The Average Hull Dimension of Negacyclic Codes over Finite Fields

: Hulls of linear codes have been extensively studied due to their wide applications and links with the efﬁciency of some algorithms in coding theory. In this paper, the average dimension of the Euclidean hull of negacyclic codes of length n over ﬁnite ﬁelds F q , denoted by E ( n , − 1, q ) , has been investigated. The formula for E ( n , − 1, q ) has been determined. Some upper and lower bounds of E ( n , − 1, q ) have been given as well. Asymptotically, it has been shown that either E ( n , − 1, q ) is zero or it grows the same rate as n .


Introduction
In practice, communication systems are not 100% reliable due to noise or other forms of interference. Coding theory is a branch of Engineering Mathematics that has been introduced and applied to solve this problem since the 1960s. Codes have later been extensively studied and linked with other problems and applications.
In 1990, the (Euclidean) hull of a linear code has been introduced to classify finite projective planes in [1]. It is defined to be the intersection of a linear code and its Euclidean dual. Hulls of linear codes have various applications and play an important role in the efficiency determination of some algorithms in coding theory such as computing permutation equivalence of two linear codes and finding the automorphism group of linear codes (see, for example, [2][3][4][5][6]). Precisely, the efficiency of these computations is limited by the hull size of codes. In [7], the hulls of linear codes have been applied in constructing good entanglement-assisted quantum error correcting codes.
Properties of hulls of codes have been extensively studied. The average dimensions of the Euclidean hull of linear codes and of cyclic codes were given in [8,9], respectively. 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. In [13], it has been shown that the Euclidean dual of λ-constacyclic code is again λ-constacyclic if and only if λ = ±1. Therefore, the average dimension of the Euclidean hull of negacyclic codes (λ = −1) is the only remaining case.
In this paper, we focus on the average dimension of the Euclidean hull of negacyclic codes of length n over finite fields F q as well as its lower and upper bounds. The paper is organized as follows. Basic properties of codes and polynomials over finite fields are recalled in Section 2. In Section 3, the expression for E(n, −1, q), the formula for the average dimension of neagcyclic codes, is given together with some upper bounds. In Section 4, upper and lower bounds on E(n, −1, q) are derived. The summary and remarks are given in Section 5.

Preliminaries
Let p be a prime and let q be a p-power. Denote by F q the finite field of order q and characteristic p. For given positive integers k ≤ n, a linear code of length n and dimension k over F q is a k-dimensional subspace of the F q -vector space F n q . The Euclidean dual of a linear code C is defined to be The Euclidean hull of a linear code C is defined to be A linear code of length n over F q is said to be negacyclic if (−c n−1 , c 0 , . . . , c n−2 ) ∈ C for all (c 0 , c 1 , . . . , c n−1 ) ∈ C.
Let C(n, −1, q) denote the set of all neagcyclic codes of length n over F q . The average dimension of the hull of negacyclic codes of length n over F q is defined to be Every non-zero negacyclic code C of length n over F q can be viewed as an ideal of the principal ideal ring F q [x]/ x n + 1 generated by a monic divisor g(x) of x n + 1 (see [10]). In this case, g(x) is called the generator polynomial for C and dim C = n − deg(g(x)).
For a polynomial f (x) = a 0 + a 1 x + · · · + a k x k ∈ F q [x] of degree k and a 0 = 0, the reciprocal . Otherwise, f (x) and f * (x) are called a reciprocal polynomial pair.
Let C be a negacyclic code of length n over F q with the generator polynomial g(x) and let h(x) = x n +1 g(x) . Then, h * (x) is a monic divisor of x n + 1 and it is the generator polynomial of C ⊥ by Lemma 2.1 of [13]. Therefore, Hull(C) is generated by the polynomial lcm(g(x), h * (x)) (see Theorem 1 of [10]).
Recall that the characteristic of F q is p. Then, a positive integer n can be written in the form of n = np ν , where p n and ν ≥ 0. Using arguments similar to those in Section 4 of [10], up to permutation, there exist nonnegative integers s and t such that where f j (x) and f * j (x) are a reciprocal polynomial pair and g i (x) is a monic irreducible self-reciprocal polynomial for all 1 ≤ i ≤ s and 1 ≤ j ≤ t.
For a given negacyclic code C of length n over F q , based on the factorization in (1), the generator polynomial of C can be viewed of the form and hence the generator polynomial of Hull(C) is Since 1 and −1 are identical when the characteristic of F q is even, in the rest of this paper, we assume that the characteristic p of F q is odd.

The Average Dimension
In this section, we focus on an explicit expression for the formula of the average dimension of the Euclidean hull of negacyclic codes of length n over F q . Employing techniques similar to those for the cyclic case in [9], slightly different results for the negacyclic case can be deduced.
Assume that x n + 1 has the factorization in the form of Equation (1) and let B n,−1,q = ∑ s i=1 deg g i (x). The expectation E( · ) in Lemma 1 can be obtained using arguments similar to those in the proof of Proposition 22 of [9] . Lemma 1. Let p be an odd prime and let ν be a nonnegative integer. Let 0 ≤ u, z, w ≤ p ν . Then, the following statements hold: The average dimension of the Euclidean hull of neagcyclic codes of length n over F q can be determined as follows. Theorem 1. Let F q be a finite field of order q and odd characteristic p and let n be a positive integer such that n = np ν , p n and ν ≥ 0. Then, the average dimension of the Euclidean hull of negacyclic codes of length n over F q is Proof. By Lemma 1, Equation (2), and arguments similar to those in the proof of Theorem 3.2 of [11], it can be deduced that This completes the proof.
The next corollary is a direct consequence of Theorem 1.

Corollary 1.
Assume the notations as in Theorem 1. Then, the following statements hold:

Properties of B n,−1,q and Bounds on E(n, −1, q)
In this section, some number theoretical tools are constructed and applied to study properties of B n,−1,q . As a consequence, lower and upper bounds for E(n, −1, q) can be derived using B n,−1,q .
For an odd prime power q, let N q := ≥ 1 : divides q i + 1 . For coprime positive integers i and j, denote by ord j (i) the multiplicative order of i modulo j. An element in N q has the following properties.
Lemma 2. Let q be an odd prime power. If ∈ N q and > 2, then ord (q) is even.
Next, we introduce a partition for the set N q . For each integer α ≥ 0, let where 2 α ||k is used if α is the integer such that 2 α |k and 2 α+1 k. Then, we have N q = P q,0 ∪ P q,1 ∪ P q,2 · · · .
Theorem 2 (Theorem 4 of [9]). Let q be an odd prime power and let be a positive integer. Then, the following statements hold: 1. Let be an odd integer. If > 1 is such that = ∏ k i=1 p e i i the prime factorization of . Then, ∈ N q if and only if there exists α > 0 such that p i ∈ P q,α for all i. In this case, we have ∈ P q,α . 2. Let β ≥ 1 be an integer. Then 2 β ∈ N q if and only if 2 β divides q + 1. Moreover, if 2 β ∈ N q , β ≥ 2, then 2 β ∈ P q,1 . 3. Let q and be odd. Then, 2 ∈ N q if and only if ∈ N q . In this case, and 2 belong to the same set P q,α . 4. Let = 2 β where is odd and β ≥ 2. Then, ∈ N q if and only if 2 β ∈ N q and ∈ P q,1 . In this case, we have ∈ P q,1 .
The characterization of elements in P q,α are given in the following corollary.

Proof. By Equation
From Equation (29) of [10], x n + 1 can be factored as are a monic irreducible-reciprocal polynomial pair of degree ord j (q), and g ij (x) is a monic irreducible self-reciprocal polynomial of degree ord j (q).
Altogether, it can be concluded that Hence, as desired.

Remark 1.
From Lemma 4, we have the following facts. The set Ω ∩ N q can be empty. For convenience, the empty summation will be regarded as 0. In this case, B n,−1,q = ∑ j∈Ω∩N q φ(j) = ∑ j∈∅ φ(j) = 0. For example, The expression of the set Ω can be simplified using the definition of n as follows.
Lemma 5. Write n = 2 β n , where n is an odd integer and β is a non-negative integer. Then, Ω = 2 β+1 k : k ∈ N and k|n .
The following result is a consequence of Lemma 5 and Theorem 2.

Statement (iii) follows immediately from (i) and (ii).
By Proposition 1, we have the following corollary.

Statement (iii) can be deduced directly from (i) and (ii).
Corollary 4. Assume the notations as above. Then, the following statements hold: Proof. The first statement can be deduced directly from Theorem 1 and Corollary 3. The second statement follows from Corollary 1 and the fact that 1 6(p ν +1) reaches its maximum value 1 12 when ν = 0.
Next, we focus on the case where β + 1 ≤ γ. Let be a positive integer relatively prime to q. Let = 2 β p e 1 1 . . . p e k k be the prime factorization of , where β ≥ 0, k ≥ 0, p 1 , p 2 , . . . , p k are distinct odd primes, and e i ≥ 1 for all i = 1, 2, . . . , k. Partition the index set {1, . . . , k} into K , K 1 , K 2 , . . . as follows: Let d = ∏ i∈K p e i i and d α = ∏ i∈K α p e i i for all 1 ≤ α ≤ k. For convenience, the empty product will be referred to as 1. Therefore, we have = 2 β d d 1 d 2 . . . which is called the N q -factorization of , where d i = 1 for all but finitely many integers i. By Theorem 2, we have d α ∈ P q,α . The characterization of ∈ N q is given in the following lemma.
The following proposition provides a simplified expression of B n,−1,q . Proposition 2. Let n = 2 β d d 1 d 2 . . . be an N q -factorization of n = 2 β n . If β + 1 ≤ γ and 2n ∈ N q , then Proof. We distinguish the proof into two cases where β = 0 and β = 0.
Since k|n if and only if k|d 1 , it follows that The results follow.