Title: On Binary Codes That Are Maximal Totally Isotropic Subspaces with Respect to an Alternating Form Show Chinese title

Title: 关于相对于交替型的最大完全迷向子空间的二元码

Abstract: Self-dual binary linear codes have been extensively studied and classified for length n <= 40. However, little attention has been paid to linear codes that coincide with their orthogonal complement when the underlying inner product is not the dot product. In this paper, we introduce an alternating form defined on F_2^n and study codes that are maximal totally isotropic with repsect to this form. We classify such codes for n <= 24 and present a MacWilliams-type identity which relates the weight enumerator of a linear code and that of its orthogonal complement with respect to our alternating inner product. As an application, we derive constraints on the weight enumerators of maximal totally isotropic codes. Show Chinese abstract

Abstract: 自对偶二元线性码在长度 n <= 40 的情况下已被广泛研究和分类。 然而，对于当基础内积不是点积时与正交补空间重合的线性码，关注较少。 在本文中，我们引入了一个定义在 F_2^n 上的交替形式，并研究了相对于该形式为极大完全迷向的码。 我们对 n <= 24 的此类码进行了分类，并提出一个类似于 MacWilliams 的恒等式，该恒等式将线性码的重量枚举多项式与其正交补的重量枚举多项式通过我们的交替内积联系起来。 作为应用，我们推导出极大完全迷向码的重量枚举多项式的约束条件。