Abstract
Let X be a complete n-dimensional simplicial toric variety over a finite field with homogeneous coordinate ring S. In this survey, we review algebraic methods for studying evaluation codes defined on subsets of the algebraic torus T X. The key object is the vanishing ideal of the subset and its Hilbert function. We also explore the nice correspondence between subgroups of the group T X and lattice ideals as their vanishing ideals. We present recent results for obtaining a basis for the lattice and for computing a minimal generating set of its ideal.
The author is supported by TÜBİTAK Project No. 114F094.
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References
Baran, E., Şahin, M.: Vanishing ideals of parameterized toric codes. http://www.arxiv.org/abs/1802.04083
Brown, G., Kasprzyk, A.M.: Seven new champion linear codes. LMS J. Comput. Math. 16, 109–117 (2013)
Cox, D.A.: The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4, 17–50 (1995)
Cox, D.A., Little, J., Schenck, H.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. AMS, Providence (2011)
Dias, E., Neves, J.: Codes over a weighted torus. Finite Fields Appl. 33, 66–79 (2015)
Duursma, I., Rentería, C., Tapia-Recillas, H.: Reed-Muller codes on complete intersections. Appl. Algebra Eng. Commun. Comput. 11, 455–462 (2001)
Gold, L., Little, J., Schenck, H.: Cayley–Bacharach and evaluation codes on complete intersections. J. Pure Appl. Algebra 196(1), 91–99 (2005)
Hansen, J.: Linkage and codes on complete intersections. Appl. Algebra Eng. Commun. Comput. 14, 175–185 (2003)
Little, J.B.: Toric codes and finite geometries. Finite Fields Appl. 45, 203–216 (2017)
Maclagan, D., Smith, G.: Uniform bounds on multigraded regularity. J. Algebraic Geom. 14(1), 137–164 (2005)
MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1978)
Martínez-Moro, E., Ruano, D.: Toric Codes. Advances in Algebraic Geometry Codes. Coding Theory and Cryptology, vol. 5, pp. 295–322. World Scientific, Hackensack (2008)
Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)
Morales, M., Thoma, A.: Complete Intersection Lattice Ideals. J. Algebra 284, 755–770 (2005)
Rentería, C., Simis, A., Villarreal, R.H.: Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields. Finite Fields Appl. 17(1), 81–104 (2011)
Şahin, M.: Toric Codes and Lattice Ideals. Finite Fields Appl. 52, 243–260 (2018)
Şahin, M., Soprunov, I.: Multigraded Hilbert functions and toric complete intersection codes. J. Algebra 459, 446–467 (2016)
Sidman, J., Van Tuyl, A.: Multigraded regularity: syzygies and fat points. Beiträge Algebra Geom. 47(1), 67–87 (2006)
Soprunov, I.: Toric complete intersection codes. J. Symb. Comput. 50, 374–385 (2013)
J.H. van Lint, Introduction to Coding Theory, Graduate Texts in Mathematics, vol. 86. Springer, Heidelberg (1982)
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Şahin, M. (2020). Lattice Ideals, Semigroups and Toric Codes. In: Barucci, V., Chapman, S., D'Anna, M., Fröberg, R. (eds) Numerical Semigroups . Springer INdAM Series, vol 40. Springer, Cham. https://doi.org/10.1007/978-3-030-40822-0_16
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