On the size of the symmetry group of a perfect code

It is shown that for every nonlinear perfect code CC of length nn and rank rr with n−log(n+1)+1≤r≤n−1n−log(n+1)+1≤r≤n−1, |Sym(C)|≤|GL(n−r,2)|⋅|GL(log(n+1)−(n−r),2)|⋅(n+12n−r)n−r, where Sym(C) denotes the group of symmetries of CC. This bound considerably improves a bound of Malyugin.