Relative generalized Hamming weights of evaluation codes

TL;DR

This paper provides an algebraic description and bounds for the relative generalized Hamming weights of evaluation codes, including specific computations for toric codes over hypersimplices.

Contribution

It introduces a footprint bound for these weights and demonstrates its sharpness, along with computing the next-to-minimal weight for certain toric codes.

Findings

Established a lower bound for relative generalized Hamming weights.

Proved the bound can be sharp in certain cases.

Computed the next-to-minimal weight of specific toric codes.

Abstract

The aim of this work is to algebraically describe the relative generalized Hamming weights of evaluation codes. We give a lower bound for these weights in terms of a footprint bound. We prove that this bound can be sharp. We compute the next-to-minimal weight of toric codes over hypersimplices of degree 1.