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On non-full-rank perfect codes over finite fields

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Abstract

The paper deals with perfect 1-error correcting codes over a finite field with q elements (briefly q-ary 1-perfect codes). We show that the orthogonal code to a q-ary non-full-rank 1-perfect code of length \(n = (q^{m}-1)/(q-1)\) is a q-ary constant-weight code with Hamming weight equal to \(q^{m - 1}\), where m is any natural number not less than two. Necessary and sufficient conditions for q-ary codes to be q-ary non-full-rank 1-perfect codes are obtained. We suggest a generalization of the concatenation construction to the q-ary case and construct a ternary 1-perfect code of length 13 and rank 12.

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Acknowledgements

The author would like to thank the anonymous referees for their helpful comments and suggestions. The results of this paper were presented at the International Conference “Mathematics in the Modern World” [22].

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  1. Sobolev Institute of Mathematics, 4 Acad. Koptyug Avenue, Novosibirsk, Russia, 630090

    Alexander M. Romanov

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  1. Alexander M. Romanov
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Correspondence to Alexander M. Romanov.

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Communicated by V. A. Zinoviev.

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Romanov, A.M. On non-full-rank perfect codes over finite fields. Des. Codes Cryptogr. 87, 995–1003 (2019). https://doi.org/10.1007/s10623-018-0506-1

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