Bringing Toric Codes to the next dimension

TL;DR

This paper investigates the minimum distance of higher dimensional toric codes, revealing multiplicative properties and providing explicit computations for a broad class of these codes, advancing understanding in algebraic coding theory.

Contribution

It introduces new multiplicative properties of minimum distance in higher dimensional toric codes and offers explicit calculations for a wide class of examples.

Findings

Minimum distance is multiplicative under polytope products.

Explicit minimum distance calculations for k-dilates of pyramids.

Provides a framework for constructing higher dimensional toric codes with known parameters.

Abstract

This paper is concerned with the minimum distance computation for higher dimensional toric codes defined by lattice polytopes. We show that the minimum distance is multiplicative with respect to taking the product of polytopes, and behaves in a simple way when one builds a k-dilate of a pyramid over a polytope. This allows us to construct a large class of examples of higher dimensional toric codes where we can compute the minimum distance explicitly.