A Bound on the number of rational points of certain Artin-Schreier families

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Let F[sub q] = F[sub p]l for some 1 > 0 and consider the extension F[sub q][sup m] with m > 1. We consider families of curves of the form F = {y[sup q] - y = λ[sub 1]x[sup i1]+ λ[sub 2]x[sup i2] + ··· + λ[sub s]x[sup is]; λ[sub j] ∈ F[sub q]m, i[sub j] > 0 }. We call such families Artin-Schreier families, even though not every curve in F need be an Artin-Schreier curve. It is easy to see that the members of such a family can have at most q[sup m+l] affine F[sub q[sup m]]-rational points. Using a well-known coding theory technique, we determine the condition under which F can attain this bound and we obtain some simple, but interesting, corollaries of this result. One of these consequences shows the existence of maximal curves of Artin-Schreier type. Our main result is important for minimum distance analysis of certain two-dimensional cyclic codes