Hulls of Projective Reed-Muller Codes

TL;DR

This paper analyzes the hulls of projective Reed-Muller codes $C_{n,k}^q$ over finite fields by combining Sørensen's duality results with a direct linear-algebraic approach to hulls. It provides complete criteria for when such codes are self-dual, self-orthogonal, or LCD, and proves that for sufficiently large $q$, the hull dimension satisfies $\dim(Hull(C_{n,k}^q))=\dim(C_{n,k}^q)-1$, along with additional hull-dimension formulas across broader parameter ranges. A key outcome is a new, direct proof of a 2024 result of Ruano and San-José for the projective plane, with extensions to higher dimensions. The work has implications for quantum codes via hull-related parameters and suggests directions for hull-variation problems and future generalizations.

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 José (2024).

Theorems & Definitions (33)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Proposition 2.4
• proof
• Corollary 2.5
• Proposition 2.6
• Theorem 2.7
• Remark 2.8
• Theorem 2.9