Binary Hamming codes and Boolean designs

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