On Classification of Toric Surface Codes of Low Dimension

TL;DR

This paper classifies low-dimensional toric surface codes, revealing their equivalence classes and illustrating how monomially equivalent codes can arise from non-equivalent polygons.

Contribution

It provides a complete classification of toric surface codes with dimension ≤ 6, except for one pair, and demonstrates the construction of equivalent codes from non-equivalent polygons.

Findings

Classification of codes with dimension ≤ 6

Identification of monomially equivalent codes from non-equivalent polygons

Examples over finite fields _7 and _8

Abstract

This work is a natural continuation of our previous work \cite{yz}. In this paper, we give a complete classification of toric surface codes of dimension less than or equal to 6, except a special pair, $C_{P_6^{(4)}}$ and $C_{P_6^{(5)}}$ over $\mathbb{F}_8$. Also, we give an example, $C_{P_6^{(5)}}$ and $C_{P_6^{(6)}}$ over $\mathbb{F}_7$, to illustrate that two monomially equivalent toric codes can be constructed from two lattice non-equivalent polygons.