caa a22 4500 463203062 CHVBK 20180405153117.0 cr unu---uuuuu 170326e20070801xx s 000 0 eng 10.1007/s00200-007-0041-1 doi (NATIONALLICENCE)springer-10.1007/s00200-007-0041-1 On toric codes and multivariate Vandermonde matrices [Elektronische Daten] [John Little, Ryan Schwarz] Toric codes are a class of m-dimensional cyclic codes introduced recently by Hansen (Coding theory, cryptography and related areas (Guanajuato, 1998), pp 132-142, Springer, Berlin, 2000; Appl Algebra Eng Commun Comput 13:289-300, 2002), and studied in Joyner (Appl Algebra Eng Commun Comput 15:63-79, 2004) and Little and Schenck (SIAM Discrete Math, 2007). They may be defined as evaluation codes obtained from monomials corresponding to integer lattice points in an integral convex polytope $$P \subseteq {\mathbb{R}}^m$$ . As such, they are in a sense a natural extension of Reed-Solomon codes. Several articles cited above use intersection theory on toric varieties to derive bounds on the minimum distance of some toric codes. In this paper, we will provide a more elementary approach that applies equally well to many toric codes for all $$m \ge 2$$ . Our methods are based on a sort of multivariate generalization of Vandermonde determinants that has also been used in the study of multivariate polynomial interpolation. We use these Vandermonde determinants to determine the minimum distance of toric codes from simplices and rectangular polytopes. We also prove a general result showing that if there is a unimodular integer affine transformation taking one polytope P 1 to a second polytope P 2, then the corresponding toric codes are monomially equivalent (hence have the same parameters). We use this to begin a classification of two-dimensional cyclic toric codes with small dimension. Springer-Verlag, 2007 Coding theory nationallicence Toric code nationallicence Vandermonde matrix nationallicence Little John Department of Mathematics and Computer Science, College of the Holy Cross, 01610, Worcester, MA, USA aut Schwarz Ryan Department of Mathematics and Computer Science, College of the Holy Cross, 01610, Worcester, MA, USA aut Applicable Algebra in Engineering, Communication and Computing Springer-Verlag 18/4(2007-08-01), 349-367 0938-1279 18:4<349 2007 18 200 https://doi.org/10.1007/s00200-007-0041-1 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00200-007-0041-1 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Little John Department of Mathematics and Computer Science, College of the Holy Cross, 01610, Worcester, MA, USA aut NATIONALLICENCE 700 1- Schwarz Ryan Department of Mathematics and Computer Science, College of the Holy Cross, 01610, Worcester, MA, USA aut NATIONALLICENCE 773 0- Applicable Algebra in Engineering, Communication and Computing Springer-Verlag 18/4(2007-08-01), 349-367 0938-1279 18:4<349 2007 18 200 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer