Abstract
We show the use of Möller’s Algorithm and related techniques for decoding and studying some combinatorial properties of linear codes. It is a concise summary of our previous results, with emphasis in illustrating the applications and comparing the developed method for computing the Gröbner basis associated with the code with the classical way to solve the same problem.
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M. Borges-Quintana, M. A. Borges-Trenard, and E. Martínez-Moro, GBLA_LC: Gröbner basis by linear algebra and linear codes, 2006a, http://www.math.arq.uva.es/~edgar/GBLAweb/.
M. Borges-Quintana, M. A. Borges-Trenard, and E. Martínez-Moro, A general framework for applying FGLM techniques to linear codes, LNCS, vol. 3857, Springer, Berlin, 2006b, pp. 76–86.
M. Borges-Quintana, M. A. Borges-Trenard, and E. Martínez-Moro, On a Gröbner bases structure associated to linear codes, J. Discrete Math. Sci. Cryptogr. 10 (2007), no. 2, 151–191.
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), no. 5, 393–411.
M. A. Borges-Trenard, M. Borges-Quintana, and T. Mora, Computing Gröbner bases by FGLM techniques in a non-commutative setting, J. Symbolic Comput. 30 (2000), no. 4, 429–449.
B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. thesis, Innsbruck, 1965.
B. Buchberger, Bruno Buchberger’s PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal, J. Symb. Comput. 41 (2006), nos. 3–4, 475–511.
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 (1993), no. 4, 329–344.
M. G. Marinari, H. M. Möller, and T. Mora, Gröbner bases of ideals defined by functionals with an application to ideals of projective points, AAECC 4 (1993), no. 2, 103–145.
T. Mora, The FGLM problem and Möller’s algorithm on zero-dimensional ideals, this volume, 2009, pp. 27–45.
The GAP Group, GAP—groups, algorithms, and programming, version 4.4.12, 2008, http://www.gap-system.org.
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Borges-Quintana, M., Borges-Trenard, M.A., Martínez-Moro, E. (2009). An Application of Möller’s Algorithm to Coding Theory. In: Sala, M., Sakata, S., Mora, T., Traverso, C., Perret, L. (eds) Gröbner Bases, Coding, and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93806-4_24
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DOI: https://doi.org/10.1007/978-3-540-93806-4_24
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