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

我们研究了q^2元的域F上的厄米曲线y^q+y=x^&lt；q^+&gt；上的两点编码，其中q^2是素数的幂。作为这些代码的两个点，我们可以选择关于方程的无穷远的点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。我们的问题是计算二次C(m，n)和寻找C(m，n)的最小距离。第一个结果是考虑0[小于等于]n[小于等于]q的两点编码C(m，n)就足够了。在本研究项目的第一年，我们成功地确定了这个范围内所有(m，n)的C(m，n)的维度，并找到了n=0和q的最小距离。第二年，我们很高兴地找到了所有n的0[小于等于]n[小于等于]q的最小距离C(m，n)。此外，作为第三个结果的推论，我们在前一篇文章中发现了两点编码的Ω构造的例子(与S.J.Kim，带有魏尔斯特拉斯对的Goppa码，纯应用代数162(2001))显示了前一篇文章中所解释的两点编码的最小距离估计的精确性。

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.