Hulls of Projective Reed-Muller Codes

TL;DR

This paper investigates the properties of projective Reed-Muller codes, focusing on their duals, hull dimensions, and conditions for self-duality, self-orthogonality, and LCD status, providing new insights and proofs.

Contribution

It characterizes the hull dimensions of projective Reed-Muller codes for various parameters and offers a new proof of recent results by Ruano and San José.

Findings

Determines when codes are self-dual, self-orthogonal, or LCD.

Shows hull dimension is one less than code dimension for large q.

Provides a new proof of a recent hull dimension result.

Abstract

Projective Reed-Muller codes are constructed from the family of projective hypersurfaces of a fixed degree over a finite field $\F_q$. We consider the relationship between projective Reed-Muller codes and their duals. We determine when these codes are self-dual, when they are self-orthogonal, and when they are LCD. We then show that when $q$ is sufficiently large, the dimension of the hull of a projective Reed-Muller code is 1 less than the dimension of the code. We determine the dimension of the hull for a wider range of parameters and describe how this leads to a new proof of a recent result of Ruano and San Jos\'e (2024).