The average dimension of the Hermitian hull of constacyclic codes over finite fields of square order

Abstract

The hulls of linear and cyclic codes have been extensively studied due to their wide applications. The dimensions and average dimension of the Euclidean hull of linear and cyclic codes have been well-studied. In this paper, the average dimension of the Hermitian hull of constacyclic codes of length n over a finite field 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 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.

The Average Hull Dimension of Negacyclic Codes over Finite Fields

“…The average dimension of the Euclidean hull of cyclic codes has been studied in [9]. A complete study of the average dimension of the Hermitian hull of cyclic and constacyclic codes was given in [11,12]. Therefore, the remaining case is the Euclidean hull of negacyclic codes (see [13]).…”

“…This paper provides a complete study for this problem. The detailed comparison for the results on the Euclidean case is given in Table 1 and the Hermitian case is given in [12]. Author Contributions: E.S.…”

“…The dimensions of the hulls of cyclic codes and negacyclic codes were determined in [10]. Later, the complete study of the average dimension of the Hermitian hull of cyclic and constacyclic codes was given in [11,12]. It is of natural interest to study the average dimension of the Euclidean hull of constacyclic codes.…”

Hulls of cyclic codes over Z4

“…Later, the hull dimensions of cyclic and negacyclic codes and the number of cyclic codes whose hulls share a given dimension have been established in [17]. The average hull dimension of constacyclic codes over finite fields have been given in [8,9,10].…”

“…3. Then E(n) < 5n 9 . The average 2-dimension E(n) of the hull of cyclic codes of odd length up to 53 over Z 4 are given in Table 2.…”