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

It is proved that for every integer n = 2 ^k - 1, with k ≥ 5, 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 ≥ 5, and any rank r, with n - log(n+ 1)+3≤r≤n-1 there exist perfect codes with a trivial symmetry group.