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Lattice polytopes in coding theory

Year 2015, Volume: 2 Issue: 2, 85 - 94, 30.04.2015

Ivan Soprunov

https://doi.org/10.13069/jacodesmath.75353

Abstract

In this paper we discuss combinatorial questions about lattice polytopes motivated by recent results on minimum distance estimation for toric codes. We also include a new inductive bound for the minimum distance of generalized toric codes. As an application, we give new formulas for the minimum distance of generalized toric codes for special lattice point configurations.

Keywords

Toric code Lattice polytope Minkowski length Sparse polynomials

References

  • O. Beckwith, M. Grimm, J. Soprunova, B. Weaver, Minkowski length of 3D lattice polytopes, Discrete and Computational Geometry 48, Issue 4, 1137-1158, 2012.
  • W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24, 235-265, 1997.
  • G. Brown, A. M. Kasprzyk, Small polygons and toric codes, Journal of Symbolic Computation, 51, 55-62, April 2013.
  • G. Brown, A. M. Kasprzyk, Seven new champion linear codes, LMS Journal of Computation and Mathematics, 16, 109-117, 2013.
  • V. Cestaro, Parameters of toric codes in small dimension, Senior undergraduate project, CSU 2011.
  • M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, online, http:// www.codetables.de/, accessed on October 1, 2013.
  • J. Hansen, Toric surfaces and error-correcting codes in Coding Theory, Cryptography, and Related Areas, Springer, 132-142, 2000.
  • J. Hansen, Toric varieties Hirzebruch surfaces and error-correcting codes, Appl. Algebra Engrg. Comm. Comput., 13, 289-300, 2002.
  • D. Joyner, Toric codes over finite fields, Appl. Algebra Engrg. Comm. Comput., 15, 63-79, 2004.
  • J. Little, H. Schenck, Toric surface codes and Minkowski sums, SIAM J. Discrete Math. 20, no. 4, (electronic) 999-1014, 2006.
  • J. Little, R. Schwarz, On toric codes and multivariate Vandermonde matrices, Appl. Algebra Engrg. Comm. Comput., 18(4), 349-367, 2007.
  • J. Little, Remarks on generalized toric codes, Finite Fields and Their Applications, 24, 1-14, Novem- ber 2013.
  • X. Luo, S. S.-T. Yau, M. Zhang, H. Zuo, On classification of toric surface codes of low dimension, arXiv:1402.0060.
  • D. Ruano, On the parameters ofr-dimensional toric codes, Finite Fields and Their Applications, 13, 962-976, 2007.
  • D. Ruano, On the structure of generalized toric codes, Journal of Symbolic Computation, 44(5), 499-506, 2009.
  • I. Soprunov, J. Soprunova, Toric surface codes and Minkowski length of polygons, SIAM J. Discrete Math., 23(1), 384-400, 2009.
  • I. Soprunov, J. Soprunova, Bringing toric codes to the next dimension, SIAM J. Discrete Math., 24(2), 655-665, 2010.
  • M. Tsfasman, S. Vlˇadut, D. Nogin, Algebraic geometric codes: Basic notions Providence, R.I.: Amer- ican Mathematical Society, 2007.
  • V. G. Umana, M. Velasco Dual toric codes and polytopes of degree one, preprint arXiv:1404.4063.
  • J. Whitney, A bound on the minimum distance of three dimensional toric codes, Ph.D. Thesis, 2010.

Year 2015, Volume: 2 Issue: 2, 85 - 94, 30.04.2015

Ivan Soprunov

https://doi.org/10.13069/jacodesmath.75353

Abstract

References

  • O. Beckwith, M. Grimm, J. Soprunova, B. Weaver, Minkowski length of 3D lattice polytopes, Discrete and Computational Geometry 48, Issue 4, 1137-1158, 2012.
  • W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24, 235-265, 1997.
  • G. Brown, A. M. Kasprzyk, Small polygons and toric codes, Journal of Symbolic Computation, 51, 55-62, April 2013.
  • G. Brown, A. M. Kasprzyk, Seven new champion linear codes, LMS Journal of Computation and Mathematics, 16, 109-117, 2013.
  • V. Cestaro, Parameters of toric codes in small dimension, Senior undergraduate project, CSU 2011.
  • M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, online, http:// www.codetables.de/, accessed on October 1, 2013.
  • J. Hansen, Toric surfaces and error-correcting codes in Coding Theory, Cryptography, and Related Areas, Springer, 132-142, 2000.
  • J. Hansen, Toric varieties Hirzebruch surfaces and error-correcting codes, Appl. Algebra Engrg. Comm. Comput., 13, 289-300, 2002.
  • D. Joyner, Toric codes over finite fields, Appl. Algebra Engrg. Comm. Comput., 15, 63-79, 2004.
  • J. Little, H. Schenck, Toric surface codes and Minkowski sums, SIAM J. Discrete Math. 20, no. 4, (electronic) 999-1014, 2006.
  • J. Little, R. Schwarz, On toric codes and multivariate Vandermonde matrices, Appl. Algebra Engrg. Comm. Comput., 18(4), 349-367, 2007.
  • J. Little, Remarks on generalized toric codes, Finite Fields and Their Applications, 24, 1-14, Novem- ber 2013.
  • X. Luo, S. S.-T. Yau, M. Zhang, H. Zuo, On classification of toric surface codes of low dimension, arXiv:1402.0060.
  • D. Ruano, On the parameters ofr-dimensional toric codes, Finite Fields and Their Applications, 13, 962-976, 2007.
  • D. Ruano, On the structure of generalized toric codes, Journal of Symbolic Computation, 44(5), 499-506, 2009.
  • I. Soprunov, J. Soprunova, Toric surface codes and Minkowski length of polygons, SIAM J. Discrete Math., 23(1), 384-400, 2009.
  • I. Soprunov, J. Soprunova, Bringing toric codes to the next dimension, SIAM J. Discrete Math., 24(2), 655-665, 2010.
  • M. Tsfasman, S. Vlˇadut, D. Nogin, Algebraic geometric codes: Basic notions Providence, R.I.: Amer- ican Mathematical Society, 2007.
  • V. G. Umana, M. Velasco Dual toric codes and polytopes of degree one, preprint arXiv:1404.4063.
  • J. Whitney, A bound on the minimum distance of three dimensional toric codes, Ph.D. Thesis, 2010.
There are 20 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Ivan Soprunov This is me

Publication Date April 30, 2015
Published in Issue Year 2015 Volume: 2 Issue: 2

Cite

APA Soprunov, I. (2015). Lattice polytopes in coding theory. Journal of Algebra Combinatorics Discrete Structures and Applications, 2(2), 85-94. https://doi.org/10.13069/jacodesmath.75353
AMA Soprunov I. Lattice polytopes in coding theory. Journal of Algebra Combinatorics Discrete Structures and Applications. April 2015;2(2):85-94. doi:10.13069/jacodesmath.75353
Chicago Soprunov, Ivan. “Lattice Polytopes in Coding Theory”. Journal of Algebra Combinatorics Discrete Structures and Applications 2, no. 2 (April 2015): 85-94. https://doi.org/10.13069/jacodesmath.75353.
EndNote Soprunov I (April 1, 2015) Lattice polytopes in coding theory. Journal of Algebra Combinatorics Discrete Structures and Applications 2 2 85–94.
IEEE I. Soprunov, “Lattice polytopes in coding theory”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 2, no. 2, pp. 85–94, 2015, doi: 10.13069/jacodesmath.75353.
ISNAD Soprunov, Ivan. “Lattice Polytopes in Coding Theory”. Journal of Algebra Combinatorics Discrete Structures and Applications 2/2 (April 2015), 85-94. https://doi.org/10.13069/jacodesmath.75353.
JAMA Soprunov I. Lattice polytopes in coding theory. Journal of Algebra Combinatorics Discrete Structures and Applications. 2015;2:85–94.
MLA Soprunov, Ivan. “Lattice Polytopes in Coding Theory”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 2, no. 2, 2015, pp. 85-94, doi:10.13069/jacodesmath.75353.
Vancouver Soprunov I. Lattice polytopes in coding theory. Journal of Algebra Combinatorics Discrete Structures and Applications. 2015;2(2):85-94.