On the Parameters of r-dimensional Toric Codes

TL;DR

This paper analyzes the parameters of r-dimensional toric codes constructed from convex polytopes, providing formulas for their dimension and estimates for their minimum distance, and challenges existing conjectures with counterexamples.

Contribution

It extends Hansen's methods to higher dimensions, computes code dimensions via cohomology, and estimates minimum distances using intersection theory and mixed volumes.

Findings

Derived explicit formulas for code dimensions.

Provided bounds for minimum distances.

Counterexample to Joyner's conjectures.

Abstract

From a rational convex polytope of dimension $r\ge 2$ J.P. Hansen constructed an error correcting code of length $n=(q-1)^r$ over the finite field $\fq$. A rational convex polytope is the same datum as a normal toric variety and a Cartier divisor. The code is obtained evaluating rational functions of the toric variety defined by the polytope at the algebraic torus, and it is an evaluation code in the sense of Goppa. We compute the dimension of the code using cohomology. The minimum distance is estimated using intersection theory and mixed volumes, extending the methods of J.P. Hansen for plane polytopes. Finally we give a counterexample to Joyner's conjectures.