Hulls of Projective Reed-Muller Codes
<|reference_start|>Hulls of Projective Reed-Muller Codes: 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).<|reference_end|>
@article{kaplan2024hulls,
    title={Hulls of Projective Reed-Muller Codes},
    author={Nathan Kaplan, Jon-Lark Kim},
    journal={arXiv preprint arXiv:2406.04757},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.04757},
    primaryClass={cs.IT cs.DM math.IT}
}