Projective Reed–Muller codes are constructed from the family of projective hypersurfaces of a fixed degree over a finite field \(\mathbb {F}_q\) . 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.
