Remarks on low weight codewords of generalized affine and projective Reed-Muller codes

TL;DR

This paper investigates the structure of low weight codewords in affine and projective generalized Reed-Muller codes, providing new theoretical insights and bounds on their weights under various conditions.

Contribution

The paper introduces new results on the structure of low weight codewords and bounds on the second distance of generalized Reed-Muller codes in affine and projective settings.

Findings

Low weight codewords are products of affine functions when the field size is large.

Certain codewords cannot be second, third, or fourth weight under specific hypotheses.

Bounds on the second distance of projective generalized Reed-Muller codes are established.

Abstract

We propose new results on low weight codewords of affine and projective generalized Reed-Muller codes. In the affine case we prove that if the size of the working finite field is large compared to the degree of the code, the low weight codewords are products of affine functions. Then in the general case we study some types of codewords and prove that they cannot be second, thirds or fourth weight depending on the hypothesis. In the projective case the second distance of generalized Reed-Muller codes is estimated, namely a lower bound and an upper bound of this weight are given.