# On the next-to-minimal weight of projective Reed-Muller codes

## TL;DR

This paper determines several values of the next-to-minimal weights for projective Reed-Muller codes over finite fields with q ≥ 3, extending previous binary results and providing examples of codewords with specific zero set properties.

## Contribution

It extends the known results on next-to-minimal weights of projective Reed-Muller codes to non-binary fields and provides explicit examples of codewords with particular zero set configurations.

## Key findings

- Several values for the next-to-minimal weights are identified.
- Examples of codewords with non-hyperplane zero sets are provided.
- Extension of binary results to q ≥ 3 fields.

## Abstract

In this paper we present several values for the next-to-minimal weights of projective Reed-Muller codes. We work over $\mathbb{F}_q$ with $q \geq 3$ since in IEEE-IT 62(11) p. 6300-6303 (2016) we have determined the complete values for the next-to-minimal weights of binary projective Reed-Muller codes. As in loc. cit. here we also find examples of codewords with next-to-minimal weight whose set of zeros is not in a hyperplane arrangement.