On the size of the symmetry group of a perfect code

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