Dual toric codes and polytopes of degree one

TL;DR

This paper introduces a statistical measure for short words in linear codes over finite fields, characterizes dual toric codes from degree one polytopes as extremal, and provides geometric insights and formulas for their parameters.

Contribution

It uniquely characterizes dual toric codes from degree one polytopes using a new statistical measure and offers geometric interpretations and explicit formulas for their parameters.

Findings

Dual toric codes from degree one polytopes are extremal with respect to the new measure.

The minimum distance of dual toric codes has a geometric interpretation.

Explicit formulas for parameters of primal and dual toric codes are derived.

Abstract

We define a statistical measure of the typical size of short words in a linear code over a finite field. We prove that the dual toric codes coming from polytopes of degree one are characterized, among all dual toric codes, by being extremal with respect to this measure. We also give a geometric interpretation of the minimum distance of dual toric codes and characterize its extremal values. Finally, we obtain formulas for the parameters of both primal and dual toric codes associated to polytopes of degree one.