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

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  • 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 $n+1$.
    Mathematics Subject Classification: Primary: 94B25.

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