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

Pages: 295 - 309, Volume 3, Issue 3, August 2009

doi:10.3934/amc.2009.3.295       Abstract        Full Text (211.3K)       Related Articles

Olof Heden - Department of Mathematics, KTH, Stockholm, Sweden S-100 44, Sweden (email)
Fabio Pasticci - Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli, 1, I-06123 Perugia, Italy (email)
Thomas Westerbäck - Department of Mathematics, KTH, S-100 44 Stockholm, Sweden (email)

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.

Received: April 2009;      Revised: July 2009;      Published: August 2009.