-
Previous Article
On self-dual cyclic codes of length $p^a$ over $GR(p^2,s)$
- AMC Home
- This Issue
-
Next Article
On tameness of Matsumoto-Imai central maps in three variables over the finite field $\mathbb F_2$
On the ideal associated to a linear code
1. | INRIA Paris-Rocquencourt, SECRET Project-Team, 78153 Le Chesnay Cedex, France |
2. | Dpto. Matemática Aplicada, Universidad de Valladolid, Castilla |
3. | Departament d'Enginyeria de la Informació i de les Comunicacions, Universitat Autònoma de Barcelona (UAB), Spain |
References:
[1] |
M. Aliasgari, M. R. Sadeghi and D. Panario, Gröbner Bases for Lattices and an Algebraic Decoding Algorithm, IEEE Trans. Commun., 61 (2013), 1222-1230. |
[2] |
A. Ashikhmin and A. Barg, Minimal vectors in linear codes, IEEE Trans. Inf. Theory, 44 (1998), 2010-2017.
doi: 10.1109/18.705584. |
[3] |
A. Barg, Complexity issues in coding theory, in Handbook of Coding Theory, North-Holland, Amsterdam, 1998, 649-754. |
[4] |
E. R. Berlekamp, R. J. McEliece and H. C. A. Van Tilborg, On the inherent intractability of certain coding problems, IEEE Trans. Inf. Theory, IT-24 (1978), 384-386. |
[5] |
M. Borges-Quintana, M. A. Borges-Trenard, P. Fitzpatrick and E. Martínez-Moro, Gröbner bases and combinatorics for binary codes, Appl. Algebra Engrg. Comm. Comput., 19 (2008), 393-411.
doi: 10.1007/s00200-008-0080-2. |
[6] |
M. Borges-Quintana, M. A. Borges-Trenard, I. Márquez-Corbella and E. Martínez-Moro, An algebraic view to gradient descent decoding, in IEEE Inf. Theory Workshop (ITW), 2010, 1-4. |
[7] |
J. Bruck and M. Naor, The hardness of decoding linear codes with preprocessing, IEEE Trans. Inf. Theory, 36 (1990), 381-385.
doi: 10.1109/18.52484. |
[8] |
D. A. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, 2007.
doi: 10.1007/978-0-387-35651-8. |
[9] |
F. Di Biase and R. Urbanke, An algorithm to calculate the kernel of certain polynomial ring homomorphisms, Exper. Math., 4 (1995), 227-234. |
[10] |
D. Eisenbud and B. Sturmfels, Binomial ideals, Duke Math. J., 84 (1996), 1-45.
doi: 10.1215/S0012-7094-96-08401-X. |
[11] |
J. C. Faugère, P. Gianni, D. Lazard and T. Mora, Efficient computation of zero-dimensional Gröbner bases by change of ordering,, J. Symbolic Comput., 16 (): 329.
doi: 10.1006/jsco.1993.1051. |
[12] |
P. Fitzpatrick, Solving a multivariable congruence by change of term order, J. Symb. Comput., 24 (1997), 575-589.
doi: 10.1006/jsco.1997.0153. |
[13] |
P. Fitzpatrick and J. Flynn, A Gröbner basis technique for Padé approximation, J. Symb. Comput., 13 (1992), 133-138.
doi: 10.1016/S0747-7171(08)80087-9. |
[14] |
D. Ikegami and Y. Kaji, Maximum likelihood decoding for linear block codes using Grobner bases, IEICE Trans. Fund. Electron. Commun. Comput. Sci., E86-A (2003), 643-651. |
[15] |
R. A. Liebler, Implementing gradient descent decoding, Michigan Math. J., 58 (2009), 285-291.
doi: 10.1307/mmj/1242071693. |
[16] |
I. Márquez-Corbella and E. Martínez-Moro, Algebraic structure of the minimal support codewords set of some linear codes, Adv. Math. Commun., 5 (2011), 233-244.
doi: 10.3934/amc.2011.5.233. |
[17] |
I. Márquez-Corbella and E. Martínez-Moro, Decomposition of modular codes for computing test sets and Graver basis, Math. Comp. Sci., 6 (2012), 147-165.
doi: 10.1007/s11786-012-0120-y. |
[18] |
E. Prange, Step-by-step decoding in groups with weight function. Part 1, Air Force Cambridge Res. Labs Hanscom AFB MA, 1961. |
[19] |
P. Samuel, Algebraic Theory of Numbers: Translated from the French by Allan J. Silberger, Dover, 2013. |
[20] |
B. Sturmfels, Gröbner Bases and Convex Polytopes, Amer. Math. Soc., Providence, 1996. |
show all references
References:
[1] |
M. Aliasgari, M. R. Sadeghi and D. Panario, Gröbner Bases for Lattices and an Algebraic Decoding Algorithm, IEEE Trans. Commun., 61 (2013), 1222-1230. |
[2] |
A. Ashikhmin and A. Barg, Minimal vectors in linear codes, IEEE Trans. Inf. Theory, 44 (1998), 2010-2017.
doi: 10.1109/18.705584. |
[3] |
A. Barg, Complexity issues in coding theory, in Handbook of Coding Theory, North-Holland, Amsterdam, 1998, 649-754. |
[4] |
E. R. Berlekamp, R. J. McEliece and H. C. A. Van Tilborg, On the inherent intractability of certain coding problems, IEEE Trans. Inf. Theory, IT-24 (1978), 384-386. |
[5] |
M. Borges-Quintana, M. A. Borges-Trenard, P. Fitzpatrick and E. Martínez-Moro, Gröbner bases and combinatorics for binary codes, Appl. Algebra Engrg. Comm. Comput., 19 (2008), 393-411.
doi: 10.1007/s00200-008-0080-2. |
[6] |
M. Borges-Quintana, M. A. Borges-Trenard, I. Márquez-Corbella and E. Martínez-Moro, An algebraic view to gradient descent decoding, in IEEE Inf. Theory Workshop (ITW), 2010, 1-4. |
[7] |
J. Bruck and M. Naor, The hardness of decoding linear codes with preprocessing, IEEE Trans. Inf. Theory, 36 (1990), 381-385.
doi: 10.1109/18.52484. |
[8] |
D. A. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, 2007.
doi: 10.1007/978-0-387-35651-8. |
[9] |
F. Di Biase and R. Urbanke, An algorithm to calculate the kernel of certain polynomial ring homomorphisms, Exper. Math., 4 (1995), 227-234. |
[10] |
D. Eisenbud and B. Sturmfels, Binomial ideals, Duke Math. J., 84 (1996), 1-45.
doi: 10.1215/S0012-7094-96-08401-X. |
[11] |
J. C. Faugère, P. Gianni, D. Lazard and T. Mora, Efficient computation of zero-dimensional Gröbner bases by change of ordering,, J. Symbolic Comput., 16 (): 329.
doi: 10.1006/jsco.1993.1051. |
[12] |
P. Fitzpatrick, Solving a multivariable congruence by change of term order, J. Symb. Comput., 24 (1997), 575-589.
doi: 10.1006/jsco.1997.0153. |
[13] |
P. Fitzpatrick and J. Flynn, A Gröbner basis technique for Padé approximation, J. Symb. Comput., 13 (1992), 133-138.
doi: 10.1016/S0747-7171(08)80087-9. |
[14] |
D. Ikegami and Y. Kaji, Maximum likelihood decoding for linear block codes using Grobner bases, IEICE Trans. Fund. Electron. Commun. Comput. Sci., E86-A (2003), 643-651. |
[15] |
R. A. Liebler, Implementing gradient descent decoding, Michigan Math. J., 58 (2009), 285-291.
doi: 10.1307/mmj/1242071693. |
[16] |
I. Márquez-Corbella and E. Martínez-Moro, Algebraic structure of the minimal support codewords set of some linear codes, Adv. Math. Commun., 5 (2011), 233-244.
doi: 10.3934/amc.2011.5.233. |
[17] |
I. Márquez-Corbella and E. Martínez-Moro, Decomposition of modular codes for computing test sets and Graver basis, Math. Comp. Sci., 6 (2012), 147-165.
doi: 10.1007/s11786-012-0120-y. |
[18] |
E. Prange, Step-by-step decoding in groups with weight function. Part 1, Air Force Cambridge Res. Labs Hanscom AFB MA, 1961. |
[19] |
P. Samuel, Algebraic Theory of Numbers: Translated from the French by Allan J. Silberger, Dover, 2013. |
[20] |
B. Sturmfels, Gröbner Bases and Convex Polytopes, Amer. Math. Soc., Providence, 1996. |
[1] |
Ismara Álvarez-Barrientos, Mijail Borges-Quintana, Miguel Angel Borges-Trenard, Daniel Panario. Computing Gröbner bases associated with lattices. Advances in Mathematics of Communications, 2016, 10 (4) : 851-860. doi: 10.3934/amc.2016045 |
[2] |
Hannes Bartz, Antonia Wachter-Zeh. Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases. Advances in Mathematics of Communications, 2018, 12 (4) : 773-804. doi: 10.3934/amc.2018046 |
[3] |
Irene Márquez-Corbella, Edgar Martínez-Moro. Algebraic structure of the minimal support codewords set of some linear codes. Advances in Mathematics of Communications, 2011, 5 (2) : 233-244. doi: 10.3934/amc.2011.5.233 |
[4] |
Arnulf Jentzen, Felix Lindner, Primož Pušnik. On the Alekseev-Gröbner formula in Banach spaces. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4475-4511. doi: 10.3934/dcdsb.2019128 |
[5] |
Romar dela Cruz, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. On the minimum number of minimal codewords. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2020130 |
[6] |
Piotr Pokora, Tomasz Szemberg. Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone. Electronic Research Announcements, 2014, 21: 126-131. doi: 10.3934/era.2014.21.126 |
[7] |
Seok-Jin Kang and Jae-Hoon Kwon. Quantum affine algebras, combinatorics of Young walls, and global bases. Electronic Research Announcements, 2002, 8: 35-46. |
[8] |
Sergei Avdonin, Julian Edward. Controllability for a string with attached masses and Riesz bases for asymmetric spaces. Mathematical Control and Related Fields, 2019, 9 (3) : 453-494. doi: 10.3934/mcrf.2019021 |
[9] |
Anna Chiara Lai, Paola Loreti. Robot's finger and expansions in non-integer bases. Networks and Heterogeneous Media, 2012, 7 (1) : 71-111. doi: 10.3934/nhm.2012.7.71 |
[10] |
Abderrazek Karoui. A note on the construction of nonseparable wavelet bases and multiwavelet matrix filters of $L^2(\R^n)$, where $n\geq 2$. Electronic Research Announcements, 2003, 9: 32-39. |
[11] |
Daniele Bartoli, Lins Denaux. Minimal codewords arising from the incidence of points and hyperplanes in projective spaces. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021061 |
[12] |
Andreas Klein, Leo Storme. On the non-minimality of the largest weight codewords in the binary Reed-Muller codes. Advances in Mathematics of Communications, 2011, 5 (2) : 333-337. doi: 10.3934/amc.2011.5.333 |
[13] |
Somphong Jitman, San Ling, Ekkasit Sangwisut. On self-dual cyclic codes of length $p^a$ over $GR(p^2,s)$. Advances in Mathematics of Communications, 2016, 10 (2) : 255-273. doi: 10.3934/amc.2016004 |
[14] |
K. Schittkowski. Optimal parameter selection in support vector machines. Journal of Industrial and Management Optimization, 2005, 1 (4) : 465-476. doi: 10.3934/jimo.2005.1.465 |
[15] |
Pooja Louhan, S. K. Suneja. On fractional vector optimization over cones with support functions. Journal of Industrial and Management Optimization, 2017, 13 (2) : 549-572. doi: 10.3934/jimo.2016031 |
[16] |
Lauri Harhanen, Nuutti Hyvönen. Convex source support in half-plane. Inverse Problems and Imaging, 2010, 4 (3) : 429-448. doi: 10.3934/ipi.2010.4.429 |
[17] |
Florian Dumpert. Quantitative robustness of localized support vector machines. Communications on Pure and Applied Analysis, 2020, 19 (8) : 3947-3956. doi: 10.3934/cpaa.2020174 |
[18] |
Yubo Yuan, Weiguo Fan, Dongmei Pu. Spline function smooth support vector machine for classification. Journal of Industrial and Management Optimization, 2007, 3 (3) : 529-542. doi: 10.3934/jimo.2007.3.529 |
[19] |
Ying Zhang, Ling Ma, Zheng-Hai Huang. On phaseless compressed sensing with partially known support. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1519-1526. doi: 10.3934/jimo.2019014 |
[20] |
Armin Lechleiter. Explicit characterization of the support of non-linear inclusions. Inverse Problems and Imaging, 2011, 5 (3) : 675-694. doi: 10.3934/ipi.2011.5.675 |
2020 Impact Factor: 0.935
Tools
Metrics
Other articles
by authors
[Back to Top]