On classification of toric surface codes of low dimension (Q2011484)

The article under review continues the works of \textit{J. Little} and \textit{R. Schwarz} [Appl. Algebra Eng. Commun. Comput. 18, No. 4, 349--367 (2007; Zbl 1166.94355)] and \textit{S. S. T. Yau} and \textit{H. Zuo} [Appl. Algebra Eng. Commun. Comput. 20, No. 2, 175--185 (2009; Zbl 1174.94029)] about the classification, up to monomially equivalence, of toric surface codes [\textit{J. P. Hansen}, in: Coding theory, cryptography and related areas. Proceedings of an international conference, Guanajuato, Mexico, April 1998. Berlin: Springer. 132--142 (2000; Zbl 1010.94014)]. That is, algebraic geometry codes from a toric surface. The classification follows from the fact that toric surface codes are defined from rational plane polytopes and lattice equivalent polytopes provide monomially equivalent toric codes. This article provides a complete classification of toric surface codes of dimension 6, excepting for a special pair. The classification is not complete since the article provides an example of two monomially equivalent toric codes defined by two polytopes that are not lattice equivalent.