TRANSLATION-INVARIANT PROPELINEAR CODES

A class of binary group codes is investigated, These codes are the pro pelinear codes, defined over the Hamming metric space F-n, F = {0, 1}, with a group structure, Generally, they are neither Abelian nor trans lation-invariant codes but they have good algebraic and combinatorial properties. Linear codes and Z(4)-linear codes can be seen as a subcla ss of propelinear codes, It is shown here that the subclass of transla tion-invariant propelinear codes is of type Z(2)(k1) + Z(4)(k2) + Q(8) (k3) where Q(8) is the non-Abelian quaternion group of eight elements. Exactly, every translation-invariant propelinear code of length n can be seen as a subgroup of Z(2)(k1) + Z(4)(k2) + Q(8)(k3) with k(1) + 2 k(2) + 4k(3) = n. For k(2) = k(3) = 0 we obtain linear binary codes an d for k(1) = k(3) = 0 we obtain Z(4)-linear codes, The class of additi ve propelinear codes-the Abelian subclass of the translation-invariant propelinear codes-is studied and a family of nonlinear binary perfect codes,vith a very simply construction and a very simply decoding algo rithm is presented.