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Indiscreet logarithms in finite fields of small characteristic
Hilbert quasi-polynomial for order domains and application to coding theory
Università degli Studi di Trento, Trento, Italy |
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.
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. |
| [2] |
M. Caboara and C. Mascia, A partial characterization of Hilbert quasi-polynomials in the non-standard case, arXiv: 1607.05468, (2016). |
| [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. |
| [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. |
| [5] |
A. Garcia, C. Güneri and H. Stichtenoth,
A generalization of the Giulietti--Korchmáros maximal curve, Advances in Geometry, 10 (2010), 427-434.
|
| [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. |
| [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. |
| [8] |
O. Geil,
Evaluation codes from an affine-variety codes perspective, Advances in Algebraic Geometry Codes, Ser. Coding Theory Cryptol, 5 (2008), 153-180.
|
| [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. |
| [10] |
J. W. L. Glaisher,
Formulae for partitions into given elements, derived from Sylvester's theorem, Quart. J. Math, 40 (1909), 275-348.
|
| [11] |
V. D. Goppa,
Codes associated with divisors, Problem of Inform. Trans., 13 (1977), 33-39.
|
| [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. |
| [13] |
M. Kreuzer and L. Robbiano, Computational Commutative Algebra 2, Springer Science & Business Media, 2005. |
| [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. |
| [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. |
| [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.
|
| [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.
|
| [18] |
S. Miura,
Linear Codes on Affine Algebraic Varieties, IEICE Trans. Fundamentals, 1996. |
| [19] |
R. Stanley, Combinatorics and Commutative Algebra, Progress in Mathematics, 41. Birkhäuser Boston, Inc., Boston, MA, 1983. |
| [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. |
| [21] |
J. J. Sylvester,
Computational methods in commutative algebra and algebraic geometry, Springer Science & Business Media, 2 (2004).
|
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. |
| [2] |
M. Caboara and C. Mascia, A partial characterization of Hilbert quasi-polynomials in the non-standard case, arXiv: 1607.05468, (2016). |
| [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. |
| [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. |
| [5] |
A. Garcia, C. Güneri and H. Stichtenoth,
A generalization of the Giulietti--Korchmáros maximal curve, Advances in Geometry, 10 (2010), 427-434.
|
| [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. |
| [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. |
| [8] |
O. Geil,
Evaluation codes from an affine-variety codes perspective, Advances in Algebraic Geometry Codes, Ser. Coding Theory Cryptol, 5 (2008), 153-180.
|
| [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. |
| [10] |
J. W. L. Glaisher,
Formulae for partitions into given elements, derived from Sylvester's theorem, Quart. J. Math, 40 (1909), 275-348.
|
| [11] |
V. D. Goppa,
Codes associated with divisors, Problem of Inform. Trans., 13 (1977), 33-39.
|
| [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. |
| [13] |
M. Kreuzer and L. Robbiano, Computational Commutative Algebra 2, Springer Science & Business Media, 2005. |
| [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. |
| [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. |
| [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.
|
| [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.
|
| [18] |
S. Miura,
Linear Codes on Affine Algebraic Varieties, IEICE Trans. Fundamentals, 1996. |
| [19] |
R. Stanley, Combinatorics and Commutative Algebra, Progress in Mathematics, 41. Birkhäuser Boston, Inc., Boston, MA, 1983. |
| [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. |
| [21] |
J. J. Sylvester,
Computational methods in commutative algebra and algebraic geometry, Springer Science & Business Media, 2 (2004).
|
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