Hulls of Projective Reed-Muller Codes

2024/06/07 by Kaplan, Nathan, Kim, Jon-Lark · 1 citation
#11T06 #11T71 #94B60 #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.2406.04757

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).

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