On the symmetry group of extended perfect binary codes of length $n+1$ and rank $n-\log(n+1)+2$

Abstract

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.