Polynomial-Time Construction of Spherical Codes

We give a simple lower bound for the dimensions of the families of polynomially constructible spherical codes of given minimal angle ~o, deduced from the analog of the Katsman-Tsfasman-Vl~du!i bound for linear codes. In particulax the supremum rpol of the numbers log 2 CardX/dim X, where X ranges over all polynomially constructible families of spherical codes with ~ _> Ir/3, is such th~.t rpot > 2/15. 1 s p h e r i c a l c o d e s We note x.y the usual scalar product of two vectors x and y in the euclidean space R n, thus II x 112= x.x is the square of the length of x. A spherical code is a set X of points lying on the unit sphere s "-1 = {~ ~ l~" [ II • II == 1}. We refer to [2] and to the references therein for the results on spherical codes. If x and y are in S n 1 we set ~(. , y) = arccos..y, ( 0 < v _ < ~ ) , p(. ,v) = I t * y tt=; thus the parameters p and T are related by v~ p : 4 sin 2 ~ , ~ : 2 aresin -~ I f X C S n 1 is a spherical code, the minimal angle of X is ~o(X) = Min{io(x , y) t z , y e X , a :/: y}, *]~quipe Arlthra~tique ¢t Th&arle de l'hfformatlon, C.I.R.M., Lumlny Case 916, 132SS Marseille cedex 9, France lI~quipe de Logique, Universlt~ Paris 7 & D.M.I., Ecole Normale Supdrieure, 45 rue d'Ulm, 75230 Paris cedex 05