2 00 6 Codes defined by forms of degree 2 on quadric and non - degenerate hermitian varieties in P 4

We study the functional codes of second order defined by G. Lachaud on X ⊂ P(Fq) a quadric of rank(X )=3,4,5 or a non-degenerate hermitian variety. We give some bounds for the number of points of quadratic sections of X , which are the best possible and show that codes defined on non-degenerate quadrics are better than those defined on degenerate quadrics. We also show the geometric structure of the minimum weight codewords and estimate the second weight of these codes. For X a non-degenerate hermitian variety, we list the first five weights and the corresponding codewords. The paper ends with two conjectures. One on the minimum distance for the functional codes of order h on X ⊂ P(Fq) a non-singular hermitian variety. The second conjecture on the distribution of the codewords of the first five weights of the functional codes of second order on X ⊂ P(Fq) the non-singular hermitian variety.