Artin-Schreier curves and weights of two dimensional cyclic codes

Let GF(q) be the finite field with q elements of characteristic p, GF(q^m) be the extension of degree m>1 and f(x) be a polynomial over GF(q^m). We determine a necessary and sufficient condition for y^q-y=f(x) to have the maximum number of affine GF(qm)-rational points. Then we study the weights of 2-D cyclic codes. For this, we give a trace representation of the codes starting with the zeros of the dual 2-D cyclic code. This leads to a relation between the weights of codewords and a family of Artin-Schreier curves.We give a lower bound on the minimum distance for a large class of 2-D cyclic codes. Then we look at some special classes that are not covered by our main result and obtain similar minimum distance bounds

Artin-Schreier families and 2-D cycle codes

We start with the study of certain Artin-Schreier families. Using coding theory techniques, we determine a necessary and sufficient condition for such families to have a nontrivial curve with the maximum possible number of rational points over the finite field in consideration. This result produces several nice corollaries, including the existence of certain maximal curves; i.e., curves meeting the Hasse-Weil bound.We then present a way to represent two-dimensional (2-D) cyclic codes as trace codes starting from a basic zero set of its dual code. This representation enables us to relate the weight of a codeword to the number of rational points on certain Artin-Schreier curves via the additive form of Hilbert’s Theorem 90. We use our results on Artin-Schreier families to give a minimum distance bound for a large class of 2-D cyclic codes. Then, we look at some specific classes of 2-D cyclic codes that are not covered by our general result. In one case, we obtain the complete weight enumerator and show that these types of codes have two nonzero weights. In the other cases, we again give minimum distance bounds. We present examples, in some of which our estimates are fairly effcient

Curves of every genus with many points, I: Abelian and toric families

Let N_q(g) denote the maximal number of F_q-rational points on any curve of genus g over the finite field F_q. Ihara (for square q) and Serre (for general q) proved that limsup_{g-->infinity} N_q(g)/g > 0 for any fixed q. In their proofs they constructed curves with many points in infinitely many genera; however, their sequences of genera are somewhat sparse. In this paper, we prove that lim_{g-->infinity} N_q(g) = infinity. More precisely, we use abelian covers of P^1 to prove that liminf_{g-->infinity} N_q(g)/(g/log g) > 0, and we use curves on toric surfaces to prove that liminf_{g-->infty} N_q(g)/g^{1/3} > 0; we also show that these results are the best possible that can be proved with these families of curves.Comment: LaTeX, 20 page

Weight distribution of cyclic codes defined by quadratic forms and related curves

We consider cyclic codes CL associated to quadratic trace forms inm variables (Formula Presented) determined by a family L of q-linearized polynomials R over Fqm, and three related codes CL,0, CL,1, and CL,2. We describe the spectra for all these codes when L is an even rank family, in terms of the distribution of ranks of the forms QR in the family L, and we also computethe complete weight enumerator for CL. In particular, considering the family L = ‹xql›, with l fixed in N, we give the weight distribution of four parametrized families of cyclic codes Cl, Cl,0,Cl,1, and Cl,2 over Fq with zeros(Formula Presented) respectively,where q = ps with p prime, α is a generator of F*qm, and m/(m,l)is even. Finally, we give simple necessary and sufficient conditions for Artin–Schreier curves yp−y = xR(x)+βx, p prime, associated to polynomials R ∈ L to be optimal. We then obtain several maximal and minimal such curves inthe case (Formula Presented).Fil: Podesta, Ricardo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; ArgentinaFil: Videla Guzman, Denis Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin