Translation-invariant propelinear codes
Josep Rifà, Jaume Pujol · IEEE Transactions on Information Theory · 1997
A class of binary group codes is investigated. These codes are the propelinear codes, defined over the Hamming metric space F/sup m/, F=(0, 1), with a group structure. Generally, they are neither Abelian nor translation-invariant codes but they have good algebraic and combinatorial properties. Linear codes and Z/sub 4/-linear codes can be seen as a subclass of propelinear codes. It is shown here that the subclass of translation-invariant propelinear codes is of type Z/sub 2//sup k1//spl oplus/Z/sub 4//sup k2//spl oplus/Q/sub 8/(k3) where Q/sub 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/sub 2//sup k1//spl oplus/Z/sub 4//sup k2//spl oplus/Q/sub 8//sup k3/ with k/sub 1/+2k/sub 2/+4k/sub 3/=n. For k/sub 2/=k/sub 3/=0 we obtain linear binary codes and for k/sub 1/=k/sub 3/=0 we obtain Z/sub 4/-linear codes. The class of additive propelinear codes-the Abelian subclass of the translation-invariant propelinear codes-is studied and a family of nonlinear binary perfect codes with a very simply construction and a very simply decoding algorithm is presented.