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Designs, Codes and Cryptography

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07.06.2018

On non-full-rank perfect codes over finite fields

verfasst von: Alexander M. Romanov

Erschienen in: Designs, Codes and Cryptography | Ausgabe 5/2019

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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.

Vorheriger Artikel Orthogonal one-factorizations of complete multipartite graphs

Nächster Artikel Identity-based encryption with hierarchical key-insulation in the standard model

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Titel: On non-full-rank perfect codes over finite fields
verfasst von: Alexander M. Romanov
Publikationsdatum: 07.06.2018
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 5/2019
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-018-0506-1

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On non-full-rank perfect codes over finite fields

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