Hull's Parameters of Projective Reed-Muller Code
Yufeng Song, Jinquan Luo
TL;DR
The paper determines the minimal distance of Hull$(PRM(q,m,v))$ for arbitrary $(q,m,v)$ and extends Kaplan–Kim's hull-dimension results to wider ranges of $v$. It analyzes dual-containing PRM codes, providing a precise necessary-and-sufficient condition, and derives explicit hull-dimension formulas in several parametric regimes, with bases for Hull$(PRM(q,m,v))$ given via monomial exclusions. It also establishes a fundamental relation between the hull distance and the minimum distances of related PRM codes, and supports its results with computational checks (MAGMA). The work broadens understanding of PRM hulls across higher-dimensional projective spaces and sets the stage for future exploration of more general parameter ranges and potential quantum-code implications.
Abstract
Projective Reed-Muller codes(PRM codes) are constructed from the family of projective hypersurfaces of a fixed degree over a finite field $\F_q$. In this paper, we completely determine the minimal distance of the hull of any Projective Reed-Muller codes. Motivated by Nathan Kaplan and Jon-Lark Kim \cite{kaplankim},we extend their results and calculate the hulls' dimension of Projective Reed-Muller Codes in a larger range. We also analyse two special classes of PRM codes apart from self-dual,self-orthgonal and LCD cases, which Kaplan and Kim \cite[section 3]{kaplankim} didn't consider.