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

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May 2011, 5(2): 333-337. doi: 10.3934/amc.2011.5.333

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, Belgium, Belgium

Received May 2010 Revised August 2010 Published August 2011

Abstract
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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: minimal codewords., Reed-Muller codes.

Mathematics Subject Classification: Primary: 94B05; Secondary: 51E2.

Citation: Andreas Klein, Leo Storme. On the non-minimality of the largest weight codewords in the binary Reed-Muller codes. Advances in Mathematics of Communications, 2011, 5 (2) : 333-337. doi: 10.3934/amc.2011.5.333

References:

[1]	A. Ashikhmin and A. Barg, Minimal vectors in linear codes, IEEE Trans. Inform. Theory, 44 (1998), 2010-2017. doi: 10.1109/18.705584.
[2]	Y. Borissov, N. L. Manev and S. Nikova, On the non-minimal codewords in binary Reed-Muller codes, Discrete Appl. Math., 128 (2003), 65-74. doi: 10.1016/S0166-218X(02)00436-5.
[3]	T. Kasami and N. Tokura, On the weight structure of Reed-Muller codes, IEEE Trans. Inform. Theory, IT-16 (1970), 752-759. doi: 10.1109/TIT.1970.1054545.
[4]	T. Kasami, N. Tokura and S. Azumi, On the weight enumeration of weight less than $2.5d$ of Reed-Muller codes, Rept. of Faculty of Eng. Sci., Osaka Univ., Japan, 1974.
[5]	T. Kasami, N. Tokura and S. Azumi, On the weight enumeration of weight less than $2.5d$ of Reed-Muller codes, Inform. Control, 30 (1976), 380-395. doi: 10.1016/S0019-9958(76)90355-7.
[6]	F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, Amsterdam, 1977.
[7]	J. L. Massey, Minimal codewords and secret sharing, in "Proceedings of the 6th Joint Swedish-Russian International Workshop on Information Theory,'' (1993), 276-279.
[8]	J. Schillewaert, L. Storme and J. A. Thas, Minimal codewords in Reed-Muller codes, Des. Codes Crypt., 54 (2010), 273-286. doi: 10.1007/s10623-009-9323-x.

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References:

[1]	A. Ashikhmin and A. Barg, Minimal vectors in linear codes, IEEE Trans. Inform. Theory, 44 (1998), 2010-2017. doi: 10.1109/18.705584.
[2]	Y. Borissov, N. L. Manev and S. Nikova, On the non-minimal codewords in binary Reed-Muller codes, Discrete Appl. Math., 128 (2003), 65-74. doi: 10.1016/S0166-218X(02)00436-5.
[3]	T. Kasami and N. Tokura, On the weight structure of Reed-Muller codes, IEEE Trans. Inform. Theory, IT-16 (1970), 752-759. doi: 10.1109/TIT.1970.1054545.
[4]	T. Kasami, N. Tokura and S. Azumi, On the weight enumeration of weight less than $2.5d$ of Reed-Muller codes, Rept. of Faculty of Eng. Sci., Osaka Univ., Japan, 1974.
[5]	T. Kasami, N. Tokura and S. Azumi, On the weight enumeration of weight less than $2.5d$ of Reed-Muller codes, Inform. Control, 30 (1976), 380-395. doi: 10.1016/S0019-9958(76)90355-7.
[6]	F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, Amsterdam, 1977.
[7]	J. L. Massey, Minimal codewords and secret sharing, in "Proceedings of the 6th Joint Swedish-Russian International Workshop on Information Theory,'' (1993), 276-279.
[8]	J. Schillewaert, L. Storme and J. A. Thas, Minimal codewords in Reed-Muller codes, Des. Codes Crypt., 54 (2010), 273-286. doi: 10.1007/s10623-009-9323-x.

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