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.

射影 Reed-Muller 码是由有限场 \（\mathbb {F}_q\） 上固定度的射影超曲面族构成的。我们考虑了射影 Reed-Muller 码与其对偶之间的关系。我们确定这些代码何时是自对偶的，何时是自正交的，以及何时是 LCD。然后，我们证明，当 q 足够大时，投影 Reed-Muller 码的船体维数比码的维数小 1。我们确定了范围更广参数的船体尺寸，并描述了这如何导致 Ruano 和 San-José 最近结果的新证明。