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August  2009, 3(3): 295-309. doi: 10.3934/amc.2009.3.295

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

Olof Heden 1, , Fabio Pasticci 2, and  Thomas Westerbäck 3,

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, Italy
3. 

Department of Mathematics, KTH, S-100 44 Stockholm, Sweden

Received  April 2009 Revised  July 2009 Published  August 2009

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: 94B2.
Citation: Olof Heden, Fabio Pasticci, Thomas Westerbäck. On the existence of extended perfect binary codes with trivial symmetry group. Advances in Mathematics of Communications, 2009, 3 (3) : 295-309. doi: 10.3934/amc.2009.3.295
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