Maximal hypersurfaces of degree d in \({{\mathbb {P}}}^n({{\mathbb {F}}}_q)\) and minimum weight spectrum of the generalized PRM(q, d, n)-code

We give an intrinsic projective configuration of the maximal hypersurfaces of degree d in \({{\mathbb {P}}}^n({{\mathbb {F}}}_q)\) ${\mathbb{P}}^n({\mathbb{F}}_q)$ . It is a particular arrangement of hyperplanes containing subarrangements of hyperplanes in general position. This characterization allows us to compute the number of minimum distance codewords of the projective Reed-Muller code PRM(q, d, n).