Towards the Complete Determination of Next-to-Minimal Weights of Projective Reed-Muller Codes

Abstract

Projective Reed-Muller codes are obtained by evaluating homogeneous polynomials of degree d in ${\mathbb {F}}_q[X_0, \ldots , X_n]$ on the points of a projective space of dimension n defined over a finite field ${\mathbb {F}}_q$. They were introduced by Lachaud, in 1986, and their minimum distance was determined by Serre and Sørensen. As for the higher Hamming weights, contributions were made by Rodier, Sboui, Ballet and Rolland, mostly for the case where $d < q$. In 2016 we succeeded in determining all next-to-minimal weights when $q = 2$, and in 2018 we determined all next-to-minimal weights for $q = 3$, and almost all of these weights for the case where $q \ge 4$. In the present paper we determine some of the missing next-to-minimal weights of projective Reed-Muller codes when $q \ge 4$. Our proofs combine results of geometric nature with techniques from Gröbner basis theory.