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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

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.

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.
There are 16 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Jean Jacques Ferdinand Randriamiarampanahy

Harinaivo Andriatahiny

Toussaint Joseph Rabeherimanana

Publication Date April 30, 2019
Submission Date August 2, 2018
Published in Issue Year 2019 Volume: 7 Issue: 1

Cite

APA Randriamiarampanahy, J. J. F., Andriatahiny, H., & Rabeherimanana, T. J. (2019). Standard Bases for Linear Codes over Prime Fields. Mathematical Sciences and Applications E-Notes, 7(1), 94-101. https://doi.org/10.36753/mathenot.559263
AMA Randriamiarampanahy JJF, Andriatahiny H, Rabeherimanana TJ. Standard Bases for Linear Codes over Prime Fields. Math. Sci. Appl. E-Notes. April 2019;7(1):94-101. doi:10.36753/mathenot.559263
Chicago Randriamiarampanahy, Jean Jacques Ferdinand, Harinaivo Andriatahiny, and Toussaint Joseph Rabeherimanana. “Standard Bases for Linear Codes over Prime Fields”. Mathematical Sciences and Applications E-Notes 7, no. 1 (April 2019): 94-101. https://doi.org/10.36753/mathenot.559263.
EndNote Randriamiarampanahy JJF, Andriatahiny H, Rabeherimanana TJ (April 1, 2019) Standard Bases for Linear Codes over Prime Fields. Mathematical Sciences and Applications E-Notes 7 1 94–101.
IEEE J. J. F. Randriamiarampanahy, H. Andriatahiny, and T. J. Rabeherimanana, “Standard Bases for Linear Codes over Prime Fields”, Math. Sci. Appl. E-Notes, vol. 7, no. 1, pp. 94–101, 2019, doi: 10.36753/mathenot.559263.
ISNAD Randriamiarampanahy, Jean Jacques Ferdinand et al. “Standard Bases for Linear Codes over Prime Fields”. Mathematical Sciences and Applications E-Notes 7/1 (April 2019), 94-101. https://doi.org/10.36753/mathenot.559263.
JAMA Randriamiarampanahy JJF, Andriatahiny H, Rabeherimanana TJ. Standard Bases for Linear Codes over Prime Fields. Math. Sci. Appl. E-Notes. 2019;7:94–101.
MLA Randriamiarampanahy, Jean Jacques Ferdinand et al. “Standard Bases for Linear Codes over Prime Fields”. Mathematical Sciences and Applications E-Notes, vol. 7, no. 1, 2019, pp. 94-101, doi:10.36753/mathenot.559263.
Vancouver Randriamiarampanahy JJF, Andriatahiny H, Rabeherimanana TJ. Standard Bases for Linear Codes over Prime Fields. Math. Sci. Appl. E-Notes. 2019;7(1):94-101.
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