$\ell$-Complementary Subspaces and Codes in Finite Bilinear Spaces

TL;DR

This paper studies the properties and typical behavior of $ ext{ell}$-complementary subspaces and codes in finite bilinear spaces, providing formulas and asymptotic analysis that generalize known results in coding theory.

Contribution

It introduces the concept of $ ext{ell}$-complementary subspaces, generalizes weight distribution formulas, and analyzes the asymptotic behavior of self-orthogonal and related codes.

Findings

Closed formula for average weight distribution of $ ext{ell}$-complementary codes.

Most self-orthogonal codes are MDS over large fields.

Self-orthogonal codes behave similarly to generic codes asymptotically.

Abstract

We consider (symmetric, non-degenerate) bilinear spaces over a finite field and investigate the properties of their $\ell$-complementary subspaces, i.e., the subspaces that intersect their dual in dimension $\ell$. This concept generalizes that of a totally isotropic subspace and, in the context of coding theory, specializes to the notions of self-orthogonal, self-dual and linear-complementary-dual (LCD) codes. In this paper, we focus on the enumerative and asymptotic combinatorics of all these objects, giving formulas for their numbers and describing their typical behavior (rather than the behavior of a single object). For example, we give a closed formula for the average weight distribution of an $\ell$-complementary code in the Hamming metric, generalizing a result by Pless and Sloane on the aggregate weight enumerator of binary self-dual codes. Our results also show that self-orthogonal codes, despite being very sparse in the set of codes of the same dimension over a large field, asymptotically behave quite similarly to a typical, not necessarily self-orthogonal, code. In particular, we prove that most self-orthogonal codes are MDS over a large field by computing the asymptotic proportion of the non-MDS ones for growing field size.