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 Sib. Èlektron. Mat. Izv., 2018, Volume 15, Pages 1455–1462 (Mi semr1007)

Discrete mathematics and mathematical cybernetics

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

I. Yu. Mogilnykhab

a Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia
b Tomsk State University, Regional Scientific and Educational Mathematical Center, pr. Lenina, 36, 634050, Tomsk, Russia

Abstract: We consider the subgroups of the automorphism group of the linear Hadamard code that act regularly on the codewords of the code. These subgroups correspond to the regular subgroups of the general affine group $\mathrm{GA}(r,2)$ with respect to the 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 $\mathrm{GA}(r,2)$ only when $r=3$. Following the approach of [13] we study the regular subgroups of the automorphism group of the Hamming code obtained from the regular subgroups of the automorphism group of the Hadamard code of length $15$.

Keywords: error-correcting code, automorphism group, regular action, affine group.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 1.12877.2018/12.1 The work was supported by the Ministry of Education and Science of Russia (state assignment No. 1.12877.2018/12.1.

DOI: https://doi.org/10.33048/semi.2018.15.119

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Document Type: Article
UDC: 519.725
MSC: 94B60
Received July 12, 2018, published November 22, 2018
Language: English

Citation: I. Yu. Mogilnykh, “A note on regular subgroups of the automorphism group of the linear Hadamard code”, Sib. Èlektron. Mat. Izv., 15 (2018), 1455–1462

Citation in format AMSBIB
\Bibitem{Mog18} \by I.~Yu.~Mogilnykh \paper A note on regular subgroups of the automorphism group of the linear Hadamard code \jour Sib. \Elektron. Mat. Izv. \yr 2018 \vol 15 \pages 1455--1462 \mathnet{http://mi.mathnet.ru/semr1007} \crossref{https://doi.org/10.33048/semi.2018.15.119} `