Functional codes arising from rank $n$ Hermitian varieties and hypersurfaces in low dimensions

We study the functional code $C_d(\mathcal{X})$, introduced by G. Lachaud in 1996, in the case where $\mathcal{X}$ is a rank $n$ degenerate Hermitian variety $P{\mathcal{U}}_{n-1}$ in ${\mathbb{P}}^n({\mathbb{F}}_{q^2})$ and $d\le q$. We establish an upper bound for the maximum number of ${\mathbb{F}}_{q^2}$-rational points in the intersection of $P{\mathcal{U}}_{n-1}$ with an ${\mathbb{F}}_{q^2}$-hypersurface of degree at most $q$ in ${\mathbb{P}}^n$. Using this bound, we determine the parameters of the codes $C_d(P{\mathcal{U}}_{n-1})$ in the cases $n=2,3,4$. We also characterize the hypersurfaces that correspond to the minimum distance of these codes in the cases $n=2,3,4$.