Reed-Muller codes associated to projective algebraic varieties

Abstract

The classical generalized Reed-Muller codes introduced by Kasami, Lin and Peterson [5], and studied also by Delsarte, Goethals and Mac Williams [2], are defined over the affine space Aⁿ(F_q) over the finite field F_q with q elements. Moreover Lachaud [6], following Manin and Vladut [7], has considered projective Reed-Muller codes, i.e. defined over the projective space Pⁿ(F_q).

In this paper, the evaluation of the forms with coefficients in the finite field F_q is made on the points of a projective algebraic variety V over the projective space Pⁿ(F_q). Firstly, we consider the case where V is a quadric hypersurface, singular or not, Parabolic, Hyperbolic or Elliptic. Some results about the number of points in a (possibly degenerate) quadric and in the hyperplane sections are given, and also is given an upper bound of the number of points in the intersection of two quadrics.

In application of these results, we obtain Reed-Muller codes of order 1 associated to quadrics with three weights and we give their parameters, as well as Reed-Muller codes of order 2 with their parameters.

Secondly, we take V as a hypersurface, which is the union of hyperplanes containing a linear variety of codimension 2 (these hypersurfaces reach the Serre bound). If V is of degree h, we give parameters of Reed-Muller codes of order d < h, associated to V.