A characterization of 1-perfect additive codes

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    Abstract

    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.
    Original languageEnglish
    Pages (from-to)1688-1697
    JournalIEEE Transactions on Information Theory
    Volume45
    DOIs
    Publication statusPublished - 1 Dec 1999

    Keywords

    • Distance-compatible additive codes
    • Perfect codes
    • Translation-invariant propelinear codes

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