# The next-to-minimal weights of binary projective Reed-Muller codes

## TL;DR

This paper determines the next-to-minimal weights of binary projective Reed-Muller codes, filling a gap in understanding their higher Hamming weights and code performance.

## Contribution

It provides a complete characterization of the next-to-minimal weights for these codes, extending prior knowledge limited to minimum distances.

## Key findings

- Next-to-minimal weights are fully determined for binary projective Reed-Muller codes.
- In most cases, these weights match those of Reed-Muller codes.
- The results reveal exceptions where weights differ from Reed-Muller codes.

## Abstract

Projective Reed-Muller codes were introduced by Lachaud, in 1988 and their dimension and minimum distance were determined by Serre and S{\o}rensen in 1991. In coding theory one is also interested in the higher Hamming weights, to study the code performance. Yet, not many values of the higher Hamming weights are known for these codes, not even the second lowest weight (also known as next-to-minimal weight) is completely determined. In this paper we determine all the values of the next-to-minimal weight for the binary projective Reed-Muller codes, which we show to be equal to the next-to-minimal weight of Reed-Muller codes in most, but not all, cases.