Hull's Parameters of Projective Reed-Muller Code

TL;DR

This paper determines the minimal distance of the hulls of Projective Reed-Muller codes, extends existing results on hull dimensions, and analyzes two special classes beyond previously studied cases.

Contribution

It fully characterizes hull minimal distances, extends hull dimension calculations, and explores two new classes of PRM codes not previously analyzed.

Findings

Minimal distance of hulls of PRM codes determined

Hull dimensions extended to larger parameter ranges

Analysis of two new classes of PRM codes

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.