The second weight of generalized Reed-Muller codes in most cases

TL;DR

This paper determines the second weight of generalized Reed-Muller codes for most cases, filling in gaps for certain values of the code order, and provides estimates for specific cases.

Contribution

It extends the knowledge of second weights of Reed-Muller codes to new parameter ranges not previously covered.

Findings

Second weight determined for most cases of d

Estimates provided for special case d=a(q-1)+1

Completes the understanding of second weights in generalized Reed-Muller codes

Abstract

The second weight of the Generalized Reed-Muller code of order $d$ over the finite field with $q$ elements is now known for $d <q$ and $d>(n-1)(q-1)$. In this paper, we determine the second weight for the other values of $d$ which are not multiple of $q-1$ plus 1. For the special case $d=a(q-1)+1$ we give an estimate.