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

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≤ r≤ 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.