Standard Bases for Linear Codes over Prime Fields

Year 2019, Volume: 7 Issue: 1, 94 - 101, 30.04.2019

Jean Jacques Ferdinand Randriamiarampanahy Harinaivo Andriatahiny Toussaint Joseph Rabeherimanana

https://doi.org/10.36753/mathenot.559263

Abstract

It is known that a linear code can be represented by a binomial ideal. In this paper, we give standard

bases for the ideals in a localization of the multivariate polynomial ring in the case of the linear codes

over prime fields.

Keywords

Linear code, semigroup order, Groebner basis, local ring, standard basis

References

• [1] Adams, W. and Loustaunau, P., An Introduction to Groebner Bases, American Mathematical Society, Vol.3, 1994.
• [2] Borges-quintana, M., Borges-Trenard, M., Fitzpatrick P. and Martinez-Moro E., Groebner bases and combinatorics for binary codes, Applicable Algebra in Engineering Communication and Computing - AAECC 19(2008), 393-411.
• [3] Buchberger, B., An Algorithm for Finding the Basis Elements of the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal, PhD thesis, University of Innsbruck, 1965.
• [4] Cooper, A., Towards a new method of decoding Algebraic codes using Groebner bases, Transactions 10th Army Conf. Appl. Math. Comp. 93(1992), 293-297.
• [5] Cox, D., Little J. and O’Shea D., Ideals, Varieties, and Algorithms, Springer, 1996.
• [6] Cox, D., Little J. and O’Shea D., Using Algebraic Geometry, Springer, 1998.
• [7] Drton, M., Sturmfels B., Sullivan S., Lectures on Algebraic Statistics, Birkhäuser, Basel, 2009.
• [8] Dück, N. and Zimmermann, K.H., Gröbner bases for perfect binary linear codes, International Journal of Pure and Applied Mathematics 91(2014), no.2, 155-167.
• [9] Dück, N. and Zimmermann, K.H., Standard Bases for binary Linear Codes, International Journal of Pure and Applied Mathematics 80(2012), no.3, 315-329.
• [10] Dück, N. and Zimmermann, K.H., Universal Groebner bases for Binary Linear Code, International Journal of Pure and Applied Mathematics 86(2013), no.2, 345-358.
• [11] Greuel, G.M. and Pfister, G., A Singular Introduction to Commutative Algebra, Springer-Verlag, Berlin, 2002.
• [12] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann.Math. 79(1964), 109-326.
• [13] Mora, T., Pfister, G. and Traverso, C., An introduction to the tangent cone algorithm, Advances in Computing Research 6(1992), 199-270.
• [14] Sala, M., Mora, T., Perret, L., Sakata, S., and Traverso, C., Groebner Bases, Coding, and Cryptography, Springer, Berlin 2009.
• [15] Saleemi, M. and Zimmermann, K.H., Groebner Bases for Linear Codes, International Journal of Pure and Applied Mathematics 62(2010), no.4, 481-491.
• [16] Saleemi, M. and Zimmermann, K.H., Linear Codes as Binomial Ideals, International Journal of Pure and Applied Mathematics 61(2010), no.2, 147-156.