The average dimension of the Hermitian hull of cyclic 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.

Good integers and some applications in coding theory

“…The above results imply that avg H p 2ν (A × Z p k ) is zero or grows as the same rate with mp k . Note that if A is a cyclic group, these results coincide with [11,Theorem 4.9].…”

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

“…From Theorem 5.1 and Table 1, the average dimension of Hermitian hull of cyclic codes of length n over F q 2 is either E H (n, 1, q 2 ) = 0 if n ∈ M q or n 8 ≤ E H (n, 1, q 2 ) < n 3 if n / ∈ M q . The results coincide with [6,Theorem 4.9]…”

“…In [9], the number of cyclic codes of length n over F q having hull of a fixed dimension has been determined together with the dimensions of the hulls of cyclic codes of length n over F q . The average dimension of the hull of cyclic codes with respect to the Euclidean and Hermitian inner products have been investigated in [15] and [6], respectively. It has been shown that either the average dimension of the hull of such codes is zero or it grows at the same rate with n. In [10], the dimensions of the Hermitian hulls of constacyclic codes of length n over F q 2 have been determined.…”

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].…”