On the structure of generalized toric codes

TL;DR

This paper explores the structure of generalized toric codes, extending classical toric codes by evaluating polynomial algebra elements at the algebraic torus, and analyzes their algebraic and metric properties.

Contribution

It introduces a comprehensive study of the multicyclic and metric structure of generalized toric codes, including duality and minimum distance estimation.

Findings

Characterization of the multicyclic structure

Expression of dual codes

Bounds on minimum distance

Abstract

Toric codes are obtained by evaluating rational functions of a nonsingular toric variety at the algebraic torus. One can extend toric codes to the so called generalized toric codes. This extension consists on evaluating elements of an arbitrary polynomial algebra at the algebraic torus instead of a linear combination of monomials whose exponents are rational points of a convex polytope. We study their multicyclic and metric structure, and we use them to express their dual and to estimate their minimum distance.