Finding a mass formula for a given class of linear codes is a fundamental problem in combinatorics and coding theory. In this paper, we consider the action of the unitary (resp. symplectic) group on the set of all Hermitian (resp. symplectic) linear complementary dual (LCD) codes, prove that all Hermitian (resp. symplectic) LCD codes are on a unique orbit under this action, and determine the formula for the size of the orbit. Based on this, we develop a general technique to obtain a closed mass formula for linear codes with prescribed Hermitian (resp. symplectic) hull dimension, and further obtain some asymptotic results.

翻译：寻找给定类别线性码的质量公式是组合学与编码理论中的一个基本问题。本文考虑酉群（或辛群）在所有厄米特（或辛）线性互补对偶码集合上的作用，证明了所有厄米特（或辛）线性互补对偶码在该作用下均位于唯一轨道上，并确定了该轨道大小的计算公式。在此基础上，我们发展了一种通用技术来获得具有指定厄米特（或辛）壳维数的线性码的闭合质量公式，并进一步得到若干渐近结果。