WEIGHT DISTRIBUTION OF CYCLIC CODES DEFINED BY QUADRATIC FORMS AND RELATED CURVES

We consider cyclic codes C-L associated to quadratic trace forms in m variables Q(R)(x) = Tr(q)m (/ q) (xR(x)) determined by a family L of q-linearized polynomials R over F(q)m, and three related codes C-L,(0,) C-L,C-1, and C-L,C-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 Q(R) in the family L, and we also compute the complete weight enumerator for C-L. In particular, considering the family L = <(x)q(l)>, with l fixed in N, we give the weight distribution of four parametrized families of cyclic codes C-l, C-l,C-0 C-l,C-1 and C-l,C-2 over F-q with zeros { alpha(-( ql+1)) }, {1, alpha(-(ql +1))}, {alpha(-1), alpha(-(ql+1)) }, and {1, alpha(-1), alpha(-(ql+1))} respec- tively, where q = p(s) with p prime, alpha is a generator of F-qm*, and m/(m,l) is even. Finally, we give simple necessary and sufficient conditions for ArtinSchreier curves y(P) - y = xR(x) + beta x, p prime, associated to polynomials R is an element of L to be optimal. We then obtain several maximal and minimal such curves in the case L = <(pl)(x)> and L = <(pl )(x), (p3l)(x)>.