Computing with toric varieties
Journal of Symbolic Computation
On the structure of generalized toric codes
Journal of Symbolic Computation
The Order Bound for Toric Codes
AAECC-18 '09 Proceedings of the 18th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Algebraic geometry codes from polyhedral divisors
Journal of Symbolic Computation
Bringing Toric Codes to the Next Dimension
SIAM Journal on Discrete Mathematics
On the parameters of r-dimensional toric codes
Finite Fields and Their Applications
Toric complete intersection codes
Journal of Symbolic Computation
Small polygons and toric codes
Journal of Symbolic Computation
Weighted Reed---Muller codes revisited
Designs, Codes and Cryptography
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Toric codes are evaluation codes obtained from an integral convex polytope $P \subset {\mathbb R}^n$ and finite field ${\mathbb F}_q$. They are, in a sense, a natural extension of Reed-Solomon codes, and have been studied recently in [V. Diaz, C. Guevara, and M. Vath, Proceedings of Simu Summer Institute, 2001], [J. Hansen, Appl. Algebra Engrg. Comm. Comput., 13 (2002), pp. 289-300; Coding Theory, Cryptography and Related Areas (Guanajuato, 1998), Springer, Berlin, pp. 132-142], and [D. Joyner, Appl. Algebra Engrg. Comm. Comput., 15 (2004), pp. 63-79]. In this paper, we obtain upper and lower bounds on the minimum distance of a toric code constructed from a polygon $P \subset {\mathbb R}^2$ by examining Minkowski sum decompositions of subpolygons of $P$. Our results give a simple and unifying explanation of bounds in Hansen’s work and empirical results of Joyner; they also apply to previously unknown cases.