A Characterization of 1-Perfect Additive Codes

J.Borges and J.Rifà

Abstract. The characterization of perfect single-error correcting codes, or 1-perfect codes, is an open question for a long time. Recently, J.Rifà has proved that a binary 1-perfect code can be viewed as a distance compatible structure in Fn and an homomorphism between Fn, and a loop (a quasigroup with identity element). In this paper, we study 1-perfect codes in the extremal case when Fn, with the distance compatible structure, and the loop are Abelian groups. More precissely, we study 1-perfect codes which are subgroups of Fn with a distance compatible Abelian structure. We compute the set of admissible parameters for such codes, and we give a construction for any case. We 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 are giving a full characterization. Easy coding and decoding algorithms are also presented.

Key words. Perfect codes, distance compatible Abelian codes, translation invariant propelinear codes.

Registration: PIRDI-1/98, January 1998.

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