Linear Codes with Certain Dimension of Hermitian Hulls

TL;DR

This paper investigates the properties and enumeration of Hermitian $ ext{ell}$-complementary codes over finite fields, revealing asymptotic behaviors and density results for Hermitian self-orthogonal codes, especially MDS codes, as the alphabet size grows.

Contribution

It provides closed-form formulas for counting Hermitian $ ext{ell}$-complementary codes and analyzes their asymptotic weight distribution and density of MDS codes.

Findings

Hermitian $ ext{ell}$-complementary codes have explicit enumeration formulas.

Asymptotic weight distribution of Hermitian self-orthogonal codes resembles that of unrestricted codes.

MDS codes in the Hermitian self-orthogonal class are asymptotically dense as alphabet size increases.

Abstract

In this paper, we study the enumerative and asymptotic properties related to Hermitian $\ell$-complementary codes on the unitary space over $\F_{q^2}$. We provide some closed form expressions for the counting formulas of Hermitian $\ell$-complementary codes. There is a similarity in the asymptotic weight distribution between Hermitian self-orthogonal codes and unrestricted codes. Furthermore, we study the asymptotic behavior of Hermitian self-orthogonal codes whose minimum distance is at least $d$. In particular, we conclude that MDS codes within the class of Hermitian self-orthogonal codes are asymptotically dense when the alphabet size approaches to infinity.