A characterization of 1-perfect additive codes

The characterization of perfect single error-correcting codes, or 1 perfect codes, has been an open question for a long time. Recently, Rifa has proved that a binary 1-perfect code can be viewed as a distance-compatible structure in Fⁿ and a homomorphism θ:Fⁿ→Ω where Ω is a loop (a quasi-group with identity element). In this correspondence, we consider 1-perfect codes that are subgroups of Fⁿ 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