# American Institute of Mathematical Sciences

November  2016, 10(4): 851-860. doi: 10.3934/amc.2016045

## Computing Gröbner bases associated with lattices

 1 Departamento de Matemática, Universidad de Oriente, Santiago de Cuba 2 School of Mathematics and Statistics, Carleton University, Ottawa

Received  March 2015 Published  November 2016

We specialize Möller's algorithm to the computation of Gröbner bases related to lattices. We give the complexity analysis of our algorithm. Then we provide experiments showing that our algorithm is more efficient than Buchberger's algorithm for computing the associated Gröbner bases. Furthermore we show that the binomial ideal associated to the lattice can be constructed from a set of binomials associated with a set of generators of the corresponding label code. This result is presented in a general way by means of three ideal constructions associated with group codes that constitute the same ideal. This generalizes earlier results for specific cases of group codes such as linear codes, codes over ${\mathbb Z}_m$ and label codes of lattices.
Citation: 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
##### References:

show all references

##### References:
 [1] 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 [2] Amin Sakzad, Mohammad-Reza Sadeghi. On cycle-free lattices with high rate label codes. Advances in Mathematics of Communications, 2010, 4 (4) : 441-452. doi: 10.3934/amc.2010.4.441 [3] Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Addendum. Advances in Mathematics of Communications, 2011, 5 (3) : 543-546. doi: 10.3934/amc.2011.5.543 [4] Michael Braun. On lattices, binary codes, and network codes. Advances in Mathematics of Communications, 2011, 5 (2) : 225-232. doi: 10.3934/amc.2011.5.225 [5] Sergio Estrada, J. R. García-Rozas, Justo Peralta, E. Sánchez-García. Group convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 83-94. doi: 10.3934/amc.2008.2.83 [6] Arnulf Jentzen, Felix Lindner, Primož Pušnik. On the Alekseev-Gröbner formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4475-4511. doi: 10.3934/dcdsb.2019128 [7] Steven T. Dougherty, Cristina Fernández-Córdoba, Roger Ten-Valls, Bahattin Yildiz. Quaternary group ring codes: Ranks, kernels and self-dual codes. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020023 [8] Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79 [9] 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 [10] Terasan Niyomsataya, Ali Miri, Monica Nevins. Decoding affine reflection group codes with trellises. Advances in Mathematics of Communications, 2012, 6 (4) : 385-400. doi: 10.3934/amc.2012.6.385 [11] Jamshid Moori, Amin Saeidi. Some designs and codes invariant under the Tits group. Advances in Mathematics of Communications, 2017, 11 (1) : 77-82. doi: 10.3934/amc.2017003 [12] Cristina García Pillado, Santos González, Victor Markov, Consuelo Martínez, Alexandr Nechaev. New examples of non-abelian group codes. Advances in Mathematics of Communications, 2016, 10 (1) : 1-10. doi: 10.3934/amc.2016.10.1 [13] Olof Heden, Fabio Pasticci, Thomas Westerbäck. On the existence of extended perfect binary codes with trivial symmetry group. Advances in Mathematics of Communications, 2009, 3 (3) : 295-309. doi: 10.3934/amc.2009.3.295 [14] Crnković Dean, Vedrana Mikulić Crnković, Bernardo G. Rodrigues. On self-orthogonal designs and codes related to Held's simple group. Advances in Mathematics of Communications, 2018, 12 (3) : 607-628. doi: 10.3934/amc.2018036 [15] Axel Kohnert, Johannes Zwanzger. New linear codes with prescribed group of automorphisms found by heuristic search. Advances in Mathematics of Communications, 2009, 3 (2) : 157-166. doi: 10.3934/amc.2009.3.157 [16] Irene I. Bouw, Sabine Kampf. Syndrome decoding for Hermite codes with a Sugiyama-type algorithm. Advances in Mathematics of Communications, 2012, 6 (4) : 419-442. doi: 10.3934/amc.2012.6.419 [17] Ettore Fornasini, Telma Pinho, Raquel Pinto, Paula Rocha. Composition codes. Advances in Mathematics of Communications, 2016, 10 (1) : 163-177. doi: 10.3934/amc.2016.10.163 [18] Bernardo Gabriel Rodrigues. Some optimal codes related to graphs invariant under the alternating group $A_8$. Advances in Mathematics of Communications, 2011, 5 (2) : 339-350. doi: 10.3934/amc.2011.5.339 [19] Steven T. Dougherty, Joe Gildea, Abidin Kaya, Bahattin Yildiz. New self-dual and formally self-dual codes from group ring constructions. Advances in Mathematics of Communications, 2020, 14 (1) : 11-22. doi: 10.3934/amc.2020002 [20] Alexis Eduardo Almendras Valdebenito, Andrea Luigi Tironi. On the dual codes of skew constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 659-679. doi: 10.3934/amc.2018039

2018 Impact Factor: 0.879