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

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$(r,m)$.
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. 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. 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. 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., (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. 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, (1977).

[7]

J. L. Massey, Minimal codewords and secret sharing,, in, (1993), 276.

[8]

J. Schillewaert, L. Storme and J. A. Thas, Minimal codewords in Reed-Muller codes,, Des. Codes Crypt., 54 (2010), 273. doi: 10.1007/s10623-009-9323-x.

show all references

References:
[1]

A. Ashikhmin and A. Barg, Minimal vectors in linear codes,, IEEE Trans. Inform. Theory, 44 (1998), 2010. 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. 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. 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., (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. 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, (1977).

[7]

J. L. Massey, Minimal codewords and secret sharing,, in, (1993), 276.

[8]

J. Schillewaert, L. Storme and J. A. Thas, Minimal codewords in Reed-Muller codes,, Des. Codes Crypt., 54 (2010), 273. doi: 10.1007/s10623-009-9323-x.

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