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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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, Amsterdam, 1977. Google Scholar

[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. Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, Amsterdam, 1977. Google Scholar

[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. Google Scholar

[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.  Google Scholar

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