New informations on the structure of the functional codes defined by forms of degree h on non - degenerate Hermitian varieties in P n ( F q )

We study the functional codes of order h defined by G. Lachaud on X ⊂ P(Fq) a nondegenerate Hermitian variety. We give a condition of divisibility of the weights of the codewords. For X a non-degenerate Hermitian surface, we list the first five weights and the corresponding codewords and give a positive answer on a conjecture formulated on this question. The paper ends with a conjecture on the minimum distance and the distribution of the codewords of the first 2h+1 weights of the functional codes for the functional codes of order h on X ⊂ P(Fq) a non-singular Hermitian variety.