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Toric Codes from Order Polytopes

Discrete & Computational Geometry volume 69, pages 834–848 (2023)Cite this article

Abstract

We investigate a class of linear error correcting codes in relation with the order polytopes. In particular we consider the order polytopes of tree posets and bipartite posets. We calculate the parameters of the associated toric variety codes.

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References

  1. Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)

    Google Scholar 

  2. Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51, 161–166 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  3. Geissinger, L.: The face structure of a poset polytope. In: 3rd Caribbean Conference on Combinatorics and Computing (Bridgetown 1981), pp. 125–133. University West Indies, Cave Hill Campus, Barbados (1981)

  4. Hansen, J.P.: Toric varieties Hirzebruch surfaces and error-correcting codes. Appl. Algebra Eng. Commun. Comput. 13(4), 289–300 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hansen, S.H.: Error-correcting codes from higher-dimensional varieties. Finite Fields Appl. 7(4), 531–552 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hibi, T.: Distributive lattices, affine semigroup rings and algebras with straightening laws. In: Commutative Algebra and Combinatorics (Kyoto 1985). Advance Studies in Pure Mathematics, vol. 11, pp. 93–109. North-Holland, Amsterdam (1987)

  7. Hibi, T., Higashitani, A.: Smooth Fano polytopes arising from finite partially ordered sets. Discrete Comput. Geom. 45(3), 449–461 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hibi, T., Li, N., Sahara, Y., Shikama, A.: The numbers of edges of the order polytope and the chain polytope of a finite partially ordered set. Discrete Math. 340(5), 991–994 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Little, J., Schenck, H.: Toric surface codes and Minkowski sums. SIAM J. Discrete Math. 20(4), 999–1014 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Little, J., Schwarz, R.: On toric codes and multivariate Vandermonde matrices. Appl. Algebra Eng. Commun. Comput. 18(4), 349–367 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ruano, D.: On the parameters of \(r\)-dimensional toric codes. Finite Fields Appl. 13(4), 962–976 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Soprunov, I., Soprunova, J.: Bringing toric codes to the next dimension. SIAM J. Discrete Math. 24(2), 655–665 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Stanley, R.P.: Two poset polytopes. Discrete Comput. Geom. 1(1), 9–23 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. Tsfasman, M., Vlăduţ, S., Nogin, D.: Algebraic Geometric Codes: Basic Notions. Mathematical Surveys and Monographs, vol. 139. American Mathematical Society, Providence (2007)

    Book  MATH  Google Scholar 

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Acknowledgements

We thank Roy Joshua and G.V. Ravindra for many useful discussions on the topics of this paper. We thank the referee for the constructive comments and for the very careful reading of our paper. Finally, we gratefully acknowledge the research of the first author was partially supported by a grant from the Louisiana Board of Regents.

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Authors and Affiliations

  1. Tulane University, New Orleans, LA, USA

    Mahir Bilen Can

  2. Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, 565-0871, Japan

    Takayuki Hibi

Authors
  1. Mahir Bilen Can
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  2. Takayuki Hibi
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Correspondence to Mahir Bilen Can.

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Can, M.B., Hibi, T. Toric Codes from Order Polytopes. Discrete Comput Geom 69, 834–848 (2023). https://doi.org/10.1007/s00454-021-00329-w

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  • DOI: https://doi.org/10.1007/s00454-021-00329-w

Keywords

  • Toric code
  • Parameter
  • Poset polytope
  • Order polytope
  • Shrub
  • Bipartite poset

Mathematics Subject Classification

  • 11T71
  • 06A07