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May  2018, 12(2): 287-301. doi: 10.3934/amc.2018018

Hilbert quasi-polynomial for order domains and application to coding theory

Carla Mascia , Giancarlo Rinaldo and  Massimiliano Sala

Università degli Studi di Trento, Trento, Italy

Received  October 2016 Revised  November 2017 Published  March 2018

Fund Project: This research was partially funded by the Italian Ministry of Education, Universities and Research, with the project PRIN 2015TW9LSR "Group theory and applications".

We present an application of Hilbert quasi-polynomials to order domains, allowing the effective check of the second order-domain condition in a direct way. We also provide an improved algorithm for the computation of the related Hilbert quasi-polynomials. This allows to identify order domain codes more easily.

Keywords: Groebner basis, Hilbert polynomial, Hilbert quasi-polynomial, order domain, order domain code.
Mathematics Subject Classification: Primary: 13P25, 11T71; Secondary: 12Y05.
Citation: Carla Mascia, Giancarlo Rinaldo, Massimiliano Sala. Hilbert quasi-polynomial for order domains and application to coding theory. Advances in Mathematics of Communications, 2018, 12 (2) : 287-301. doi: 10.3934/amc.2018018
References:
[1]

H. E. Andersen and O. Geil, Evaluation codes from order domain theory, Finite Fields Appl., 14 (2008), 92-123.  doi: 10.1016/j.ffa.2006.12.004.  Google Scholar

[2]

M. Caboara and C. Mascia, A partial characterization of Hilbert quasi-polynomials in the non-standard case, arXiv: 1607.05468, (2016). Google Scholar

[3]

S. Fanali, M. Giulietti and I. Platoni, On maximal curves over finite fields of small order, Adv. Math. Commun., 6 (2012), 107-120.  doi: 10.3934/amc.2012.6.107.  Google Scholar

[4]

J. Fitzgerald and R. F. Lax, Decoding affine variety codes using Gröbner bases, Des. Codes Cryptogr., 13 (1998), 147-158.  doi: 10.1023/A:1008274212057.  Google Scholar

[5]

A. Garcia, C. Güneri and H. Stichtenoth, A generalization of the Giulietti--Korchmáros maximal curve, Advances in Geometry, 10 (2010), 427-434.   Google Scholar

[6]

O. Geil, Algebraic geometry codes from order domains, In M. Sala, T. Mora, L. Perret, S. Sakata and C. Traverso, Groebner Bases, Coding, and Cryptography, RISC Book Series, Springer, (2009), 121–141. doi: 10.1007/978-3-540-93806-4_8.  Google Scholar

[7]

O. Geil and R. Pellikaan, On the structure of order domains, Finite Fields Appl., 8 (2002), 369-396.  doi: 10.1006/ffta.2001.0347.  Google Scholar

[8]

O. Geil, Evaluation codes from an affine-variety codes perspective, Advances in Algebraic Geometry Codes, Ser. Coding Theory Cryptol, 5 (2008), 153-180.   Google Scholar

[9]

M. Giulietti and G. Korchmáros, A new family of maximal curves over a finite field, Mathematische Annalen, 343 (2009), 229-245.  doi: 10.1007/s00208-008-0270-z.  Google Scholar

[10]

J. W. L. Glaisher, Formulae for partitions into given elements, derived from Sylvester's theorem, Quart. J. Math, 40 (1909), 275-348.   Google Scholar

[11]

V. D. Goppa, Codes associated with divisors, Problem of Inform. Trans., 13 (1977), 33-39.   Google Scholar

[12]

T. Høholdt, J. van Lint and R. Pellikaan, Algebraic geometry of codes, In Handbook of Coding Theory, V. S. Pless and W. C. Huffman, 1/2 (1998), 871–961.  Google Scholar

[13]

M. Kreuzer and L. Robbiano, Computational Commutative Algebra 2, Springer Science & Business Media, 2005.  Google Scholar

[14]

D. V. Lee, On the power-series expansion of a rational function, Acta Arithmetica, 62 (1992), 229-255.  doi: 10.4064/aa-62-3-229-255.  Google Scholar

[15]

C. Marcolla, E. Orsini and M. Sala, Improved decoding of affine-variety codes, Journal of Pure and Applied Algebra, 216 (2012), 1533-1565.  doi: 10.1016/j.jpaa.2012.01.002.  Google Scholar

[16]

R. Matsumoto, Miura's Generalization of One-Point AG codes is Equivalent to Høholdt, van Lint and Pellikaan's generalization, IEICE Trans. Fund., E82-A.10 (1999), 2007-2010.   Google Scholar

[17]

R. Matsumoto and S. Miura, On the Feng-Rao bound for the L-construction of algebraic geometry codes, IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences, 83 (2000), 923-926.   Google Scholar

[18]

S. Miura, Linear Codes on Affine Algebraic Varieties, IEICE Trans. Fundamentals, 1996. Google Scholar

[19]

R. Stanley, Combinatorics and Commutative Algebra, Progress in Mathematics, 41. Birkhäuser Boston, Inc., Boston, MA, 1983.  Google Scholar

[20]

J. J. Sylvester, On subvariants, ie semi-invariants to binary quantics of an unlimited order, American Journal of Mathematics, 5 (1882), 79-136.  doi: 10.2307/2369536.  Google Scholar

[21]

J. J. Sylvester, Computational methods in commutative algebra and algebraic geometry, Springer Science & Business Media, 2 (2004).   Google Scholar

show all references

References:
[1]

H. E. Andersen and O. Geil, Evaluation codes from order domain theory, Finite Fields Appl., 14 (2008), 92-123.  doi: 10.1016/j.ffa.2006.12.004.  Google Scholar

[2]

M. Caboara and C. Mascia, A partial characterization of Hilbert quasi-polynomials in the non-standard case, arXiv: 1607.05468, (2016). Google Scholar

[3]

S. Fanali, M. Giulietti and I. Platoni, On maximal curves over finite fields of small order, Adv. Math. Commun., 6 (2012), 107-120.  doi: 10.3934/amc.2012.6.107.  Google Scholar

[4]

J. Fitzgerald and R. F. Lax, Decoding affine variety codes using Gröbner bases, Des. Codes Cryptogr., 13 (1998), 147-158.  doi: 10.1023/A:1008274212057.  Google Scholar

[5]

A. Garcia, C. Güneri and H. Stichtenoth, A generalization of the Giulietti--Korchmáros maximal curve, Advances in Geometry, 10 (2010), 427-434.   Google Scholar

[6]

O. Geil, Algebraic geometry codes from order domains, In M. Sala, T. Mora, L. Perret, S. Sakata and C. Traverso, Groebner Bases, Coding, and Cryptography, RISC Book Series, Springer, (2009), 121–141. doi: 10.1007/978-3-540-93806-4_8.  Google Scholar

[7]

O. Geil and R. Pellikaan, On the structure of order domains, Finite Fields Appl., 8 (2002), 369-396.  doi: 10.1006/ffta.2001.0347.  Google Scholar

[8]

O. Geil, Evaluation codes from an affine-variety codes perspective, Advances in Algebraic Geometry Codes, Ser. Coding Theory Cryptol, 5 (2008), 153-180.   Google Scholar

[9]

M. Giulietti and G. Korchmáros, A new family of maximal curves over a finite field, Mathematische Annalen, 343 (2009), 229-245.  doi: 10.1007/s00208-008-0270-z.  Google Scholar

[10]

J. W. L. Glaisher, Formulae for partitions into given elements, derived from Sylvester's theorem, Quart. J. Math, 40 (1909), 275-348.   Google Scholar

[11]

V. D. Goppa, Codes associated with divisors, Problem of Inform. Trans., 13 (1977), 33-39.   Google Scholar

[12]

T. Høholdt, J. van Lint and R. Pellikaan, Algebraic geometry of codes, In Handbook of Coding Theory, V. S. Pless and W. C. Huffman, 1/2 (1998), 871–961.  Google Scholar

[13]

M. Kreuzer and L. Robbiano, Computational Commutative Algebra 2, Springer Science & Business Media, 2005.  Google Scholar

[14]

D. V. Lee, On the power-series expansion of a rational function, Acta Arithmetica, 62 (1992), 229-255.  doi: 10.4064/aa-62-3-229-255.  Google Scholar

[15]

C. Marcolla, E. Orsini and M. Sala, Improved decoding of affine-variety codes, Journal of Pure and Applied Algebra, 216 (2012), 1533-1565.  doi: 10.1016/j.jpaa.2012.01.002.  Google Scholar

[16]

R. Matsumoto, Miura's Generalization of One-Point AG codes is Equivalent to Høholdt, van Lint and Pellikaan's generalization, IEICE Trans. Fund., E82-A.10 (1999), 2007-2010.   Google Scholar

[17]

R. Matsumoto and S. Miura, On the Feng-Rao bound for the L-construction of algebraic geometry codes, IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences, 83 (2000), 923-926.   Google Scholar

[18]

S. Miura, Linear Codes on Affine Algebraic Varieties, IEICE Trans. Fundamentals, 1996. Google Scholar

[19]

R. Stanley, Combinatorics and Commutative Algebra, Progress in Mathematics, 41. Birkhäuser Boston, Inc., Boston, MA, 1983.  Google Scholar

[20]

J. J. Sylvester, On subvariants, ie semi-invariants to binary quantics of an unlimited order, American Journal of Mathematics, 5 (1882), 79-136.  doi: 10.2307/2369536.  Google Scholar

[21]

J. J. Sylvester, Computational methods in commutative algebra and algebraic geometry, Springer Science & Business Media, 2 (2004).   Google Scholar

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