Translation-invariant propelinear codes

A class of binary group codes is investigated. These codes are the propelinear codes, deened over the Hamming metric space F n , F = f0; 1g, with a group structure. Generally, they are neither abelian nor translation invariant codes but they have good algebraic and com-binatorial properties. Linear codes and Z 4-linear codes can be seen as a subclass of prope-linear codes. Exactly, it is shown here that the subclass of translation invariant propelinear codes is of type Z k 1 2 Z k 2 4 Q k 3 8 where Q 8 is the non abelian quaternion group of eight elements. For k 2 = k 3 = 0 we obtain linear binary codes and for k 1 = k 3 = 0 we obtain Z 4-linear codes. The McWilliams Identity for the class of additive propelinear codes {the abelian subclass of the translation invariant propelinear codes{ is stablished. Finally, a family of non-linear binary perfect codes with a very simply construction and a very simply decoding algorithm is presented.