Abstract
Toric codes are examples of evaluation codes. They are produced by evaluating homogeous polynomials of a fixed degree at the \({\mathbb {F}}_q\)-rational points of a subset Y of a toric variety X. These codes reveal how algebraic geometry and coding theory are interrelated. The minimum distance of a code is the minimum number of nonzero entries in the codewords of the code. Let I(Y) be the ideal generated by all homogeneous polynomials vanishing at all the points of Y, which is also known as the vanishing ideal of Y. We give three algebraic algorithms computing the minimum distance by using commutative algebraic tools such as the multigraded Hilbert polynomials of ideals obtained from I(Y) and zero divisors f of I(Y), and primary decomposition of I(Y), for finding a homogeneous polynomial f among all homogeneous polynomials of the same degree which has the maximum number of roots on Y.
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References
Baran, E., Sahin, M.: On parameterized Toric codes. Appl. Algebra Eng. Commun. Comput. (2023). https://doi.org/10.1007/s00200-021-00513-8
Birmpilis, S., Labahn, G., Storjohann, A.: A fast algorithm for computing the Smith normal form with multipliers for a nonsingular integer matrix. J. Symbolic Comput. 116, 146–182 (2023). https://doi.org/10.1016/j.jsc.2022.09.002
Bosch, S.: Algebraic Geometry and Commutative Algebra, Universitext, p. 504. Springer, London (2013). https://doi.org/10.1007/978-1-4471-4829-6
Brown, G., Kasprzyk, A.M.: Seven new champion linear codes. LMS J. Comput. Math. 16, 109–117 (2013). https://doi.org/10.1112/S1461157013000041
Brown, G., Kasprzyk, A.M.: Small polygons and Toric codes. J. Symb. Comput. 51, 55–62 (2013). https://doi.org/10.1016/j.jsc.2012.07.001
Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics, vol. 124, p. 841. American Mathematical Society, Providence, RI (2011). https://doi.org/10.1090/gsm/124
Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
Hansen, J.P.: Secret sharing schemes with strong multiplication and a large number of players from toric varieties. In: Arithmetic, Geometry, Cryptography and Coding Theory. Contemporary Mathematics, vol. 686, pp. 171–185 (2017). https://doi.org/10.1090/conm/686/13783
Hansen, J.P.: Toric surfaces, linear and quantum codes secret sharing and decoding. Facets of Algebraic Geometry: Volume 1: A Collection in Honor of William Fulton’s 80th Birthday 472, 371 (2022)
Lachaud, G.: The parameters of projective Reed-Muller codes. Discrete Math. 81(2), 217–221 (1990). https://doi.org/10.1016/0012-365X(90)90155-B
Little, J.B.: Remarks on generalized Toric codes. Finite Fields Appl. 24, 1–14 (2013). https://doi.org/10.1016/j.ffa.2013.05.004
Maclagan, D., Smith, G.: Uniform bounds on multigraded regularity. J. Algebr. Geom 14(1), 137–164 (2005). https://doi.org/10.1090/S1056-3911-04-00385-6
Martinez-Bernal, J., Pitones, Y., Villarreal, R.H.: Minimum distance functions of graded ideals and reed-muller-type codes. J. Pure Appl. Algebra 221(2), 251–275 (2017). https://doi.org/10.1016/j.jpaa.2016.06.006
Nardi, J.: Projective Toric codes. Int. J. Number Theory 18(01), 179–204 (2022). https://doi.org/10.1142/S1793042122500142
Nardi, J.: Algebraic geometric codes on minimal Hirzebruch surfaces. J. Algebra 535, 556–597 (2019). https://doi.org/10.1016/j.jalgebra.2019.06.022
Şahin, M.: Computing vanishing ideals for Toric codes. arXiv preprint arXiv:2207.01061 (2022)
Şahin, M., Yayla, O.: Codes on subgroups of weighted projective tori. arXiv preprint arXiv:2210.07550 (2022)
Şahin, M.: Toric codes and lattice ideals. Finite Fields Appl. 52, 243–260 (2018). https://doi.org/10.1016/j.ffa.2018.04.007
Şahin, M.: In: Barucci, V., Chapman, S., D’Anna, M., Fröberg, R. (eds.) Lattice Ideals, Semigroups and Toric Codes, pp. 285–302. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-40822-0_16
Smith, G.G.: NormalToricVarieties: A Macaulay2 package. Version 1.9. A Macaulay2 package available at https://github.com/Macaulay2/M2/blob/master/M2/Macaulay2/packages/NormalToricVarieties.m2
Stillman, M., Yackel, C., Chen, J., Sayrafi, M.: PrimaryDecomposition: A Macaulay2 package. Version 2.0. A Macaulay2 package available at https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.21/share/doc/Macaulay2/PrimaryDecomposition/html/index.html
Acknowledgements
Fadime Baldemir is supported by TÜBİTAK (TÜBİTAK-2211-A National PhD Scholarship Program). Mesut Şahin is supported by TÜBİTAK Project No:119F177. This article is part of Fadime Baldemir’s PhD thesis.
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Baldemir, F., Şahin, M. Calculating the Minimum Distance of a Toric Code via Algebraic Algorithms. Math.Comput.Sci. 17, 20 (2023). https://doi.org/10.1007/s11786-023-00566-7
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DOI: https://doi.org/10.1007/s11786-023-00566-7