On Binary Codes That Are Maximal Totally Isotropic Subspaces with Respect to an Alternating Form
Patrick King, Mikhail Kotchetov
TL;DR
This work studies maximal totally isotropic subspaces of $\mathbb{F}_2^n$ under an alternating bilinear form and their connection to binary codes that are equal to their orthogonal complement. It establishes a MacWilliams-type identity tailored to this alternating inner product, enabling the weight enumerator of the orthogonal complement to be determined from the original code’s weight distribution, with special structure when the code is even or Type II. The authors provide a complete classification of odd Lagrangians for $n \le 24$ (up to $S_n$-equivalence) and derive weight-enumerator constraints that sharpen our understanding of how these subspaces can behave. The results illuminate the interplay between symplectic-like geometry over $\mathbb{F}_2$, invariant theory, and binary coding theory, and yield concrete tools for constraining possible weight distributions of maximal totally isotropic codes.
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.