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arXiv · 2406.04757

Hulls of Projective Reed-Muller Codes

Nathan Kaplan·Jon-Lark Kim
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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).

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Nathan Kaplan, Jon-Lark Kim. 2024-06-07. Hulls of Projective Reed-Muller Codes. https://arxiv.org/abs/2406.04757

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