A study of algebraic curves from viewpoints of the coding theory and the finite geometry

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We found the following facts. Let Fq be the finite field of q-elements, and F(X, Y, Z) a homogeneous polynomial of degree d over Fq,. We consider the plane curve C defined by F(X, Y, Z)=0.Suppose that C has no Fq-linear component. Let Nq(C) denote the number of the set of Fq. points of C.Then we have Theorem. Unless C is a curve defined over F4 which is projectively equivalent to(X+Y+Z) 4+(XY+YZ+ZX) 2+XYZ(X+Y+Z), the inequality Nq(C).(d-1) q+1 holds true.Moreover, there exists a nonsingular curve of degree d which attains the bound above if and only if d is one of the following numbers{q+2, q+1, q, q-1,√<q+1>(if q is a square), 2}.