The Average Dimension of the Hermitian Hull of Constayclic Codes over Finite Fields

TL;DR

This paper investigates the average dimension of the Hermitian hull of constacyclic codes over finite fields, providing bounds and analyzing its growth relative to code length.

Contribution

It determines the average dimension of the Hermitian hull of constacyclic codes and compares it with Euclidean hulls, offering new insights into their properties.

Findings

Average dimension is either zero or grows linearly with code length.

Provides upper and lower bounds for the average dimension.

Discusses comparison with Euclidean hulls of cyclic codes.

Abstract

The hulls of linear and cyclic codes have been extensively studied due to their wide applications. In this paper, the average dimension of the Hermitian hull of constacyclic codes of length $n$ over a finite field $\mathbb{F}_{q^2}$ is determined together with some upper and lower bounds. It turns out that either the average dimension of the Hermitian hull of constacyclic codes of length $n$ over $\mathbb{F}_{q^2}$ is zero or it grows the same rate as $n$. Comparison to the average dimension of the Euclidean hull of cyclic codes is discussed as well.