Nested code pairs play a crucial role in the construction of ramp secret sharing schemes [15] and in the CSS construction of quantum codes [14]. The important parameters are (1) the codimension, (2) the relative minimum distance of the codes, and (3) the relative minimum distance of the dual set of codes. Given values for two of them, one aims at finding a set of nested codes having parameters with these values and with the remaining parameter being as large as possible. In this work we study nested codes from the Hermitian curve. For not too small codimension, we present improved constructions and provide closed formula estimates on their performance. For small codimension we show how to choose pairs of one-point algebraic geometric codes in such a way that one of the relative minimum distances is larger than the corresponding non-relative minimum distance.

嵌套代码对在构建斜坡秘密共享方案中[15]和在量子代码的CSS中起关键作用[14]。重要的参数是（1）余维，（2）代码的相对最小距离，以及（3）对偶代码集的相对最小距离。给定其中两个的值，一个目标是找到一组嵌套代码，这些嵌套代码的参数与这些值相同，而其余参数则尽可能大。在这项工作中，我们从Hermitian曲线研究嵌套代码。对于较小的余量，我们提出了改进的结构并提供了封闭式的性能估算。