Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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FOOTPRINT AND MINIMUM DISTANCE FUNCTIONS

  • Nunez-Betancourt, Luis (Centro de Investigacion en Matematicas) ;
  • Pitones, Yuriko (Departamento de Matematicas Centro de Investigacion y de Estudios Avanzados del IPN) ;
  • Villarreal, Rafael H. (Departamento de Matematicas Centro de Investigacion y de Estudios Avanzados del IPN)
  • Received : 2017.04.04
  • Accepted : 2017.06.21
  • Published : 2018.01.31
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Abstract

Let S be a polynomial ring over a field K, with a monomial order ${\prec}$, and let I be an unmixed graded ideal of S. In this paper we study two functions associated to I: The minimum distance function ${\delta}_I$ and the footprint function $fp_I$. It is shown that ${\delta}_I$ is positive and that $fp_I$ is positive if the initial ideal of I is unmixed. Then we show that if I is radical and its associated primes are generated by linear forms, then ${\delta}_I$ is strictly decreasing until it reaches the asymptotic value 1. If I is the edge ideal of a Cohen-Macaulay bipartite graph, we show that ${\delta}_I(d)=1$ for d greater than or equal to the regularity of S/I. For a graded ideal of dimension ${\geq}1$, whose initial ideal is a complete intersection, we give an exact sharp lower bound for the corresponding minimum distance function.

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References

  1. N. Alon and Z. Furedi, Covering the cube by affine hyperplanes, European J. Combin. 14 (1993), no. 2, 79-83. https://doi.org/10.1006/eujc.1993.1011
  2. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Revised Edition, Cambridge University Press, 1997.
  3. C. Carvalho, On the second Hamming weight of some Reed-Muller type codes, Finite Fields Appl. 24 (2013), 88-94. https://doi.org/10.1016/j.ffa.2013.06.004
  4. M. Chardin and G. Moreno-Socas, Regularity of lex-segment ideals: some closed for-mulas and applications, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1093-1102. https://doi.org/10.1090/S0002-9939-02-06647-9
  5. D. Cox, J. Little, and D. O'Shea, Ideals, Varieties, and Algorithms, Springer-Verlag, 1992.
  6. D. Eisenbud, The geometry of syzygies: A second course in commutative algebra and algebraic geometry, Graduate Texts in Mathematics 229, Springer-Verlag, New York, 2005.
  7. O. Geil, On the second weight of generalized Reed-Muller codes, Des. Codes Cryptogr. 48 (2008), no. 3, 323-330. https://doi.org/10.1007/s10623-008-9211-9
  8. O. Geil, Evaluation codes from an affine variety code perspective, Advances in algebraic geometry codes, 153-180, Ser. Coding Theory Cryptol., 5, World Sci. Publ., Hackensack, NJ, 2008.
  9. O. Geil and T. Hholdt, Footprints or generalized Bezout's theorem, IEEE Trans. Inform. Theory 46 (2000), no. 2, 635-641. https://doi.org/10.1109/18.825832
  10. O. Geil and R. Pellikaan, On the structure of order domains, Finite Fields Appl. 8 (2002), no. 3, 369-396. https://doi.org/10.1006/ffta.2001.0347
  11. M. Gonzalez-Sarabia, C. Rentera, and H. Tapia-Recillas, Reed-Muller-type codes over the Segre variety, Finite Fields Appl. 8 (2002), no. 4, 511-518. https://doi.org/10.1016/S1071-5797(02)90360-6
  12. D. Grayson and M. Stillman, Macaulay, Available via anonymous ftp from math. uiuc.edu, 1996.
  13. J. Herzog and T. Hibi, Distributive lattices, bipartite graphs and Alexander duality, J. Algebraic Combin. 22 (2005), no. 3, 289-302. https://doi.org/10.1007/s10801-005-4528-1
  14. I. Kaplansky, Commutative Rings, revised ed., The University of Chicago Press, Chicago, Ill.-London, 1974.
  15. M. Katzman, Characteristic-independence of Betti numbers of graph ideals, J. Combin. Theory Ser. A 113 (2006), no. 3, 435-454. https://doi.org/10.1016/j.jcta.2005.04.005
  16. M. Kummini, Regularity, depth and arithmetic rank of bipartite edge ideals, J. Algebraic Combin. 30 (2009), no. 4, 429-445. https://doi.org/10.1007/s10801-009-0171-6
  17. J. Martnez-Bernal, Y. Pitones, and R. H. Villarreal, Minimum distance functions of complete intersections, Preprint, arXiv:1601.07604, 2016.
  18. J. Martnez-Bernal, Y. Pitones, and R. H. Villarreal, Minimum distance functions of graded ideals and Reed-Muller-type codes, J. Pure Appl. Algebra 221 (2017), no. 2, 251-275. https://doi.org/10.1016/j.jpaa.2016.06.006
  19. J. C. Migliore, Introduction to liaison theory and De ciency Modules, Progress in Mathematics 165, Birkhauser Boston, Inc., Boston, MA, 1998.
  20. L. O'Carroll, F. Planas-Vilanova, and R. H. Villarreal, Degree and algebraic properties of lattice and matrix ideals, SIAM J. Discrete Math. 28 (2014), no. 1, 394-427. https://doi.org/10.1137/130922094
  21. C. Rentera, A. Simis, and R. H. Villarreal, Algebraic methods for parameterized codes and invariants of vanishing ideals over nite fields, Finite Fields Appl. 17 (2011), no. 1, 81-104. https://doi.org/10.1016/j.ffa.2010.09.007
  22. R. Stanley, Hilbert functions of graded algebras, Adv. Math. 28 (1978), no. 1, 57-83. https://doi.org/10.1016/0001-8708(78)90045-2
  23. W. V. Vasconcelos, Computational Methods in Commutative Algebra and Algebraic Geometry, Springer-Verlag, 1998.
  24. R. H. Villarreal, Cohen-Macaulay graphs, Manuscripta Math. 66 (1990), no. 3, 277-293. https://doi.org/10.1007/BF02568497
  25. R. H. Villarreal, Monomial Algebras, Second Edition, Monographs and Research Notes in Mathematics, Chapman and Hall/CRC, 2015.