Binary Hamming codes and Boolean designs

Abstract

In this paper we consider a finite-dimensional vector space P over the Galois field GF(2), and the family Bk (respectively, B∗k) of all the k-sets of elements of P (respectively, of P∗=P∖{0}) summing up to zero. We compute the parameters of the 3-design (P,Bk) for any (necessarily even) k, and of the 2-design (P∗,B∗k) for any k. Also, we find a new proof for the weight distribution of the binary Hamming code. Moreover, we find the automorphism groups of the above designs by characterizing the permutations of P, respectively of P∗, that induce permutations of Bk, respectively of B∗k. In particular, this allows one to relax the definitions of the permutation automorphism groups of the binary Hamming code and of the extended binary Hamming code as the groups of permutations that preserve just the codewords of a given Hamming weight.