Advances in Mathematics of Communications (AMC)

A survey of perfect codes

Pages: 223 - 247, Volume 2, Issue 2, May 2008      doi:10.3934/amc.2008.2.223

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Olof Heden - Department of Mathematics, KTH, Stockholm, Sweden S-100 44, Sweden (email)

Abstract: The first examples of perfect $e$-error correcting $q$-ary codes were given in the 1940's by Hamming and Golay. In 1973 Tietäväinen, and independently Zinoviev and Leontiev, proved that if q is a power of a prime number then there are no unknown multiple error correcting perfect $q$-ary codes. The case of single error correcting perfect codes is quite different. The number of different such codes is very large and the classification, enumeration and description of all perfect 1-error correcting codes is still an open problem.
   This survey paper is devoted to the rather many recent results, that have appeared during the last ten years, on perfect 1-error correcting binary codes. The following topics are considered: Constructions, connections with tilings of groups and with Steiner Triple Systems, enumeration, classification by rank and kernel dimension and by linear equivalence, reconstructions, isometric properties and the automorphism group of perfect codes.

Keywords:  Perfect codes.
Mathematics Subject Classification:  Primary: 94B25.

Received: December 2007;      Revised: April 2008;      Available Online: April 2008.