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Designs, Codes and Cryptography
Erschienen in: Designs, Codes and Cryptography 2/2014

01.11.2014

On low weight codewords of generalized affine and projective Reed–Muller codes

verfasst von: Stéphane Ballet, Robert Rolland

Erschienen in: Designs, Codes and Cryptography | Ausgabe 2/2014

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Abstract

We propose new results on low weight codewords of affine and projective generalized Reed–Muller (GRM) codes. In the affine case we prove that if the cardinality of the ground field is large compared to the degree of the code, the low weight codewords are products of affine functions. Then, without this assumption on the cardinality of the field, we study codewords associated to an irreducible but not absolutely irreducible polynomial, and prove that they cannot be second, third or fourth weight depending on the hypothesis. In the projective case the second distance of GRM codes is estimated, namely a lower bound and an upper bound on this weight are given.
Vorheriger Artikel Editorial: special issue on coding and cryptography
Nächster Artikel On the arithmetic Walsh coefficients of Boolean functions
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Metadaten
Titel
On low weight codewords of generalized affine and projective Reed–Muller codes
verfasst von
Stéphane Ballet
Robert Rolland
Publikationsdatum
01.11.2014
Verlag
Springer US
Erschienen in
Designs, Codes and Cryptography / Ausgabe 2/2014
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-013-9911-7

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