AMC
On the existence of extended perfect binary codes with trivial symmetry group
Olof Heden Fabio Pasticci Thomas Westerbäck
Advances in Mathematics of Communications 2009, 3(3): 295-309 doi: 10.3934/amc.2009.3.295
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.
AMC
On the symmetry group of extended perfect binary codes of length $n+1$ and rank $n-\log(n+1)+2$
Olof Heden Fabio Pasticci Thomas Westerbäck
Advances in Mathematics of Communications 2012, 6(2): 121-130 doi: 10.3934/amc.2012.6.121
It is proved that for every integer $n=2^k-1$, with $k\geq5$, there exists a perfect code $C$ of length $n$, of rank $r=n-\log(n+1)+2$ and with a trivial symmetry group. This result extends an earlier result by the authors that says that for any length $n=2^k-1$, with $k\geq5$, and any rank $r$, with $n-\log(n+1)+3\leq r\leq n-1$ there exist perfect codes with a trivial symmetry group.
keywords: symmetry group. Perfect codes

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