We prove the nonexistence of [g3(6, d), 6, d]3 codes for d = 86, 87, 88, where g3(k, d) = ∑⌈d/3i⌉ and i=0 ... k−1. This determines n3(6, d) for d = 86, 87, 88, where nq(k, d) is the minimum length n for which an [n, k, d]q code exists.

We study the functional codes $C_2\left(X\right)$ defined on a projective algebraic variety $X$, in the case where $X\subset {\mathbb{P}}^3\left({𝔽}_q\right)$ is a non-degenerate Hermitian surface. We first give some bounds for $\#X_{Z\left(𝒬\right)}\left({𝔽}_q\right)$, which are better than the ones known. We compute the number of codewords reaching the second weight. We also estimate the third weight, show the geometrical structure of the codewords reaching this third weight and compute their number. The paper ends with a conjecture on the fourth weight and the fifth weight of the code $C_2\left(X\right)$.