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The Classification of Some Perfect Codes

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Abstract

Perfect 1-error correcting codes C in Z 2 n, where n=2m−1, are considered. Let \( \left\langle C \right\rangle \); denote the linear span of the words of C and let the rank of C be the dimension of the vector space\( \left\langle C \right\rangle \). It is shown that if the rank of C is n−m+2 then C is equivalent to a code given by a construction of Phelps. These codes are, in case of rank n−m+2, described by a Hamming code H and a set of MDS-codes D h , h \( \in \) H, over an alphabet with four symbols. The case of rank n−m+1 is much simpler: Any such code is a Vasil'ev code.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, pr. Koptyuga 4, Novosibirsk, 630090, Russia

    Sergey V. Avgustinovich & Faina I. Solov'eva

  2. Department of Mathematics, KTH, S-100 44, Stockholm, Sweden

    Olof Heden

Authors
  1. Sergey V. Avgustinovich
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  2. Olof Heden
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  3. Faina I. Solov'eva
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Avgustinovich, S.V., Heden, O. & Solov'eva, F.I. The Classification of Some Perfect Codes. Designs, Codes and Cryptography 31, 313–318 (2004). https://doi.org/10.1023/B:DESI.0000015891.01562.c1

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  • DOI: https://doi.org/10.1023/B:DESI.0000015891.01562.c1

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