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 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. © 1999 IEEE.
- Distance-compatible additive codes
- Perfect codes
- Translation-invariant propelinear codes