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

研究了q^2元域F上厄米曲线y^q+y=x^<q^+>上的两点码，其中q^2是素数的幂。作为这些码的两个点，我们可以选择关于方程的无穷远点P和原点Q。我们用C（m，n）表示由线性系统L（mP+nQ）产生的码，我们的问题是计算dim C（m，n）和寻找C（m，n）的最小距离。第一个结果是，对于范围0[小于或等于]n[小于或等于]q，考虑两点码C（m，n）就足够了。在这个研究项目的第一年，我们成功地确定了C（m，n）的维数，并找到了n=0和q的最小距离。在第二年，我们成功地找到了所有满足0[小于或等于]n[小于或等于]q的n的最小距离C（m，n）。此外，作为第三个结果的推论，我们找到了具有Ω-结构的两点码的例子（与S. JKim，Goppa codes with Weierstrass pairs，Pure Appl.Algebra 162（2001））示出了在先前论文中解释的两点码的最小距离的估计的清晰度。

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.