A recursive construction for projective Reed-Muller codes

TL;DR

This paper introduces a recursive method to construct projective Reed-Muller codes using affine Reed-Muller codes and smaller projective codes, enabling better understanding of their parameters and weight properties.

Contribution

It presents a novel recursive construction for projective Reed-Muller codes and derives new bounds for their generalized Hamming weights.

Findings

Derived the dimension of subfield subcodes for specific degrees.

Established a sharp lower bound for generalized Hamming weights.

Demonstrated the construction's effectiveness through various cases.

Abstract

We give a recursive construction for projective Reed-Muller codes in terms of affine Reed-Muller codes and projective Reed-Muller codes in fewer variables. From this construction, we obtain the dimension of the subfield subcodes of projective Reed-Muller codes for some particular degrees that give codes with good parameters. Moreover, from this recursive construction we derive a lower bound for the generalized Hamming weights of projective Reed-Muller codes which is sharp in most of the cases we have checked.