de Mathématiques de Luminy
Codes defined by forms of degree 2 on quadric and non-degenerate hermitian varieties in.
We study the functional codes of second order defined by G. Lachaud on
a quadric of rank()=3,4,5 or a non-degenerate hermitian variety. We give some bounds for the number of points of quadratic sections of , 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 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 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 the non-singular hermitian variety.
Keywords: functional codes, hermitian surface, hermitian variety, hermitian curve, quadric, weight.
Mathematics Subject Classification: 05B25, 11T71, 14J29.
Last update : december 4, 2006, EL.