A code C is Z 2 Z 4 -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y ) is a binary linear code (respectively, a quaternary linear code). In this paper Z 2 Z 4 -additive codes are studied. Their corresponding binary images, via the Gray map, are Z 2 Z 4 -linear codes, which seem to be a very distinguished class of binary group codes. As for binary and quaternary linear codes, for these codes the fundamental parameters are found and standard forms for generator and parity-check matrices are given. In order to do this, the appropriate concept of duality for Z 2 Z 4 -additive codes is defined and the parameters of their dual codes are computed.
The characterization of perfect single error-correcting codes, or 1-perfect codes, has been an open question for a long time. Recently, Rifà has proved that a binary 1-perfect code can be viewed as a distance-compatible structure in F n and a homomorphism : F n ! , where is a loop (a quasi-group with identity element). In this correspondence, we consider 1-perfect codes that are subgroups of F n with a distance-compatible Abelian structure. We compute the set of admissible parameters and give a construction for each case. We also prove that two such codes are different if they have different parameters. The resulting codes are always systematic, and we prove their unicity. Therefore, we give a full characterization. Easy coding and decoding algorithms are also presented.
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