On the size of the symmetry group of a perfect code

Abstract

a b s t r a c tIt is shown that for every nonlinear perfect code C of length n and rank r with n − log(n + 1)where Sym(C ) denotes the group of symmetries of C . This bound considerably improves a bound of Malyugin.

On the symmetry group of the Mollard code

“…[2] considered a fundamental partition to show that any perfect code of rank n − log 2 (n + 1) + 2 is obtained by Phelps construction. Utilizing a fundamental partition Heden in [6] established an upper bound on the size of the symmetry group of a perfect code as a function of the rank of the code. In Section 5 we apply the idea of the work [6] to prove our result on the symmetry group of a Mollard code.…”

“…Utilizing a fundamental partition Heden in [6] established an upper bound on the size of the symmetry group of a perfect code as a function of the rank of the code. In Section 5 we apply the idea of the work [6] to prove our result on the symmetry group of a Mollard code.…”

“…The order of the automorphism group of an arbitrary nonlinear perfect binary code was investigated by several authors, see the papers [19,20,11,6,7]. The main definitions concerning this paper see in [9].…”

“…It is well known [9] that the symmetry group Sym(H) of the Hamming code H of length n is isomorphic to the general linear group GL(log(n + 1), 2). By the linearity of the Hamming code H of length n, we have The order of the automorphism group of an arbitrary nonlinear perfect binary code was investigated by several authors, see the papers [19,20,11,6,7]. The main definitions concerning this paper see in [9].…”

“…By Lemma 8 we are now focused on the description of T . In the next lemma we use the idea similar to that of work [6]. Proof.…”