The Classification of Some Perfect Codes |
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Authors: | Sergey V Avgustinovich Olof Heden Faina I Solov'eva |
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Institution: | (1) Sobolev Institute of Mathematics, pr. Koptyuga 4, Novosibirsk, 630090, Russia;(2) Department of Mathematics, KTH, S-100 44 Stockholm, Sweden |
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Abstract: | Perfect 1-error correcting codes C in Z
2
n
, where n=2
m–1, are considered. Let
; denote the linear span of the words of C and let the rank of C be the dimension of the vector space
. 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
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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Keywords: | perfect codes |
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