Classifying toric 3-fold codes of dimensions 4 and 5

TL;DR

This paper classifies 4- and 5-dimensional toric 3-fold codes by computing their minimum distances, providing a complete classification for the 4-dimensional case and partial results for the 5-dimensional case.

Contribution

It offers a full classification of 4-dimensional toric 3-fold codes and advances the understanding of 5-dimensional codes, especially those from polytopes of width 1.

Findings

Complete classification of 4-dimensional codes.

Progress on 5-dimensional codes classification.

Identification of codes from polytopes of width 1.

Abstract

A toric code is an error-correcting code determined by a toric variety or its associated integral convex polytope. We investigate $4$- and $5$-dimensional toric $3$-fold codes, which are codes arising from polytopes in $\mathbf{R}^3$ with four and five lattice points, respectively. By computing the minimum distances of each code, we fully classify the $4$-dimensional codes. We further present progress toward the same goal for dimension $5$ codes. In particular, we classify the $5$-dimensional toric $3$-fold codes arising from polytopes of width 1.