Projective Reed - Muller Codes

R~sum~. On inlroduit une classe de codes lintaires de la famille des codes de Reed-Muller, les codes de Reed-Muller projectifs. Ces codes sont des extensions des codes de Reed-Muller gtntralists ; les codes de Reed-Muller projectifs d'ordre I atteignent la borne de Plotkin. On donne les traram&res des codes de Reed-Muller projecfifs d'ordre 2. 1.Introduction We note pm(Fq) the projective space of dimension m over the finite field Fq with q elements, so that # pm(Fq) = qm + qm-l+ ... + 1. We consider the following data : a) a finite dimensional vector space L over Fq ; b) for every point x of a subset V c pm(Fq), an evaluation map q~x : L-~ Fq which is a linear form and associates to every f ~ L an element q~x(f) ~ Fq. With these data, we define the code C L c Fq #v as the image of the map • from L to Fq #V defined by ~(f) = (q~x(f))xeV. The geometric codes are constructed in this way (cf. [8], § 3.1). As a particular case, we can take for L certain finite dimensional spaces of rational functions on a smooth curve X, and define, for a point x of X def'med over Fq which is not a pole of the members of L the