May  2016, 10(2): 229-254. doi: 10.3934/amc.2016003

On the ideal associated to a linear code

1. 

INRIA Paris-Rocquencourt, SECRET Project-Team, 78153 Le Chesnay Cedex, France

2. 

Dpto. Matemática Aplicada, Universidad de Valladolid, Castilla

3. 

Departament d'Enginyeria de la Informació i de les Comunicacions, Universitat Autònoma de Barcelona (UAB), Spain

Received  June 2013 Revised  February 2015 Published  April 2016

This article aims to explore the bridge between the algebraic structure of a linear code and the complete decoding process. To this end, we associate a specific binomial ideal $I_+(\mathcal C)$ to an arbitrary linear code. The binomials involved in the reduced Gröbner basis of such an ideal relative to a degree-compatible ordering induce a uniquely defined test-set for the code, and this allows the description of a Hamming metric decoding procedure. Moreover, the binomials involved in the Graver basis of $I_+(\mathcal C)$ provide a universal test-set which turns out to be a set containing the set of codewords of minimal support of the code.
Citation: Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003
References:
[1]

M. Aliasgari, M. R. Sadeghi and D. Panario, Gröbner Bases for Lattices and an Algebraic Decoding Algorithm,, IEEE Trans. Commun., 61 (2013), 1222. Google Scholar

[2]

A. Ashikhmin and A. Barg, Minimal vectors in linear codes,, IEEE Trans. Inf. Theory, 44 (1998), 2010. doi: 10.1109/18.705584. Google Scholar

[3]

A. Barg, Complexity issues in coding theory,, in Handbook of Coding Theory, (1998), 649. Google Scholar

[4]

E. R. Berlekamp, R. J. McEliece and H. C. A. Van Tilborg, On the inherent intractability of certain coding problems,, IEEE Trans. Inf. Theory, IT-24 (1978), 384. Google Scholar

[5]

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

[6]

M. Borges-Quintana, M. A. Borges-Trenard, I. Márquez-Corbella and E. Martínez-Moro, An algebraic view to gradient descent decoding,, in IEEE Inf. Theory Workshop (ITW), (2010), 1. Google Scholar

[7]

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

[8]

D. A. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra,, Springer, (2007). doi: 10.1007/978-0-387-35651-8. Google Scholar

[9]

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

[10]

D. Eisenbud and B. Sturmfels, Binomial ideals,, Duke Math. J., 84 (1996), 1. doi: 10.1215/S0012-7094-96-08401-X. Google Scholar

[11]

J. C. Faugère, P. Gianni, D. Lazard and T. Mora, Efficient computation of zero-dimensional Gröbner bases by change of ordering,, J. Symbolic Comput., 16 (): 329. doi: 10.1006/jsco.1993.1051. Google Scholar

[12]

P. Fitzpatrick, Solving a multivariable congruence by change of term order,, J. Symb. Comput., 24 (1997), 575. doi: 10.1006/jsco.1997.0153. Google Scholar

[13]

P. Fitzpatrick and J. Flynn, A Gröbner basis technique for Padé approximation,, J. Symb. Comput., 13 (1992), 133. doi: 10.1016/S0747-7171(08)80087-9. Google Scholar

[14]

D. Ikegami and Y. Kaji, Maximum likelihood decoding for linear block codes using Grobner bases,, IEICE Trans. Fund. Electron. Commun. Comput. Sci., E86-A (2003), 643. Google Scholar

[15]

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

[16]

I. Márquez-Corbella and E. Martínez-Moro, Algebraic structure of the minimal support codewords set of some linear codes,, Adv. Math. Commun., 5 (2011), 233. doi: 10.3934/amc.2011.5.233. Google Scholar

[17]

I. Márquez-Corbella and E. Martínez-Moro, Decomposition of modular codes for computing test sets and Graver basis,, Math. Comp. Sci., 6 (2012), 147. doi: 10.1007/s11786-012-0120-y. Google Scholar

[18]

E. Prange, Step-by-step decoding in groups with weight function. Part 1,, Air Force Cambridge Res. Labs Hanscom AFB MA, (1961). Google Scholar

[19]

P. Samuel, Algebraic Theory of Numbers: Translated from the French by Allan J. Silberger,, Dover, (2013). Google Scholar

[20]

B. Sturmfels, Gröbner Bases and Convex Polytopes,, Amer. Math. Soc., (1996). Google Scholar

show all references

References:
[1]

M. Aliasgari, M. R. Sadeghi and D. Panario, Gröbner Bases for Lattices and an Algebraic Decoding Algorithm,, IEEE Trans. Commun., 61 (2013), 1222. Google Scholar

[2]

A. Ashikhmin and A. Barg, Minimal vectors in linear codes,, IEEE Trans. Inf. Theory, 44 (1998), 2010. doi: 10.1109/18.705584. Google Scholar

[3]

A. Barg, Complexity issues in coding theory,, in Handbook of Coding Theory, (1998), 649. Google Scholar

[4]

E. R. Berlekamp, R. J. McEliece and H. C. A. Van Tilborg, On the inherent intractability of certain coding problems,, IEEE Trans. Inf. Theory, IT-24 (1978), 384. Google Scholar

[5]

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

[6]

M. Borges-Quintana, M. A. Borges-Trenard, I. Márquez-Corbella and E. Martínez-Moro, An algebraic view to gradient descent decoding,, in IEEE Inf. Theory Workshop (ITW), (2010), 1. Google Scholar

[7]

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

[8]

D. A. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra,, Springer, (2007). doi: 10.1007/978-0-387-35651-8. Google Scholar

[9]

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

[10]

D. Eisenbud and B. Sturmfels, Binomial ideals,, Duke Math. J., 84 (1996), 1. doi: 10.1215/S0012-7094-96-08401-X. Google Scholar

[11]

J. C. Faugère, P. Gianni, D. Lazard and T. Mora, Efficient computation of zero-dimensional Gröbner bases by change of ordering,, J. Symbolic Comput., 16 (): 329. doi: 10.1006/jsco.1993.1051. Google Scholar

[12]

P. Fitzpatrick, Solving a multivariable congruence by change of term order,, J. Symb. Comput., 24 (1997), 575. doi: 10.1006/jsco.1997.0153. Google Scholar

[13]

P. Fitzpatrick and J. Flynn, A Gröbner basis technique for Padé approximation,, J. Symb. Comput., 13 (1992), 133. doi: 10.1016/S0747-7171(08)80087-9. Google Scholar

[14]

D. Ikegami and Y. Kaji, Maximum likelihood decoding for linear block codes using Grobner bases,, IEICE Trans. Fund. Electron. Commun. Comput. Sci., E86-A (2003), 643. Google Scholar

[15]

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

[16]

I. Márquez-Corbella and E. Martínez-Moro, Algebraic structure of the minimal support codewords set of some linear codes,, Adv. Math. Commun., 5 (2011), 233. doi: 10.3934/amc.2011.5.233. Google Scholar

[17]

I. Márquez-Corbella and E. Martínez-Moro, Decomposition of modular codes for computing test sets and Graver basis,, Math. Comp. Sci., 6 (2012), 147. doi: 10.1007/s11786-012-0120-y. Google Scholar

[18]

E. Prange, Step-by-step decoding in groups with weight function. Part 1,, Air Force Cambridge Res. Labs Hanscom AFB MA, (1961). Google Scholar

[19]

P. Samuel, Algebraic Theory of Numbers: Translated from the French by Allan J. Silberger,, Dover, (2013). Google Scholar

[20]

B. Sturmfels, Gröbner Bases and Convex Polytopes,, Amer. Math. Soc., (1996). Google Scholar

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