1}}$ is a regular subgroup of $GA(r,2)$ only when $r=3$. Following the approach of \cite{M} we study the regular subgroups of the Hamming code obtained from the regular subgroups of the automorphism group of the Hadamard code of length 15.