Construction and classification of Z2s-linear Hadamard codes

The Z2s-additive and Z2Z4-additive codes are subgroups of Z2sn and Z2α×Z4β, respectively. Both families can be seen as generalizations of linear codes over Z2 and Z4. A Z2s-linear (resp. Z2Z4-linear) Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive (resp. Z2Z4-additive) code. It is known that there are exactly ⌊t−12⌋ and ⌊t2⌋ nonequivalent Z2Z4-linear Hadamard codes of length 2t, with α=0 and α≠0, respectively, for all t≥3. In this paper, new Z2s-linear Hadamard codes are constructed for s>2, which are not equivalent to any Z2Z4-linear Hadamard code. Moreover, for each s>2, it is claimed that the new constructed nonlinear Z2s-linear Hadamard codes of length 2t are pairwise nonequivalent.