May  2011, 5(2): 233-244. doi: 10.3934/amc.2011.5.233

Algebraic structure of the minimal support codewords set of some linear codes

1. 

Dpto. Álgebra, Geometría y Topología, Universidad de Valladolid, Castilla, Spain

2. 

Dpto. Matemática Aplicada, Universidad de Valladolid, Castilla, Spain

Received  April 2010 Revised  December 2010 Published  May 2011

It has been widely known that complete decoding for binary linear codes can be regarded as a linear integer programming problem with binary arithmetic conditions. Conti and Traverso [9] have proposed an algorithm which uses Gröbner bases to solve integer programming with ordinary integer arithmetic conditions. Ikegami and Kaji [12] extended the Conti-Traverso algorithm to solve integer programming with modulo arithmetic conditions. It is natural to consider for those problems the Graver basis associated to them which turns out to be the minimal cycles of the matroid associated to the code, i.e. minimal support codewords in the binary case and its geometry. This provides us a universal test set for the programs considered.
Citation: Irene Márquez-Corbella, Edgar Martínez-Moro. Algebraic structure of the minimal support codewords set of some linear codes. Advances in Mathematics of Communications, 2011, 5 (2) : 233-244. doi: 10.3934/amc.2011.5.233
References:
[1]

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[2]

E. R. Berlekamp, R. J. McEliece and H. C. A. van Tilborg, On the inherent intractability of certain coding problems,, IEEE Trans. Inform. Theory, IT-24 (1978), 384.  doi: 10.1109/TIT.1978.1055873.  Google Scholar

[3]

T. Bogart, A. N. Jensen and R. R. Thomas, The circuit ideal of a vector configuration,, J. Algebra, 308 (2007), 518.  doi: 10.1016/j.jalgebra.2006.07.025.  Google Scholar

[4]

M. Borges-Quintana, M. A. Borges-Trenard, P. Fitzpatrick and E. Martínez-Moro, Gröbner bases and combinatorics for binary codes,, Appl. Algebra Engrg. Comm. Comput., 19 (2008), 393.  doi: 10.1007/s00200-008-0080-2.  Google Scholar

[5]

M. Borges-Quintana, M. A. Borges-Trenard, I. Márquez-Corbella and E. Martínez-Moro, An algebraic view to gradient descent decoding,, in, (2010).   Google Scholar

[6]

M. Borges-Quintana, M. A. Borges-Trenard and E. Martínez-Moro, A Gröbner bases structure associated to linear codes,, J. Discrete Math. Sci. Cryptogr., 10 (2007), 151.   Google Scholar

[7]

Y. Borissov and N. Manev, Minimal codewords in linear codes,, Serdica Math. J., 30 (2004), 303.   Google Scholar

[8]

J. Bruck and M. Naor, The hardness of decoding linear codes with preprocessing,, IEEE Trans. Inform. Theory, 36 (1990), 381.  doi: 10.1109/18.52484.  Google Scholar

[9]

P. Conti and C. Traverso, Buchberger algorithm and integer programming,, in, (1991), 130.   Google Scholar

[10]

F. Di Biase and R. Urbanke, An algorithm to calculate the kernel of certain polynomial ring homomorphisms,, Experiment. Math., 4 (1995), 227.   Google Scholar

[11]

T. Y. Hwang, Decoding linear block codes for minimizing word error rate,, IEEE Trans. Inform. Theory, 25 (1979), 733.  doi: 10.1109/TIT.1979.1056120.  Google Scholar

[12]

D. Ikegami and Y. Kaji, Maximum likelihood decoding for linear block codes using Gröbner bases,, IEICE Trans. Fund. Electron. Commun. Comput. Sci., 3 (2003), 643.   Google Scholar

[13]

R. Liebler, Implementing gradient descent decoding,, Michigan Math. J., 58 (2009), 285.  doi: 10.1307/mmj/1242071693.  Google Scholar

[14]

J. L. Massey, Minimal Codewords and Secret Sharing,, in, (1993), 246.   Google Scholar

[15]

H. Ohsugi, D. Ikegami, T. Kitamura and T. Hibi, Gröbner bases bases of certain zero-dimensional ideals arising in coding theory,, Adv. Appl. Math., 31 (2003), 420.  doi: 10.1016/S0196-8858(03)00019-8.  Google Scholar

[16]

P. Pisón-Casares and A. Vigneron-Tenorio, On Lawrence semigroups,, J. Symb. Comput., 43 (2008), 804.  doi: 10.1016/j.jsc.2008.02.003.  Google Scholar

[17]

A. Schrijver, "Theory of Linear and Integer Programming,'', Wiley-Interscience, (1996).   Google Scholar

[18]

B. Sturmfels, "Gröbner Bases and Convex Polytopes,'', American Mathematical Society, (1996).   Google Scholar

show all references

References:
[1]

A. Barg, Complexity issues in coding theory,, in, I (1998), 649.   Google Scholar

[2]

E. R. Berlekamp, R. J. McEliece and H. C. A. van Tilborg, On the inherent intractability of certain coding problems,, IEEE Trans. Inform. Theory, IT-24 (1978), 384.  doi: 10.1109/TIT.1978.1055873.  Google Scholar

[3]

T. Bogart, A. N. Jensen and R. R. Thomas, The circuit ideal of a vector configuration,, J. Algebra, 308 (2007), 518.  doi: 10.1016/j.jalgebra.2006.07.025.  Google Scholar

[4]

M. Borges-Quintana, M. A. Borges-Trenard, P. Fitzpatrick and E. Martínez-Moro, Gröbner bases and combinatorics for binary codes,, Appl. Algebra Engrg. Comm. Comput., 19 (2008), 393.  doi: 10.1007/s00200-008-0080-2.  Google Scholar

[5]

M. Borges-Quintana, M. A. Borges-Trenard, I. Márquez-Corbella and E. Martínez-Moro, An algebraic view to gradient descent decoding,, in, (2010).   Google Scholar

[6]

M. Borges-Quintana, M. A. Borges-Trenard and E. Martínez-Moro, A Gröbner bases structure associated to linear codes,, J. Discrete Math. Sci. Cryptogr., 10 (2007), 151.   Google Scholar

[7]

Y. Borissov and N. Manev, Minimal codewords in linear codes,, Serdica Math. J., 30 (2004), 303.   Google Scholar

[8]

J. Bruck and M. Naor, The hardness of decoding linear codes with preprocessing,, IEEE Trans. Inform. Theory, 36 (1990), 381.  doi: 10.1109/18.52484.  Google Scholar

[9]

P. Conti and C. Traverso, Buchberger algorithm and integer programming,, in, (1991), 130.   Google Scholar

[10]

F. Di Biase and R. Urbanke, An algorithm to calculate the kernel of certain polynomial ring homomorphisms,, Experiment. Math., 4 (1995), 227.   Google Scholar

[11]

T. Y. Hwang, Decoding linear block codes for minimizing word error rate,, IEEE Trans. Inform. Theory, 25 (1979), 733.  doi: 10.1109/TIT.1979.1056120.  Google Scholar

[12]

D. Ikegami and Y. Kaji, Maximum likelihood decoding for linear block codes using Gröbner bases,, IEICE Trans. Fund. Electron. Commun. Comput. Sci., 3 (2003), 643.   Google Scholar

[13]

R. Liebler, Implementing gradient descent decoding,, Michigan Math. J., 58 (2009), 285.  doi: 10.1307/mmj/1242071693.  Google Scholar

[14]

J. L. Massey, Minimal Codewords and Secret Sharing,, in, (1993), 246.   Google Scholar

[15]

H. Ohsugi, D. Ikegami, T. Kitamura and T. Hibi, Gröbner bases bases of certain zero-dimensional ideals arising in coding theory,, Adv. Appl. Math., 31 (2003), 420.  doi: 10.1016/S0196-8858(03)00019-8.  Google Scholar

[16]

P. Pisón-Casares and A. Vigneron-Tenorio, On Lawrence semigroups,, J. Symb. Comput., 43 (2008), 804.  doi: 10.1016/j.jsc.2008.02.003.  Google Scholar

[17]

A. Schrijver, "Theory of Linear and Integer Programming,'', Wiley-Interscience, (1996).   Google Scholar

[18]

B. Sturmfels, "Gröbner Bases and Convex Polytopes,'', American Mathematical Society, (1996).   Google Scholar

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