Binary Spherical Geometric Codes

Let q be a power of an odd prime number and Fq be the finite field with q elements. We will construct a binary spherical code from an algebraic curve C defined over Fq and a rational divisor G on C, as the twist by the quadratic character 11 of the Goppa code L(G). The computation of the parameters of this code is based on the study of some character sums. 0. Introduction. In a previous paper ([3]), we have constructed some non linear geometric codes on any alphabet having n > 3 elements from a smooth projective irreducible algebraic curve C defined over Fq, a rational divisor G on C and a multiplicative character X of order n 1 of Fq as a twist of the Goppa code L(G). Unfortunately, we were unable to construct a binary code in this way : this is what we will do here. In order to compute the parameters of these codes, it is convenient to define them as spherical codes. As in the case studied in [3], these parameters can be computed from estimations on certain character sums. In the first section, we give the useful definitions and result (proposition t) on character sums ; the proofs can be found in [2]. In the second section, we recall the basic concepts for spherical codes. For more informations on this topic, see [1] for example. In the third section, we defme the geometric spherical code, and give its parameters in the fourth section (theorem 1). This theorem follows from an estimation for the angle between two codewords in term of a character sum (proposition 2). In the fifth section, we deduce from this study the Hamming parameters of this code (theorem 2). Finally, we ask in the sixth section some open questions. in this paper, #X will denote the cardinality of the set X. 1. Multiplicative Character Sums. Let C be a smooth projective irreducible curve of genus g(C) defined over Fq, G a rational divisor on C prime to X(Fq), and rl the quadratic character of the multiptieative group Fq (that is, rl(x) = 1 if x ~ Fq is a square, and ~(x) = 1 if not). If P e C and f a K = K(C) the function field of C, we clef'me the P-adic valuation Vp(f) as follows :