On the non-minimality of the largest weight codewords in the binary Reed-Muller codes

Andreas Klein; Leo Storme

doi:10.3934/amc.2011.5.333

Article Contents

2011, Volume 5, Issue 2: 333-337. Doi: 10.3934/amc.2011.5.333

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On the non-minimality of the largest weight codewords in the binary Reed-Muller codes

Andreas Klein^1, and
Leo Storme^1,

1.
Department of Mathematics, Ghent University, Krijgslaan 281 - S22, 9000 Ghent

Received: April 30, 2010

Revised: July 31, 2010

Published: April 2011

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Abstract

The study of minimal codewords in linear codes was motivated by Massey who described how minimal codewords of a linear code define access structures for secret sharing schemes. As a consequence of his article, Borissov, Manev, and Nikova initiated the study of minimal codewords in the binary Reed-Muller codes. They counted the number of non-minimal codewords of weight $2d$ in the binary Reed-Muller codes RM$(r,m)$, and also gave results on the non-minimality of codewords of large weight in the binary Reed-Muller codes RM$(r,m)$. The results of Borissov, Manev, and Nikova regarding the counting of the number of non-minimal codewords of small weight in RM$(r,m)$ were improved by Schillewaert, Storme, and Thas who counted the number of non-minimal codewords of weight smaller than $3d$ in RM$(r,m)$. This article now presents new results on the non-minimality of large weight codewords in RM$(r,m)$.

Keywords:

Reed-Muller codes,
minimal codewords.

Mathematics Subject Classification: Primary: 94B05; Secondary: 51E20.

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References

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