On the existence of extended perfect binary codes with trivial symmetry group

Olof Heden; Fabio Pasticci; Thomas Westerbäck

doi:10.3934/amc.2009.3.295

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2009, Volume 3, Issue 3: 295-309. Doi: 10.3934/amc.2009.3.295

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On the existence of extended perfect binary codes with trivial symmetry group

1.
Department of Mathematics, KTH, Stockholm, Sweden S-100 44
2.
Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli, 1, I-06123 Perugia
3.
Department of Mathematics, KTH, S-100 44 Stockholm

Received: March 31, 2009

Revised: June 30, 2009

Published: July 2009

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Abstract

The set of permutations of the coordinate set that maps a perfect code $C$ into itself is called the symmetry group of $C$ and is denoted by Sym$(C)$. It is proved that for all integers $n=2^m-1$, where $m=4,5,6,...$, and for any integer $r$, where $n-$log$(n+1)+3\leq r\leq n-1$, there are perfect codes of length $n$ and rank $r$ with a trivial symmetry group, i.e. Sym$(C)=${id}. The result is shown to be true, more generally, for the extended perfect codes of length $n+1$.

Keywords:

Perfect codes,
symmetry group.

Mathematics Subject Classification: Primary: 94B25.

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