Error correcting codes from the viewpoints of algebraic curves and finite geometry

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We studied two-point codes on the Hermitian curve y^q+y=x^<q^+> over the field F of q^2 elements, where q^2 is a power of a prime number. As the two points of those codes, we may choose the point at infinity P and the origin Q with respect to the equation. We denote by C(m, n) the code arising from the linear system L(mP+nQ).Our problems were to compute dim C(m, n) and to find the minimum distance of C(m, n). First result is that it is enough to consider the two-point codes C(m, n) for the range 0【less than or equal】n【less than or equal】q. In the first year of this research project, we succeeded in determining the dimension of C(m, n) for all (m, n) in this range and finding the minimum distance for n=0 and q. In the second year, we happily succeeded in finding the minimum distance C(m, n) for all n with 0【less than or equal】n【less than or equal】q.Moreover, as a corollary of the third result, we found the example of two-point code with Ω-construction in our previous paper (with S.J Kim, Goppa codes with Weierstrass pairs, Pure Appl.Algebra 162(2001)) showed the sharpness of the estimation of the minimum distance of a two-point code that explained in the previous paper.