D Boucher

E K Nouetowa

Some constructions and existence conditions for Hermitian self-dual skew codes

Keywords: skew polynomial ring, coding theory, duality Acknowledgments

In this text, we first consider the existence conditions and the construction of Hermitian self-dual θ-cyclic and θ-negacyclic codes over IF p 2 , where p is a prime number and θ is the Frobenius automorphism over IF p 2 . We then give necessary and sufficient conditions for the existence of Hermitian self-dual θ-cyclic and θ-negacyclic codes over IFpe where e is an even integer greater than 2.

Introduction

For θ an automorphism of a finite field IF q , θ-cyclic codes (also called skew cyclic codes) of length n were defined in [START_REF] Boucher | Skew-cyclic codes[END_REF]. These codes are such that a right circular shift of each codeword gives another word which belongs to the code after application of θ to each of its n coordinates. If θ is the identity, θ-cyclic codes are cyclic codes.

Skew cyclic codes have an interpretation in the Ore ring R = IF q [X; θ] of skew polynomials where multiplication is defined by the rule X • a = θ(a)X for a in IF q . Euclidean self-dual skew cyclic codes of length n over IF q have been considered in many previous works among which [START_REF] Caruso | Selfdual skew cyclic codes[END_REF][START_REF] Irwansyah | A note on the construction and enumeration of Euclidean selfdual skew-cyclic codes[END_REF]. Their existence conditions were first established in [START_REF] Boucher | A note on the existence of self-dual skew codes over finite fields[END_REF], then their construction and enumeration were obtained over IF p 2 when p is a prime number ( [START_REF] Boucher | Construction and number of self-dual skew codes over IF p 2[END_REF]) and over IF p n when p is a prime number and θ is the Frobenius automorphism ( [START_REF] Batoul | A construction of self-dual skew cyclic and negacyclic codes of length n over IF p n[END_REF]). In this text we are concerned with Hermitian self-dual skew cyclic and skew negacyclic codes whose study was briefly initiated in [START_REF] Boucher | A first step towards the skew duadic codes[END_REF]. We first consider Hermitian self-dual θ-cyclic and θ-negacyclic codes over IF p 2 where p is a prime number and θ is the Frobenius automorphism over IF p 2 . Then we give necessary and sufficient existence conditions for Hermitian self-dual skew cyclic and skew negacyclic codes over IF q .

The text is organized as follows. In Section 2, we first give generalities on θ-cyclic and θ-negacyclic codes. In Section 3, we consider Hermitian self-dual θ-cyclic and θnegacyclic codes over IF p 2 when θ is the Frobenius automorphism. In subsection 3.1, we first characterize these codes by a system of homogeneous polynomial equations of degree p + 1 (Algorithm 1). Then, in subsection 3.2, by using factorization properties of skew polynomials, we prove that there exists no Hermitian self-dual θ-cyclic code of any dimension over IF p 2 (Theorem 2). We also provide a construction and an exact formula for the number of Hermitian self-dual θ-negacyclic codes (Algorithm 5 and Theorem 1). We give many examples of Hermitian self-dual codes over IF 4 and IF 9 , including a [68,34,[START_REF] Yang | On self-dual constacyclic codes over finite fields[END_REF] Hermitian self-dual code over IF 4 , which improves the best known Hermitian self-dual code of length 68 over IF 4 (according to [START_REF] Gaborit | Table of Hermitian self-dual codes over IF 4[END_REF]). All the computations were made with the Magma algebra system ( [START_REF] Bosma | The Magma algebra system. I. The user language[END_REF]). In Section 4, we prove that there is no Hermitian self-dual θ-cyclic code over any finite field IF p e with e even and p odd (Theorem 2). Then we give necessary and sufficient existence conditions for Hermitian self-dual θ-negacyclic codes defined over IF p e with e > 2 (Theorem 3).

Generalities on self-dual skew constacyclic codes

For a finite field IF q and θ an automorphism of IF q one considers the ring R = IF q [X; θ] where addition is defined to be the usual addition of polynomials and where multiplication is defined by the rule: for a in IF q X • a = θ(a) X.

(

) 1
The ring R is called a skew polynomial ring or Ore ring (cf. [START_REF] Ore | Theory of non-commutative polynomials[END_REF]) and its elements are skew polynomials. When θ is not the identity, the ring R is not commutative, it is a left and right Euclidean ring whose left and right ideals are principal. Left and right gcd (gcld, gcrd) and lcm (lclm, lcrm) exist in R and can be computed using the left and right Euclidean algorithms. The center of R is the commutative polynomial ring Z(R) = IF θ q [X ] where IF θ q is the fixed field of θ and is the order of θ. Definition 1 ([6, Definition 1]). Consider a non-zero element a of IF q and two integers n, k such that 0 ≤ k ≤ n. A (θ, a)-constacyclic code or skew constacyclic code C of length n is a left R-submodule Rg/R(X n -a) ⊂ R/R(X n -a) in the basis 1, X, . . . , X n-1 where g is a monic skew polynomial dividing X n -a on the right in R with degree n -k. If a = 1, the code is θ-cyclic and if a = -1, it is θ-negacyclic. The skew polynomial g is called skew generator polynomial of C.

If θ is the identity then θ-cyclic and θ-negacylic codes are respectively cyclic and negacyclic codes.

Definition 2 ([6, Definition 2]). Consider an integer d and h =

d i=0 h i X i in R of degree d. The skew reciprocal polynomial of h is h * = d i=0 X d-i • h i = d i=0 θ i (h d-i ) X i .
If m is the degree of the trailing term of h, the left monic skew reciprocal polynomial of h is h :=

1 θ d-m (hm) • h * .
The Euclidean dual of a linear code C of length n over IF q is defined as

C ⊥ = {x ∈ IF n q | ∀y ∈ C, x, y = 0} where for x, y in IF n q , x, y := n i=1 x i y i is the (Euclidean) scalar product of x and y. The code C is Euclidean self-dual if C is equal to C ⊥ .
Assume that q is an even power of an arbitrary prime and denote σ the automorphism of IF q defined by σ(a) = a √ q for a in IF q . The Hermitian dual of a linear code C of length n over IF q is defined as

C ⊥ H = {x ∈ IF n q | ∀y ∈ C, x, y H = 0} where for x, y in IF n q , x, y H := n i=1 x i σ(y i ) is the (Hermitian) scalar product of x and y. The code C is Hermitian self-dual if C is equal to C ⊥ H .
In what follows we will only consider (θ, ε)-constacyclic codes with ε 2 = 1 and with length n divisible by the order of θ. That implies that the skew polynomial X n -ε is a central polynomial. Following [START_REF] Boucher | Autour de codes définis à l'aide de polynômes tordus[END_REF]Proposition 2], the Hermitian dual of a (θ, ε)constacyclic code C of length n and skew generator polynomial g is a (θ, ε)-constacyclic code of skew generator polynomial σ(h ) where h is defined by h•g = g•h = X n -ε and where the automorphism σ is extended to R by a i X i → σ(a i )X i . In particular the code C is Hermitian self-dual if, and only if,

σ(h ) • h = X n -ε. (2)
The equation ( 2) is called Hermitian self-dual skew equation.

For F = F (X ) in IF θ q [X ], we define

H F := {h ∈ R | h is monic and σ(h ) • h = F (X )}.
3 Hermitian self-dual θ-cyclic and θ-negacyclic codes over IF p 2

We consider here the existence and the construction of Hermitian self-dual θ-cyclic and θ-negacyclic codes of length n = 2k defined over IF p 2 where θ is the Frobenius automorphism over IF p 2 . As q = p 2 , the automorphisms θ and σ are both equal to the Frobenius automorphism. The fixed field IF θ q of IF q is therefore IF p and the order of θ is = 2.

A first result was obtained in [START_REF] Boucher | A first step towards the skew duadic codes[END_REF] for θ-cyclic codes with dimension coprime with p. Namely, there exists a Hermitian self-dual θ-cyclic code of dimension coprime to p over IF p 2 if, and only if, p = 2. Furthermore, the number of Hermitian self-dual θ-cyclic codes with odd dimension over IF 4 was computed in [9, Theorem 3.7].

Polynomial system strategy

Hermitian self-dual θ-cyclic (resp. θ-negacyclic) codes of dimension k are completely determined by the set H X 2k -ε where = 1 (resp. = -1). In order to compute H X 2k -ε , a first strategy consists of solving the polynomial system satisfied by the coefficients of the solutions h of the equation θ(h ) • h = X n -(the automorphism θ being naturally extended to R by :

a i X i → θ(a i )X i ). Namely consider h = k i=0 h i X i in R with h 0 = 0, then θ(h ) • h = 1 θ k+1 (h 0 ) 2k =0   min(k, ) i=max(0, -k) θ i (θ(h k-i )h -i )   X .
Therefore we get that θ(h

) • h = X 2k -if, and only if,                          θ k+1 (h 0 ) = -h 0 = 0, ∀ ∈ {1, . . . , k -1}, i=0 θ i (θ(h k-i )h -i ) = 0, k i=0 θ i (θ(h k-i )h k-i ) = 0, ∀ ∈ {k + 1, . . . , 2k -1}, k i= -k θ i (θ(h k-i )h -i ) = 0.
The symmetries of the system enable to get rid of the last k -1 equations and we get :

θ(h )•h = X 2k -⇔                θ k+1 (h 0 ) = -h 0 = 0, ∀ ∈ {1, . . . , k -1}, 0≤i≤ i mod 2=0 h p k-i h -i + 0≤i≤ i mod 2=1 h k-i h p -i = 0, k i=0 h p+1 i = 0.
Remark 1. The solutions of the above system belong to the non-degenerate Hermitian variety defined by h p+1 0

+ • • • + h p+1 k-1 + h p+1 k = 0.
Therefore according to Theorem 8.1 of [START_REF] Bose | Hermitian varieties in a finite projective space PG(N, q 2 )[END_REF], there are at most

(p k+1 -(-1) k+1 )(p k -(-1) k ) p 2 -1
solutions. Furthermore, according to Theorem 6.3 of [START_REF] Datta | Number of solutions of systems of homogeneous polynomial equations over finite fields[END_REF], the number of solutions of this homogeneous polynomial system of k linearly independent equations into k + 1 variables with degree p + 1 is at most

p 2k-1 + p 2k-2 -1 p 2 -1 .
The aim of this section is to give an exact formula for the number of solutions of this system by using an approach based on the factorization of skew polynomials (Theorem 1). and assume that h k = 1. Then we get

θ(h )•h = X 2k -⇔            ∀ ∈ {N + 1, . . . , k -1}, 0≤i≤ i mod 2=0 h p k-i h -i + 0≤i≤ i mod 2=1 h k-i h p -i = 0, k i=0 h p+1 i = 0, where      θ k+1 (h 0 ) = -h 0 = 0, ∀ ∈ {1, . . . , N }, h = - 1≤i≤ i mod 2=0 h p k-i × h -i - 1≤i≤ i mod 2=1 h k-i × h p -i .
Therefore we get a polynomial system of N equations with k -N unknowns which we solve in Algorithm 1 by using an exhaustive search. Example 1. We consider the Hermitian self-dual θ-cyclic codes of dimension k = 3 over IF 4 = IF 2 (a) where a 2 +a+1 = 0. Their skew check polynomials h = X 3 +h 2 X 2 + h 1 X + h 0 satisfy the polynomial system :

h 2 + h 2 h 2 1 + h 2 1 h 0 = 0, h 3 0 + h 3 1 + h 3 2 + 1 = 0, where h 0 ∈ IF * 4 , h 1 + h 2 h 2 0 = 0.
We get 9 solutions :(a, 0, 0), (a 2 , 0, 0), (1, 0, 0) which give three Hermitian self-dual [START_REF] Boucher | Self-dual skew codes and factorization of skew polynomials[END_REF][START_REF] Bose | Hermitian varieties in a finite projective space PG(N, q 2 )[END_REF][START_REF] Blake | Explicit factorization of x 2 k + 1 over IF p with prime p ≡ 3 mod 4[END_REF] codes and (1, a, a), (1, a 2 , a 2 ), (a, 1, a 2 ), (a, a 2 , a 2 ), (a 2 , 1, a), (a 2 , a, a), which give 6 Hermitian self-dual [START_REF] Boucher | Self-dual skew codes and factorization of skew polynomials[END_REF][START_REF] Bose | Hermitian varieties in a finite projective space PG(N, q 2 )[END_REF][START_REF] Bosma | The Magma algebra system. I. The user language[END_REF] codes. Example 2. We consider the Hermitian self-dual θ-negacyclic codes of dimension k = 2, 3, 4 over IF 9 = IF 3 (a) where a 2 + 2a + 2 = 0.

• k = 2
The skew check polynomials h = X 2 + h 1 X + h 0 of the Hermitian self-dual codes of dimension 2 satisfy the polynomial system

   h 3 0 -h 0 = 0, h 4 0 + h 4 1 + 1 = 0, h 1 + h 3 1 h 0 = 0.
The set of solutions is {(-1, -1), (-1, 1), (-1, a 6 ), (-1, a 2 )}. The corresponding codes are [START_REF] Bosma | The Magma algebra system. I. The user language[END_REF][START_REF] Blake | Explicit factorization of x 2 k + 1 over IF p with prime p ≡ 3 mod 4[END_REF][START_REF] Bose | Hermitian varieties in a finite projective space PG(N, q 2 )[END_REF] 9 Hermitian self-dual codes.

• k = 3 Algorithm 1 Computation of H X n -wit 2 = 1 (polynomial system strategy) Require: k, such that 2 = 1 Ensure: H X 2k - 1: h k ← 1 2: F 0 ← {h 0 ∈ IF * p 2 | θ k+1 (h 0 ) = -h 0 } 3: M ← k-1 2 4: E ← F 0 × IF k-N -1 p 2 5: while E = ∅ do 6: Pick (h 0 , h N +1 , . . . , h k-1 ) in E 7: for = 1, . . . , N do 8: h ← - 1≤i≤ i mod 2=0 h p k-i × h -i - 1≤i≤ i mod 2=1 h k-i × h p -i 9:
end for 10:

if k i=0 h p+1 i = 0 then 11:
go to 6:

12: end if 13: for = 1, . . . , k -1 -N do 14: if 0≤i≤ +N i mod 2=0 h p k-i × h +N -i + 0≤i≤ +N i mod 2=1 h k-i × h p +N -i = 0 then 15:
go to 6:

H ← H ∪ {X k + k-1 i=0 h i X i } 19: E ← E \ {(h 0 , h N +1 , . . . , h k-1
)} 20: end while 21: return H We look for (h 0 , h 1 , h 2 ) ∈ IF 3 9 such that :

h 2 + h 2 h 3 1 + h 3 1 h 0 = 0, h 4 0 + h 4 1 + h 4 2 + 1 = 0,
where

h 0 ∈ IF * 9 , h 1 + h 2 h 3 0 = 0.
We get the set of solutions {(a, 0, 0), (a 3 , 0, 0), (a 5 , 0, 0), (a 7 , 0, 0), (a, 1, a), (a, a, a 2 ), (a, a 3 , 2), (a 3 , 1, a 3 ), (a 3 , a, 2), (a 3 , a 3 , a 6 ), (a 5 , a 3 , 1), (a 5 , 1, a 5 ), (a 5 , a, a 6 ), (a 7 , a, 1), (a 7 , a 3 , a 2 ), (a 7 , 1, a 7 )}, therefore we have 16 Hermitian self-dual θ-negacyclic codes of length 6. For example the Hermitian self-dual θ-negacyclic code of length 6 with skew check polynomial h = X 3 + X 2 + aX + a 7 is a [START_REF] Boucher | Self-dual skew codes and factorization of skew polynomials[END_REF][START_REF] Bose | Hermitian varieties in a finite projective space PG(N, q 2 )[END_REF][START_REF] Bosma | The Magma algebra system. I. The user language[END_REF] code.

• k = 4

We look for (h 0 , h 1 , h 2 , h 3 ) ∈ IF 4 9 such that :

   h 3 + h 3 h 3 2 + h 3 2 h 1 + h 3 1 h 0 = 0, h 2 + h 3 h 3 1 + h 3 2 h 0 = 0, h 4 0 + h 4 1 + h 4 2 + h 4 3 + 1 = 0,
where

h 0 ∈ IF * 3 , h 1 = -h 3 h 0 .
We get the set {(1, 0, a 6 , 0), (1, 0, a 2 , 0), (1, 2, a, 1), (1, a 2 , a 3 , a 6 ), (1, a 6 , a, a 2 ), (2, 0, 2, 0), (1, 1, a, 2), (2, 0, 1, 0), (1, a 6 , a 3 , a 2 ), (1, a 2 , a, a 6 ), (1, 1, a 3 , 2), (1, 2, a 3 , 1)}, therefore we have 12 Hermitian self-dual θ-negacyclic codes of length 8. For example the Hermitian self-dual θ-negacyclic code of length 8 with skew check polynomial h = X 4 + X 3 + a 3 X 2 + 2X + 1 is a [START_REF] Boucher | Construction and number of self-dual skew codes over IF p 2[END_REF][START_REF] Bosma | The Magma algebra system. I. The user language[END_REF][START_REF] Bosma | The Magma algebra system. I. The user language[END_REF] code.

Factorization strategy.

In what follows we propose a strategy which was already used for Euclidean self-dual skew codes over IF p 2 ([8, Theorem 6.1] ) and for Hermitian self-dual θ-negacyclic codes of dimension coprime to p ([9, Theorem 3.7]). We give a practical way to construct the solutions which avoids the resolution of a polynomial system and we provide a counting formula for the number of solutions. This construction is based on Proposition 2 of [START_REF] Boucher | A first step towards the skew duadic codes[END_REF] that we recall below. Proposition 1. Consider IF q a finite field with q = p 2 elements where p is a prime number, θ : x → x p the Frobenius automorphism over

IF p 2 and R = IF q [X; θ]. Consider F (X 2 ) = f 1 (X 2 ) • • • f r (X 2 ) where f 1 (X 2 ), . . . , f r (X 2 ) are pairwise coprime polynomials of IF p [X 2 ] satisfying f i = f i . The map φ : H f1(X 2 ) × • • • × H fr(X 2 ) → H F (X 2 ) (h 1 , . . . , h r ) → lcrm(h 1 , . . . , h r )
is bijective.

We introduce the following notations: for

f = f (X 2 ) ∈ IF p [X 2 ], H f := {h ∈ H f | no non-constant divisor of f (X 2 ) in IF p [X 2 ] divides h in R}, F := {f = f (X 2 ) ∈ IF p [X 2 ] | f = f is irreducible in IF p [X 2 ] and deg X 2 (f ) > 1}, G := {f = f (X 2 ) ∈ IF p [X 2 ] | f = gg with g = g irreducible in IF p [X 2 ]}.

Non-existence of Hermitian self-dual θ-cyclic codes for p odd prime

Recall that there exists a Hermitian self-dual θ-cyclic code of dimension coprime to p over IF p 2 if, and only if, p = 2 (Theorem 3.7 of [START_REF] Boucher | A first step towards the skew duadic codes[END_REF]). In what follows we prove that this result remains true if the dimension of the code is divisible by p (Proposition 2). Lemma 1. If k and p are odd then

H X 2k -1 = ∅. Proof. Assume that H X 2k -1 = ∅. Consider h in H X 2k -1 with constant coefficient h 0 . As θ(h ) • h = X 2k -1, we have θ(1/θ k (h 0 )) × h 0 = -1.
As the order of θ is 2 and k is odd, we get 1/h 0 × h 0 = -1, which is impossible over IF p 2 because p is an odd prime number. Proposition 2. If p is odd, then there exists no Hermitian self-dual θ-cyclic code over

IF p 2 .
Proof. Let us prove that for any k, H X 2k -1 = ∅. Consider s, t in IN such that k = p s ×t and p does not divide t. According to Lemma 1, the set

H X 2p s -1 = H (X 2 -1) p s is empty. We have X 2k -1 = f 1 (X 2 )f 2 (X 2 ) where f 1 (X 2 ) = (X 2 -1) p s = f 1 (X 2 ) and f 2 (X 2 ) = ( t-1 i=0 X 2i ) p s = f 2 (X 2
). As p does not divide t, these two polynomialrs are coprime polynomials of IF p [X 2 ]. Therefore, according to Proposition 1, the set H X 2k -1 is empty.

Construction of H (X 2 +1) p s

The aim of this section is to construct and to enumerate Hermitian self-dual θ-negacyclic codes over IF p 2 whose dimension is p s where θ is the Frobenius automorphism (Proposition 4 and Algorithm 2). In order to construct the set H X 2k +1 = H (X 2 +1) p s , factorization properties specific to IF p 2 [X; θ] will be useful. The following proposition enables to characterize the skew polynomials that have a unique factorization into the product of monic linear skew polynomials dividing X 2 + 1 (see also [START_REF] Boucher | Self-dual skew codes and factorization of skew polynomials[END_REF]Proposition 16]). Proposition 3. Consider p a prime number, θ the Frobenius automorphism over

IF p 2 , R = IF p 2 [X; θ], m a non-negative integer, f (X 2 ) in IF p [X 2 ] irreducible and h = h 1 • • • h m in R
where for all i in {1, . . . , m}, h i is irreducible in R, monic, and divides f (X 2 ). The following assertions are equivalent :

(i) The above factorization of h is not unique. (ii) f (X 2 ) divides h. (iii) There exists i in {1, . . . , m -1} such that h i • h i+1 = f (X 2 ).
Corollary 1. Consider p a prime number, θ the Frobenius automorphism over

IF p 2 , R = IF p 2 [X; θ], m a non-negative integer and h = (X + λ 1 ) • • • (X + λ m ) in R where for all i in {1, . . . , m}, λ p+1 i = -1.
The following assertions are equivalent :

(i) The above factorization of h is not unique. (ii) X 2 + 1 divides h. (iii) There exists i in {1, . . . , m -1} such that (X + λ i ) • (X + λ i+1 ) = X 2 + 1 i.e. λ i λ i+1 = 1.
This proposition and its corollary motivate the following partition (see Lemma 3.2 of [START_REF] Boucher | Construction and number of self-dual skew codes over IF p 2[END_REF]) : Lemma 2. Consider p a prime number, θ the Frobenius automorphism over

IF p 2 , R = IF p 2 [X; θ], s a non-negative integer and f = f (X 2 ) ∈ {X 2 + 1} ∪ F.
One has the following partition :

H f p s = p s 2 i=0 f i • H f p s -2i . (3)
Lemma 3. Consider p an odd prime number, θ the Frobenius automorphism, R = IF p 2 [X; θ], m a non-negative integer and M = m-1

2

. The number of elements of the set

H (X 2 +1) m is #H (X 2 +1) m =    (p + 1) × p M if p is odd, 3 if p = 2 and m = 1, 0 if p = 2 and m > 1.
If m ≡ 0 (mod 2), then

H (X 2 +1) m = {(X 2 + 2α 1 X -1) • • • (X 2 + 2α M +1 X -1) | α p+1 i = -1, α i+1 = -α i }. If m ≡ 1 (mod 2), then H (X 2 +1) m = {(X 2 +2α 1 X-1) • • • (X 2 +2α M X-1)•(X+α M +1 ) | α p+1 i = -1, α i+1 = -α i }.
Proof.

• One first proves that the elements of H (X 2 +1) m are obtained as products of linear monic skew polynomials (X + λ 1 )

• • • (X + λ m ) where λ 1 , . . . , λ m are elements of IF p 2 such that    ∀i ∈ {1, . . . , m}, λ p+1 i = -1, ∀i ∈ {1, . . . , m -1}, λ i λ i+1 = 1, ∀j ∈ {1, . . . , m 2 }, (λ 2j-1 λ 2j ) 2 = 1. (4)
Namely, consider h in H (X 2 +1) m . As h divides (X 2 +1) m and as

X 2 +1 is irreducible with degree 1 in IF p [X 2 ]
, h is a (not necessarily commutative) product of linear monic skew polynomials dividing X 2 + 1 ([6, Lemma 13 (2) ] or [16, page 6]). Furthermore, the degree of h is equal to m (because deg(θ(h ) • h) = 2m), therefore there exists λ 1 , . . . , λ m in IF p 2 such that :

h = (X + λ 1 ) • • • (X + λ m ) where ∀i ∈ {1, . . . , m}, λ p+1 i = -1.
In particular, the first relation of ( 4) is satisfied. As X 2 + 1 does not divide h, according to Corollary 1, we have :

∀i ∈ {1, . . . , m -1}, (X + λ i ) • (X + λ i+1 ) = X 2 + 1. (5)
Therefore ∀i ∈ {1, . . . , m -1}, λ i λ i+1 = 1, which is the second relation of (4). The following expression of h can be obtained using an induction argument (left to the reader) :

h = (X + λm ) • • • (X + λ1 )
where for i in {1, . . . , m}, λi is defined by : λi

:=    -1/λ i × (λ 1 • • • λ i ) 2 if i ≡ 1 (mod 2), -1/λ i × 1 (λ 1 • • • λ i-1 ) 2 if i ≡ 0 (mod 2). (6) Furthermore, X 2 + 1 does not divide h , otherwise X 2 + 1 would divide h. Therefore ∀i ∈ {1, . . . , m -1}, (X + λi+1 ) • (X + λi ) = X 2 + 1. ( 7
)
The relation θ(h ) • h = (X 2 + 1) m can be written

(X + θ( λm )) • • • (X + θ( λ1 )) • (X + λ 1 ) • • • (X + λ m ) = (X 2 + 1) m . ( 8
)
As X 2 + 1 is central, the factorization of the skew polynomial (X 2 + 1) m into the product of monic skew polynomials dividing X 2 + 1 is not unique, therefore, according to Corollary 1, X 2 + 1 is necessarily the product of two consecutive monic linear factors of the left hand side of (8). According to ( 5) and ( 7), the only possibility is

(X + θ( λ1 )) • (X + λ 1 ) = X 2 + 1.
As X 2 + 1 is central, the relation ( 8) can be simplified and one gets

(X + θ( λm )) • • • (X + θ( λ2 )) • (X + λ 2 ) • • • (X + λ m ) = (X 2 + 1) m-1 .
By repeating the same argument we get :

(X + θ( λ2 )) • (X + λ 2 ) = X 2 + 1, . . . (X + θ( λm )) • (X + λ m ) = X 2 + 1.
Considering the constant coefficients of the skew polynomials involved in the above equalities, we get that ∀i ∈ {1, . . . , m}, λ i θ( λi ) = 1 ⇔ θ(λ i ) λi = 1, and using the definition of λi given in (6), one gets, for i odd, (λ i λ i+1 ) 2 = 1 (third relation of ( 4)).

Conversely, consider h = (X + λ 1 ) • • • (X + λ m ) where λ 1 , . . . , λ m are elements of IF p 2 satisfying (4). According to the first relation of ( 4), the monic skew polynomials X + λ i divide X 2 + 1. According to the second relation of ( 4) and to Corollary 1,

X 2 + 1 does not divide h. The skew polynomial h is equal to (X + λm ) • • • (X + λ1 )
where λi is defined by the relations [START_REF] Boucher | Self-dual skew codes and factorization of skew polynomials[END_REF]. Furthermore, according to the third relation of ( 4)

, if i is odd, (λ 1 • • • λ i ) 2 = -1
, therefore for all i in {1, . . . , m}, λ i θ( λi ) = 1 and

X 2 + 1 = (X + θ( λi )) • (X + λ i ).
The product θ(h ) • h can be simplified as follows :

θ(h ) • h = (X + θ( λm )) • • • (X + θ( λ1 )) • (X + λ 1 ) • • • (X + λ m ) = (X 2 + 1) • (X + θ( λm )) • • • (X + θ( λ2 )) • (X + λ 2 ) • • • (X + λ m ) (because X 2 + 1 is central) . . . = (X 2 + 1) m-1 • (X + θ( λm )) • (X + λ m ) = (X 2 + 1) m
and one concludes that h belongs to H (X 2 +1) m .

• The relations (4) enable to count the number of elements of H (X 2 +1) m . Namely according to Corollary 1, the elements of H (X 2 +1) m have a unique factorization into the product of linear monic skew polynomials dividing X 2 +1. Therefore the number of elements of the set H (X 2 +1) m is the number of m-tuples (λ 1 , . . . , λ m ) of (IF p 2 ) m satisfying the conditions (4). Assume that p = 2 and that m is an integer greater than 1. Then the conditions λ 1 λ 2 = 1 and (λ 1 λ 2 ) 2 = 1 are not compatible, therefore the set H (X 2 -1) m is empty. If m = 1, it is reduced to {X + 1, X + a, X + a 2 } where a 2 + a + 1 = 0. Assume that p is odd, then according to conditions (4), we have p + 1 choices for λ 1 , 1 choice for λ 2 , p choices for λ 3 , etc ... therefore one gets (p + 1)p (m-1)/2 elements.

• Lastly, thanks to (4), the expression of h = (X +λ 1 ) • • • (X +λ m ) can be simplified as

h = (X+λ 1 )•(X-1 λ1 ) • • • (X+λ 3 )•(X-1 λ3 ) • • • = (X 2 +2λ 1 X-1)•(X 2 +2λ 3 X-1) • • • . Remark 2. Consider p = 2, f (X 2 ) = X 2 + 1
and s a positive integer. According to Lemma 2, the set H (X 2 +1) 2 s can be written as :

H (X 2 +1) 2 s = 2 s-1 i=0 (X 2 + 1) i • H (X 2 +1) 2 s -2i .
According to Lemma 3, the sets H (X 2 +1) 2 s -2i are empty when 2 s -2i ≥ 2. Therefore the above equality can be simplified as follows :

H (X 2 +1) 2 s = (X 2 + 1) 2 s-1 • H (X 2 +1) 0 , = {(X + 1) 2 s }.
One gets that for s > 0 there is only one Hermitian self-dual θ-cyclic code of dimension

M ← p s -2i-1 2 4: L ← {α ∈ IF M +1 p 2 | α p+1 i = -1, α i+1 = -α i } 5: for α ∈ L do 6: H ← {(X 2 + 1) i • (X 2 + 2α 1 X -1) • • • (X 2 + 2α M X -1) • (X + α M +1 )} ∪ H 7:
end for 8: end for 9: return H

(p + 1) p p s +1 2 -1 p -1 . Proof. Consider R = IF p 2 [X; θ].
The number of Hermitian self-dual θ-negacyclic codes of dimension p s over IF p 2 is equal to #H X 2p s +1 . According to Lemma 2, one has the following partition :

H X 2p s +1 = M i=0 (X 2 + 1) i • H (X 2 +1) p s -2i , where M = p s -1 2 . According to Lemma 3, one has #H (X 2 +1) p s -2i = (p + 1) × p M -i , therefore #H X 2p s +1 = (p + 1) M i=0 p M -i = (p + 1) p M +1 -1 p -1 .
(h ) • h = X 6 + 1 are h = (X 2 + 1) • (X + α),
where

α 4 = -1 and h = (X 2 + 2α 1 X -1) • (X + α 2 ),
where α 4 1 = α 4 2 = -1 and α 2 = -α 1 . We get the 16 Hermitian self-dual θ-negacyclic codes of dimension 3 over IF 9 obtained in Example 2. Example 4. Consider IF 9 = IF 3 (a) where a 2 + 2a + 2 = 0 and θ the Frobenius automorphism over IF 9 . Consider α = (a, a, a 3 , a, a 3 , a, a, a 3 , a 5 , a 5 , a 7 , a 5 , a 5 , a 3 ) ∈ IF 14 9 .

For all i in {1, . . . , 14}, we have α 4 i = -1 and for all i in {1, . . . , 13}, α i+1 = -α i , therefore according to Lemma 3, the skew polynomial

h = (X 2 + 2α 1 X -1) • • • (X 2 + 2α 13 X -1) • (X + α 14 ) ∈ IF 9 [X; θ],
satisfies θ(h ) • h = X 54 + 1, and the skew polynomial g = θ(h ) generates a [54, 27] 9 Hermitian self-dual code. Furthermore, its minimum distance is 18, which is the best known minimum distance of [54, 27] linear codes over IF 9 .

Construction of H f p s for f in F

The aim of this subsection is to construct H f p s for f in F and to compute its number of elements.

Consider f = f (X 2 ) in F. Recall that according to Lemma 2, one has the partition:

H f p s = p s 2 i=0 f i • H f p s -2i ,
where for m in IN, the set H f m is defined by

H f m = {h ∈ H f m | f does not divide h}.
Lemma 4 below generalizes Lemma 3 and uses the same type of arguments linked to the factorization of skew polynomials. Lemma 4. Consider p a prime number, θ the Frobenius automorphism over

IF p 2 , R = IF p 2 [X; θ], m a non-negative integer and f = f (X 2 ) in F with degree d = 2δ > 1 in X 2 . The set H f m is equal to h 1 • • • h m | h j ∈ H f , h j = θ(h j-1 )
and has (1 + p δ )p δ(m-1) elements.

Proof. To simplify the presentation, the following notations will be used in this proof: h = h(X), f = f (X 2 ).

Consider h in H f m . As h divides f m and f is irreducible in IF p [X 2 ], all the irreducible factors of h divide f and have the same degree d ([6, Lemma 13 (2)] or [16, page 6]):

h = m i=1 h i , h i monic, deg(h i ) = d, h i |f, h i irreducible.
Furthermore, f does not divide h, therefore according to Proposition 3, for all j in {1, . . . , m -1}, h j • h j+1 is distinct from f . Using an induction argument (left to the reader), one gets the following expression of h :

h = m-1 i=0 1 µ m-i h m-i • µ m-i ,
where µ 1 = 1 and for i ∈ {2, . . . , m},

µ i = (h 1 • • • h i-1 ) 0 is defined as the constant coef- ficient of h 1 • • • h i-1 .
Furthermore, this factorization (into the product of irreducible monic polynomials of same degree d dividing f ) is unique (because the factorization of h is unique).

As the factorization of f m into the product of irreducible factors is not unique (because f m is central), according to Proposition 3, f m = θ(h ) • h must have two consecutive irreducible monic factors whose product is f . As h and h do not possess two consecutive factors whose product is f , necessarily, θ( 1 µ1 h 1 • µ 1 ) • h 1 = f , and proceeding by induction, one gets ∀j ∈ {1, . . . , m}, θ

1 µ j h j • µ j • h j = f and h j+1 = θ( 1 µ j h j • µ j ) when j = m. (9) Conversely, consider h = h 1 • • • h m with θ( 1 µj h j • µ j ) • h j = f, h j+1 = θ( 1 µj h j • µ j ) and µ j constant coefficient of h 1 • • • h j-1 . Then θ(h ) • h = f m and h j • h j+1 = f .
Furthermore, the skew polynomials h j are all irreducible because they are non-trivial factors of f , and f is irreducible in IF p [X 2 ]. Therefore according to Proposition 3, the skew polynomial h is not divisible by f and it belongs to H f m .

To conclude, we get that

h ∈ H f m ⇔              h = h 1 • • • h m , θ( 1 µj h j • µ j ) • h j = f, h j+1 = θ( 1 µj h j • µ j ), µ 1 = 1, µ j = (h 1 • • • h j-1 ) 0 , j = 1. Lastly, as f (X 2 ) = f (X 2
), the degree of f (X 2 ) in X 2 is even, therefore, from the equality (9), one gets that for all j ∈ {1, . . . , m}, the degree of h j is even and the constant coefficient of h j is

1 θ deg(h j ) ((hj )0) = 1
(hj )0 . Furthermore, the constant coefficient of f is equal to 1 because f (X 2 ) = f (X 2 ), therefore, following (9), we get ∀j ∈ {1, . . . , m}, θ 1 µ j 1 (h j ) 0 µ j (h j ) 0 = 1 and ∀j ∈ {1, . . . , m}, θ((h j ) 0 ) = (h j ) 0 . As µ j is defined as the constant coefficient of h 1 • • • h j-1 , it is fixed by θ, therefore we get :

h ∈ H f m ⇔    h = h 1 • • • h m , θ(h j ) • h j = f, h j+1 = θ(h j ).
The number of elements of H f m follows from the fact that H f has 1 + p δ elements ([9, Lemma 3.3]).

Algorithm 3 Computation of H

f p s for f ∈ F Require: f, s, Ensure: H f p s 1: H f ← {h ∈ R | θ(h ) • h = f (X 2 )} 2: H ← ∅ 3: for i = 0, . . . , p s -1 2 do 4: m ← p s -2i 5: L ← {u = (u 1 , . . . , u m ) ∈ H f m | u i+1 = θ(u i )} 6:
for u ∈ L do 7:

H ← {f (X 2 ) i • u 1 • • • u m } ∪ H 8:
end for 9: end for 10: return H The construction of the set H f p s for f in F is deduced from Lemma 2 and Lemma 4. Proposition 5. Consider p a prime number, θ the Frobenius automorphism over

IF p 2 , R = IF p 2 [X; θ], s a non-negative integer and f = f (X 2 ) in F with degree d = 2δ > 1 in X 2 . The set H f p s has p δ(p s +1) -1 p δ -1 elements. Proof. According to Lemma 2, H f p s = p s 2 i=0 f i • H f p s -2i and according to Lemma 4, H f m has (1 + p δ )(p δ ) m-1 if m = 0 and 1 element if m = 0. Therefore H f p s has (p s -1)/2 i=0 (1 + p δ )(p δ ) p s -2i-1 elements if p is odd and 1 + 2 s-1 -1 i=0 (1 + 2 δ )(2 δ ) 2 s -2i-1
elements otherwise. In both cases one gets #H f p s = p δ(p s +1) -1

p δ -1
.

Corollary 2. If X 4δ + 1 belongs to F, then there are p δ(p s +1) -1

p δ -1
Hermitian self-dual θ-negacyclic codes of length n = 4δp s over IF p 2 .

Proof. We apply Proposition 5 to f = f (X 2 ) = X 4δ + 1.

Example 5. We give here an example of a [36, [START_REF] Yang | On self-dual constacyclic codes over finite fields[END_REF][START_REF] Gaborit | Table of Hermitian self-dual codes over IF 4[END_REF] Hermitian self-dual code over IF 9 = IF 3 (a) where a 2 + 2a + 2 = 0. Consider θ the Frobenius automorphism over IF 9 and the skew polynomial ring R = IF 9 [X; θ]. We have X 4 + 1 ∈ F, and the set

H X 4 +1 of the irreducible skew polynomials H satisfying θ(H ) • H = X 4 + 1 is equal to {X 2 + X + 2, X 2 + a 2 X + 2, X 2 + 2X + 2, X 2 + a 6 X + 2}.
Consider the product of 9 elements of

H X 4 +1 : h = (X 2 + X + 2) • (X 2 + X + 2) • (X 2 + a 2 X + 2) • (X 2 + a 2 X + 2) • (X 2 + X + 2) • (X 2 + a 2 X + 2) • (X 2 + a 2 X + 2) • (X 2 + a 2 X + 2) • (X 2 + 2X + 2) = X 18 + a 3 X 17 + a 5 X 16 + 2X 15 + a 2 X 14 + a 2 X 13 + aX 12 + a 7 X 11 + a 7 X 10 + a 3 X 9 + a 3 X 8 + a 7 X 7 + a 5 X 6 + a 2 X 5 + a 6 X 4 + 2X 3 + aX 2 + a 3 X + 2 in R.
The skew polynomial h belongs to H (X 4 +1) 9 and the skew polynomial g = θ(h ) generates a Hermitian self-dual θ-negacyclic [36, [START_REF] Yang | On self-dual constacyclic codes over finite fields[END_REF][START_REF] Gaborit | Table of Hermitian self-dual codes over IF 4[END_REF] code over IF 9 .

Construction of H f p s for f in G

In this section we propose an algorithm for the construction of H f p s for f = gg in G which relies on the construction of all the irreducible skew polynomials dividing g(X 2 ) in R (see Appendix A of [START_REF] Boucher | A first step towards the skew duadic codes[END_REF] ).

We first recall that the set H f p s can be partitioned as follows (see Lemma 3.2 of [START_REF] Boucher | Construction and number of self-dual skew codes over IF p 2[END_REF]). Lemma 5. Consider p a prime number, θ the Frobenius automorphism over

IF p 2 , R = IF p 2 [X; θ], s a non-negative integer and f = f (X 2 ) = g(X 2 )g (X 2 ) in G with g = g(X 2 ) = g (X 2 ) irreducible in IF p [X 2
]. One has the following partition

H f p s = p s i=0 p s -i j=0 g i g j • H f p s -(i+j) .
The construction of each set H f m will rely on the construction of a new set of special factors of g(X 2 ) m . Lemma 6. Consider p a prime number, θ the Frobenius automorphism over

IF p 2 , R = IF p 2 [X; θ]. Consider g(X 2 ) an irreducible polynomial of IF p [X 2 ] and R g(X 2
) m the set of all right factors u of g(X 2 ) m which are not divisible by g(X 2 ) and which do not divide g(X 2 ) m-1 . We have

R g(X 2 ) m = {u 1 • • • u m | u i monic irreducible, u i |g(X 2 ) and u i • u i+1 = g(X 2 )}. Proof. 1. As g(X 2 ) is irreducible in IF p [X 2 ]
, it is the product of two irreducible skew polynomials of degree equal to deg X 2 g(X 2 ), therefore g(X 2 ) m is the product of 2m irreducible skew polynomials of same degree. Assume that g(X 2

) m = u 1 • • • u r • v 1 • • • v s with r + s = 2m and r ≥ m + 1; then g(X 2 ) divides u := u 1 • • • u r . Namely, assume that g(X 2 ) does not divide u. As g(X 2 ) divides u 1 • • • u r • v 1 • • • v s then, according to Proposition 3, g(X 2 ) = u r • v 1 or there exists i in {1, . . . , s -1} such that g(X 2 ) = v i • v i+1 , therefore g(X 2 ) m-1 = ũ1 • • • ũr • ṽ1 • •
• ṽs with r ≥ m and r + s = 2(m -1); using an induction argument, we get a contradiction. 2. Assume that u = u 1 • • • u m where for i in {1, . . . , m}, u i is a monic irreducible skew polynomial dividing g(X 2 ) and u i • u i+1 = g(X 2 ) for i ∈ {1, . . . , m -1}. As u i divides g(X 2 ), u divides g(X 2 ) m . Furthermore, according to Proposition 3, g(X 2 ) does not divide u because

u i • u i+1 = g(X 2 ). Assume that u divides g(X 2 ) m-1 , then there exist m -2 irreducible skew polynomials v 1 , . . . , v m-2 such that g(X 2 ) m-1 = u 1 • • • u m • v 1 • • • v m-2 = u 1 • • • u r • v 1 • • • v s with r + s = 2(m -1)
and r ≥ m. Then according to point 1 of the proof, g(X 2 ) divides u, which is impossible. 3. Assume that u divides g(X 2 ) m , u does not divide g(X 2 ) m-1 and g(X 2 ) does not divide u. Consider r ≤ 2m such that u = u 1 • • • u r where u i is irreducible and divides g(X 2 ). Necessarily, u divides g(X 2 ) r , therefore, as u does not divide g(X 2 ) m-1 , we have r ≥ m. Assume that r ≥ m + 1, then according to point 1 of the proof, g(X 2 ) divides u, which is impossible. Therefore, r = m and as g(X 2 ) does not divide u, according to Lemma 3, we have for all i in {1, . . . , r -1},

u i • u i+1 = g(X 2 ).
Proposition 6. Consider p a prime number, θ the Frobenius automorphism over

IF p 2 , R = IF p 2 [X; θ]. Consider f (X 2 ) = g(X 2 )g (X 2 ) where g(X 2 ) = g (X 2 ) is an irreducible polynomial of IF p [X 2
] and the map φ defined by :

φ : R g(X 2 ) m → H f (X 2 ) m u → lcrm(u, v) where θ(u ) • v = g (X 2 ) m .
The map φ is a bijection and the set H f (X 2 ) m has (1 + p δ )p δ(m-1) elements where δ is the degree of g(X 2 ) in X 2 .

Proof.

• First the map φ is well defined. Namely, consider u in R g(X 2 ) m , then u divides g(X 2 ) m , it does not divide g(X 2 ) m-1 and it is not divisible by g(X 2 ). Furthermore, according to Lemma 6, the degree of u is equal to δm. Consider v in R such that g (X 2 ) m = θ(u ) • v and h = lcrm(u, v), then the degree of v is δm and as u and v are leftcoprime, the degree of h is equal to 2δm. As g(X 2 ) does not divide u and g (X 2 ) does not divide v, g(X 2 ) and g (X 2 ) do not divide h. Consider ũ and ṽ in R

such that h = u • ũ = v • ṽ. Then, θ(h ) • h = V • θ(v ) • u • ũ = U • θ(u ) • v • ṽ is divisible by the central polynomials θ(v ) • u = g(X 2 ) m and θ(u ) • v = g (X 2 ) m , therefore f (X 2 ) m divides θ(h ) • h. Considerations on the degrees lead to θ(h ) • h = f (X 2 ) m . • Consider h in H f (X 2 ) m , then θ(h ) • h = f (X 2 ) m with g(X 2 ) h and g (X 2 ) h. Then h = lcrm(u, v) where u = gcld(h, g(X 2 ) m ) and v = gcld(h, g(X 2 ) m ).
The skew polynomial u belongs to the set R g(X 2 ) m . Namely, u divides g(X 2 ) m by construction; g(X 2 ) does not divide h, therefore it does not divide u; lastly u does not divide g(X 2 ) m-1 , otherwise g(X 2 ) would divide θ(h ), which is impossible. Therefore, according to Lemma 6, u = u 1 • • • u m where u 1 , . . . , u m are m irreducible skew polynomials dividing g(X 2 ) and such that u i •u i+1 = g(X 2 ) for i in {1, . . . , m-1}.

In the same way, one gets that v = v 1 • • • v m where v 1 , . . . , v m are m irreducible skew polynomials dividing g (X 2 ) and such that

v i • v i+1 = g (X 2 ) for i in {1, . . . , m -1}. Now consider U and V in R such that h = u • U = v • V , then θ(h ) = Ṽ • θ(v ) and θ(h ) • h = Ṽ • θ(v ) • u • U . As θ(h ) • h = g(X 2 ) m g (X 2 ) m is central, we get θ(v ) • u • U • Ṽ = g(X 2 ) m g (X 2
) m . We deduce from this equality that lclm(g

(X 2 ) m , U • Ṽ ) divides θ(v ) • u • U • Ṽ . Therefore g(X 2 ) m divides θ(v )
• u and considerations on the degrees of the polynomials lead to g(X 2 ) m = θ(v ) • u. • The elements of R g(X 2 ) m are of the form g 1 • • • g m where g 1 , . . . , g m are irreducible monic factors of g(X 2 ) and g i • g i+1 = g(X 2 ). As g(X 2 ) has 1 + p δ monic irreducible factors ( [START_REF] Odoni | On additive polynomials over a finite field[END_REF] or [9, Appendix A]), we get that #H f (X 2 ) m = (1 + p δ )p δ(m-1) .

Proposition 7. Consider p a prime number, θ the Frobenius automorphism over

IF p 2 , R = IF p 2 [X; θ], s a non-negative integer and f = f (X 2 ) in G with degree d = 2δ > 1 in X 2 . The set H f p s has (p δ(p s +1) -2p s -3)(1+p δ )+4p s +4 (p δ -1) 2 elements. Algorithm 4 Computation of H f p s for f = g • g ∈ G Require: f ∈ G, s Ensure: H f p s 1: I ← set of monic irreducible right divisors of g(X 2 ) in R 2: H ← ∅ 3: for i = 0, . . . , p s do 4:
for j = 0, . . . , p s -i do

5: m ← p s -i -j 6: R ← {u 1 • • • u m | u i ∈ I, u i • u i+1 = g(X 2 )} 7:
for u ∈ R do 8:

v ← quotient of the division of g (X 2 ) m by θ(u )

9:

H ← {g(X 2 ) i • g (X 2 ) j • lcrm(u, v)} ∪ H 10:
end for 11:

end for 12: end for 13: return H Corollary 3. If X 4δ + 1 belongs to G, then there are

(p δ(p s +1) -2p s -3)(1+p δ )+4p s +4 (p δ -1) 2 Hermitian self-dual θ-negacyclic codes of length n = 4δp s over IF p 2 .
Proof. We apply Proposition 7 to

f = f (X 2 ) = X 4δ + 1. Example 6. The polynomial f (X 2 ) = X 16 + 1 factorizes over IF 3 [X 2 ] as f (X 2 ) = g(X 2 )g (X 2 ) where g(X 2 ) = X 8 + X 4 + 2 and g (X 2 ) = X 8 + 2X 4 + 2 are irreducible in IF 3 [X 2 ]. Consider IF 9 = IF3(a)
with a 2 + 2a + 2 = 0 and θ Frobenius automorphism over IF 9 . Consider the following irreducible skew polynomials dividing 3 and h = lcrm(u, v). Then the skew polynomial g = θ(h ) = X 24 +aX 23 +a 5 X 22 +X 21 +X 20 +aX 19 +a 6 X 18 +a 2 X 17 +a 3 X 15 +X 14 + a 7 X 13 +a 6 X 12 +a 3 X 11 +X 10 +a 7 X 9 +a 6 X 7 +a 6 X 6 +a 5 X 5 +X 4 +2X 3 +a 5 X 2 +a 5 X +1 generates a [48, 24, 16] Hermitian self-dual θ-negacyclic code over IF 9 .

X 8 + X 4 + 2 in R = IF 9 [X; θ] : u 1 = X 4 + a 3 X 3 + a 7 X 2 + 2X + a 7 , u 2 = X 4 + a 6 X 3 + 2X 2 + a 2 X + a 7 and u 3 = X 4 + a 2 X 2 + a 5 X + a. As u 1 • u 2 and u 2 • u 3 are distinct of X 8 + X 4 + 2, the skew polynomial u = u 1 • u 2 • u 3 belongs to R (X 8 +X 4 +2) 3 . Consider v in R such that θ(u ) • v = (X 8 + 2X 4 + 2)

Construction and enumeration of Hermitian self-dual

θ-negacyclic codes.

Recall that there is a counting formula for Hermitian self-dual θ-cyclic codes of odd dimension over IF

X 2k +1 = N × f ∈F ,f |X 2k +1 deg(f )=2δ p δ(p s +1) -1 p δ -1 × f ∈G,f |X 2k +1 deg(f )=2δ (p δ(p s +1) -2p s -3)(1 + p δ ) + 4p s + 4 (p δ -1) 2
where

N =      3 if p = 2
: Compute f 1 (X 2 ), . . . , f r (X 2 ) ∈ {X 2 + 1} ∪ F ∪ G such that X 2t + 1 = f 1 (X 2 ) • • • f r (X 2 ) in IF p [X 2 ] and gcd(f i (X 2 ), f j (X 2 )) = 1 in IF p [X 2
] for i = j. H ← H ∪ {lcrm(h 1 , . . . , h r )} 7: end for 8: return H Example 7. We give here an example of a [30, [START_REF] Jacobson | The Theory of Rings[END_REF][START_REF] Datta | Number of solutions of systems of homogeneous polynomial equations over finite fields[END_REF] Hermitian self-dual code over IF 9 = IF 3 (a) where a 2 + 2a + 2 = 0. Consider θ, the Frobenius automorphism over IF 9 and the skew polynomial ring R = IF 9 [X; θ]. We have X 30 + 1 = ((X 2 ) 5 + 1) 3 = (X 2 + 1) 3 (X 8 -X 6 + X 4 -X 2 + 1) 3 .

Consider 3 and h 2 = X 12 + aX 11 + a 7 X 10 + X 9 +a 5 X 8 +a 6 X 7 +2X 6 +a 2 X 5 +a 5 X 4 +2X 3 +a 7 X 2 +a 5 X+1 ∈ H (X 8 -X 6 +X 4 -X 2 +1) 3 , then the skew polynomial h = lcrm(h 1 , h 2 ) belongs to H X 30 +1 and the skew polynomial g = θ(h ) generates a [30, [START_REF] Jacobson | The Theory of Rings[END_REF][START_REF] Datta | Number of solutions of systems of homogeneous polynomial equations over finite fields[END_REF] Hermitian self-dual θ-negacyclic code over IF 9 .

h 1 = X 3 + a 2 X 2 + a 3 X + a 7 ∈ H (X 2 +1)
Example 8. We consider here Hermitian self-dual θ-cyclic codes with length 68 over IF 4 = IF 2 (a) where a 2 + a + 1 = 0 and θ is the Frobenius automorphism over IF 4 . According to [START_REF] Gaborit | Table of Hermitian self-dual codes over IF 4[END_REF], the best known minimum distance of Hermitian self-dual codes of length 68 over IF 4 is equal to 16. We provide here an example of a [68, 34, 18] Hermitian self-dual code over IF 4 . Consider h 1 = X 2 + 1, h 2 = (X 8 +X 5 +X 4 +X 3 +1)•(X 8 +a 2 X 7 +aX 6 +a 2 X 5 +a 2 X 4 +a 2 X 3 +aX 2 +a 2 X +1) and 2 , therefore h = lcrm(h 1 , h 2 , h 3 ) belongs to H (X 34 +1) 2 . One checks that the θ-cyclic code of length 68 and skew generator polynomial g = Θ(h * ) is a Hermitian self-dual [68,34,[START_REF] Yang | On self-dual constacyclic codes over finite fields[END_REF] code. We give below the expression of g : g = X 34 + aX 33 + a 2 X 32 + aX 31 + aX 30 + a 2 X 26 + aX 23 + a 2 X 22 + X 21 + a 2 X 20 + a 2 X 19 + a 2 X 15 + a 2 X 14 + X 13 + a 2 X 12 + aX 11 + a 2 X 8 + aX 4 + aX 3 + a 2 X 2 + aX + 1. In what follows, we assume that q = p e where p is a prime number and e is an even integer. Therefore the automorphism σ is defined by σ = F rob e 2 : x → x p e 2 where F rob : x → x p is the Frobenius automorphism over IF q . We consider the non-negative integer r defined by θ = F rob r : x → x p r , and the ring R = IF q [X; θ] of skew polynomials. We denote the order of θ and IF θ q the fixed field of θ. We denote µ the integer such that 2 µ || p + 1, which means that 2 µ divides p + 1 and 2 µ+1 does not divide p + 1.

h 3 = (X 8 +a 2 X 7 +aX 5 +X 4 +aX 3 +a 2 X +1)•(X 8 +a 2 X 7 +X 5 +X 4 +X 3 +a 2 X +1). We have h 1 ∈ H (X 2 +1)
We recall that for F = F (X ) in IF θ q [X ], the set H F is defined by

H F := {h ∈ R := IF q [X; θ] | h is monic and σ(h ) • h = F (X )}.
The Hermitian self-dual θ-cyclic codes of length n over IF q are completely determined by the set H X n -1 while the Hermitian self-dual θ-negacyclic codes are determined by H X n +1 . Recall also that we assume here that the order of θ divides the length n of the considered codes.

Existence conditions of Hermitian self-dual θ-cyclic codes

over IF p e with e > 2 even.

Recall that over IF p 2 , there is no Hermitian self-dual θ-cyclic code if p is an odd prime number. In what follows we prove that this non-existence result can be extended to IF q = IF p e . We start with an intermediate lemma :

Lemma 7. Consider F (X ) = F 1 (X )F 2 (X ) ∈ IF θ q [X ] where F 1 (X ) and F 2 (X ) are coprime in IF θ q [X ] and such that σ(F i ) = F i . If H F = ∅ then H Fi = ∅. Proof. Consider h in R such that σ(h ) • h = F (X ). Consider, for i ∈ {1, 2}, h i = gcld(h, F i (X )) and H i in R such that h = h i •H i . We have σ(h ) = Hi •σ(h i ) where Hi is in R. Therefore F (X ) = σ(h ) • h = H1 • σ(h 1 ) • h 1 • H 1 . As F (X ) is central, we get that σ(h 1 ) • h 1 divides F (X ). As F 2 (X ) also divides F (X ), lcrm(σ(h 1 ) • h 1 , F 2 (X )) divides F (X ). But F 2 (X ) is a central polynomial which has no common factor with σ(h 1 ) • h 1 because h 1 and σ(h 1 ) both divide F 1 (X ) = σ(F 1 (X )) and F 1 (X ) is co- prime with F 2 (X ). Therefore lcrm(σ(h 1 ) • h 1 , F 2 (X )) = σ(h 1 ) • h 1 • F 2 (X ) divides F (X ) = F 1 (X )F 2 (X ) and σ(h 1 ) • h 1 divides F 1 (X ). In the same way, σ(h 2 ) • h 2 divides F 2 (X ). We conclude by noticing that 2 deg(h) = 2(deg(h 1 ) + deg(h 2 )) = deg(F 1 ) + deg(F 2 ). Assume that 2 deg(h 1 ) < deg(F 1 ), then deg(F 1 ) + deg(F 2 ) < deg(F 1 ) + 2 deg(h 2 ), therefore deg(F 2 ) < 2 deg(h 2 ), which is impossible. Therefore we get that σ(h 1 ) • h 1 = F 1 (X ).
Theorem 2. If p is an odd prime number then there is no Hermitian self-dual θ-cyclic code over IF p e .

Proof.

Assume that there exists a Hermitian self-dual θ-cyclic code of dimension k over IF p e . Then the set H X 2k -1 is non-empty. Consider f 1 (X ) = (X -1) p s and f 2 (X ) = (X 2k -1)/(X -1) p s . As f 1 (X ) is a central polynomial coprime with f 2 (X ) and satisfying σ(f 1 ) = f 1 , according to Proposition 7, the set H f1(X ) is non-empty. Consider H ∈ IF p e [X; θ] with degree d and constant coefficient α such that

σ(H ) • H = X -1 p s . Then we have σ 1 θ d (α) α = -1.
Furthermore, we have = e gcd(e, r) therefore r = eµ where µ = r gcd(e, r) and we have :

σ • θ d = F rob e 2 + r p s 2 = F rob e 2 + eµp s 2 = F rob e 2 (1+µp s ) = σ 1+µp s .
As the order of σ is 2, if µ is odd, then σ • θ d is the identity and we have -

1 = σ 1 θ d (α)
α = 1 which is not possible because the characteristic of the field is odd. If µ is even, we have σ • θ d = σ. Therefore θ d is the identity, so divide d = 2 p s , which is impossible because p is odd. In what follows, we will give necessary and sufficient conditions for the existence of Hermitian self-dual θ-negacyclic codes over IF p e where p is an odd prime number. Lemma 8. If one of the following conditions is satisfied, then there exists a Hermitian self-dual θ-negacyclic code of dimension k over IF p e :

Existence conditions of

(i) e 2 ≡ 1 (mod 2) and r ≡ 1 (mod 2); (ii) p ≡ 1 (mod 4); (iii) p ≡ 3 (mod 4) and e 2 ≡ 0 (mod 2); (iv) p ≡ 3 (mod 4), e 2 ≡ 1 (mod 2), r ≡ 0 (mod 2) and k ≡ 0 (mod 2 µ-1 ). Proof. Assume that (i) is satisfied, then the restrictions of θ and of σ to IF p 2 are equal to the Frobenius automorphism, therefore according to Theorem 1, there is a monic skew polynomial h in IF p 2 [X; θ] ⊂ R such that σ(h ) • h = X 2k + 1.

Assume that one of the assertions (ii), (iii) or (iv) is satisfied. As the order of θ is equal to e gcd(e,r) , if (iii) is satisfied, we have k × gcd(e, r) ≡ 0 (mod 2) and r × k ≡ 0 (mod 2). Therefore, according to [7, Proposition 4 and Proposition 5 or Table 1], the equation h •h = X 2k +1 has a solution in IF p e/2 [X; θ]. Furthermore, as σ is the identity over IF p e/2 , we get that there exists h in IF p e/2 [X; θ] ⊂ R such that σ(h ) • h = X 2k + 1.

Lemma 9. Assume that p ≡ 3 (mod 4), e 2 ≡ 1 (mod 2), r ≡ 0 (mod 2). If k ≡ 0 (mod 2 µ-1 ), then there is no Hermitian self-dual θ-negacyclic of length n = 2k over IF p e . Proof. Assume that there exists a Hermitian self-dual θ-negacyclic code of length n over IF p e , then H X n +1 = ∅. Consider s ∈ IN, t ∈ IN * not divisible by p such that n = tp s and a in IN * such that 2 a-1 divides exactly k. That means that 2 a-1 divides k but 2 a does not divide k. Furthermore, p ≡ 3 (mod 4), e 2 ≡ 1 (mod 2), r ≡ 0 (mod 2), therefore p and = e gcd(e,r) are odd numbers, and 2 a divides exactly t. As a consequence, we have

X n + 1 = X t + 1 p s = X 2 a + 1 p s F (X ) p s ,
where F (X ) ∈ IF p [X ] is coprime with X

Y 2 a + 1 = f 1 (Y ) • • • f 2 a-1 (Y ),
where

f i = Y 2 + b i Y + 1 is irreducible in IF p [Y ]. Furthermore f i (Y ) = g i g i over IF p 2
where g i = Y + a i . One has g i = Y + 1 a i and σ(g i ) = g i .

Therefore, according to Lemma 7 applied to F 1 (X ) = g i (X )

p s
and F 2 (X ) = (X n + 1)/F 1 (X ), there exists a skew polynomial H i ∈ IF p e [X; θ] such that σ(H i ) • H i = g i (X )

p s = X + a i p s .
The degree of σ(H i ) • H i is even and the degree of X + a i p s is odd, a contradiction. Theorem 3. Consider p an odd prime number, e an even positive integer, r a non-negative integer, k a positive integer and θ the automorphism over IF q = IF p e defined by: a → a p r . There exists a Hermitian self-dual θ-negacyclic code over IF p e of dimension k if, and only if, one of the following conditions is satisfied :

1. p ≡ 1 (mod 4); 2. p ≡ 3 (mod 4) and e 2 ≡ 0 (mod 2); 3. p ≡ 3 (mod 4), e 2 ≡ 1 (mod 2) and r ≡ 1 (mod 2); 4. p ≡ 3 (mod 4), e 2 ≡ 1 (mod 2), r ≡ 0 (mod 2) and k ≡ 0 (mod 2 µ-1 ). Proof. The proof is directly deduced from Lemma 8 and Lemma 9.

Remark 3. Recall that if θ = id (i.e. r = 0), and e = 2, we get Hermitian self-dual negacyclic codes over IF p 2 . According to Theorem 3, there exists a Hermitian self-dual negacyclic code of dimension k over IF p 2 if, and only if, one of the following conditions is satisfied 1. p ≡ 1 (mod 4) 2. p ≡ 3 (mod 4) and k ≡ 0 (mod 2 µ-1 ) where 2 µ ||p + 1. This is equivalent to p + 1 ≡ 0 (mod 2 a+1 ) where 2 a || n and we get the existence conditions given in [START_REF] Yang | On self-dual constacyclic codes over finite fields[END_REF]Theorem 3.9]. Remark 4. If θ is the Frobenius automorphism (i.e. r = 1), then there always exists a Hermitian self-dual θ-negacyclic code of any dimension over IF p e .

Conclusion.

This text gives an overview of the construction of Hermitian self-dual θ-cyclic and θ-negacyclic codes of length n over IF p 2 when θ is the Frobenius automorphism. Existence conditions for these codes over any finite finite field IF q where q is an even power of a prime number are also given. The construction of this family of codes could be generalized over IF p n following [START_REF] Batoul | A construction of self-dual skew cyclic and negacyclic codes of length n over IF p n[END_REF]. Furthermore it happens that one can also consider Hermitian self-dual skew constacyclic codes which are not skew negacyclic nor skew cyclic, on the contrary to Euclidean self-dual skew constacyclic codes which are necessarily skew cyclic or skew negacyclic.

Example 3 .

 3 Consider IF 9 = IF 3 (a) where a 2 + 2a + 2 = 0 and θ the Frobenius automorphism over IF 9 . The monic solutions h ∈ IF 9 [X; θ] to the self-dual skew equation θ

4

  Existence conditions of Hermitian self-dual θ-cyclic and θ-negacyclic codes over IF p e with e > 2 even.

  Hermitian self-dual θ-negacyclic codes over IF p e with e > 2 even.

.

  As k ≡ 0 (mod 2 µ-1 ), we have µ > a, therefore, according to [2, Theorem 1], the factorization of Y 2 a + 1 ∈ IF p [Y ] over IF p into the product of distinct irreducible polynomials is of the form

  2 s over IF 4 . Proposition 4 below gives a formula for the number of Hermitian self-dual θ-negacyclic codes whose dimension is a power of p when p is an odd prime number. Proposition 4. Consider p an odd prime number, s a non-negative integer and θ the Frobenius automorphism over IF p 2 . The number of Hermitian self-dual θ-negacyclic codes of dimension p s over IF p 2 is Algorithm 2 Computation of H X 2p s +1 for p odd prime Require: s Ensure: H X 2p s +1 1: H ← ∅ 2: for i = 0, . . . , p s -1

	2	do
	3:

  [START_REF] Bosma | The Magma algebra system. I. The user language[END_REF] . In what follows we give a formula for the number of Hermitian self-dual θ-negacyclic codes of any dimension over IF p 2 where p is a prime number and θ is the Frobenius automorphism over IF p 2 . When p is even, θ-negacyclic codes are θ-cyclic codes. Theorem 1. Consider p a prime number, θ the Frobenius automorphism over IF p 2 , k a positive integer, s, t two integers such that k = p s × t and p does not divide t. The number of self-dual θ-negacyclic codes over IF p 2 with dimension k is #H

  + 1 does not divide X 2k + 1, #H (X 2 +1) p s otherwise. Therefore, N = 1 if t is even and p odd; otherwise N = #H (X 2 +1) p s is given by Remark 2 and Proposition 4. Furthermore, for f in F ∪ G, #H f p s is given by Proposition 5 and Proposition 7. Computation of H X n +1 (factorization strategy) Require: k Ensure: H X 2k +1 1: Compute s, t such that k = p s × t and p t 2

			and s = 0,
	1		if p = 2 and s > 0 or p odd and t even,
		p s +1
	(p + 1) p	2 p-1 -1	if p odd and t is odd.
	Proof. According to Proposition 1 we have
	#H X 2k +1 = N ×	#H f p s
			f ∈F ∪G,f |X 2k +1,
			deg(f )=2δ
	where
	N = if X 2 Algorithm 5 1

3 :

 3 Using Algorithms 2, 3 and 4, computeH fi(X 2 ) p s 4: H ← ∅ 5: for (h 1 , . . . , h r ) ∈ H f1(X 2 ) p s × • • • × H fr(X 2 ) p s do

	6:

  2 , h 2 ∈ H (X 16 +X 10 +X 8 +X 6 +1) 2 and h 3 ∈ H (X 16 +X 14 +X 12 +X 8 +X 4 +X 2 +1)

The authors benefit from the support of the French government Investissements d'Avenir program integrated to France 2030, bearing the


Noureddine Mhadhbi

Souad Dgachi

Soufienne Belgacem

Ali Ben Ahmed

Natacha Henry

Thierry Loiseau

Samia Nasr

Riadh Badraoui

Houcine Naϊli
email: houcine_naili@yahoo.com

Pr Houcine Naïli

Design, theoretical study, druggability, pharmacokinetics and properties evolution of a new organo-bromocadmate compound as prospective anticancer agent

Keywords: Organic-inorganic complex, X-ray diffraction, vibrational spectra, DFT calculations, optical absorption, Pharmacokinetics, Anticancer effect

Crystals of the organic-inorganic hybrid complex [H3NCH2(C5H4NH)]2CdBr6•2H2O ([H3NCH2(C5H4NH)] = diprotonated 3-picolylamine or 3-ampH2), were grown successfully by slow evaporation solution growth method at room temperature and characterized by singlecrystal X-ray diffraction, infrared absorption, optical absorption and thermal analysis. The title compound belongs to the triclinic space group P1 ̅ , with a crystal structure consisting of inorganic layers built up from hexabromocadmate anions [CdBr6] 4-and free water molecules, linked together by O-H•••Br hydrogen bonds and halogen•••halogen interactions, along the b axis. The organic cations, 3-ampH2, are intercalated between the mineral layers via N-H•••Br hydrogen bonds. The optimized molecular structure and the vibrational spectra were calculated thanks to the Density Functional Theory (DFT) method resorting to the B3LYP function with the LanL2DZ basis set. Infrared spectrum was used to gain more information of the title compound, with the assignment of the observed vibration modes. Good agreement has been found between the calculated results and the experimental data. Subsequently, the optical properties were diagnosed throughout optical absorption explaining the possible charge transfer interactions that happen within the molecules. The thermal behavior was studied by TG/DT analyses. In silico analyses showed that the synthesized compound bound several anticancer receptors (TNF-α, COX-2, VEGF...) with high affinities, which together with molecular interactions, druggability and pharmacokinetics satisfactory explain the anticancer potential of the compound.

Introduction

In recent years, the chemistry of organo-metal halides has developed rapidly and become of significant importance. These compounds are now applied in a wide range of research fields, including catalysis, biochemistry, magnetism, luminescence, nonlinear optical (NLO) and materials science [START_REF] Zhao | Metal Chlorides in Ionic Liquid Solvents Convert Sugars to 5-Hydroxymethylfurfural[END_REF][START_REF] Chatani | Ru(II)-and Pt(II)-Catalyzed Cycloisomerization of ω-Aryl-1-alkynes. Generation of Carbocationic Species from Alkynes and Transition Metal Halides and Its Interception by an Aromatic Ring[END_REF][START_REF] Férey | Microporous Solids: From Organically Templated Inorganic Skeletons to Hybrid Frameworks[END_REF][4][5][START_REF] Protesescu | Nanocrystals of Cesium Lead Halide Perovskites (CsPbX3, X = Cl, Br, and I): Novel Optoelectronic Materials Showing Bright Emission with Wide Color Gamut[END_REF]. These properties emanate from their diverse structures that can be well predicted and designed under the guidance of coordination chemistry and crystal engineering [7][8][9]. Such materials have recently been the interest of many studies due to their attractive potential for application as insulators in the electronics industry. A very important class consists of organo-metal halides. In fact, many compounds of this type have been synthesized such as ([H3NCH2(C5H4NH)] [ZnCl4] [10], (C6N2H10) [CuCl4] [11], (C6N2H10)

[CuBr4] [11],

[Bz-4-NH2Py]4[CdCl4]2•H2O [12], (C8H14N2)2[CdCl6] [13],

(C5H6N2Cl)2[CdBr4]•H2O [14], (C9H24N2)CdCl4 [15], [Bz-4-NH2Py]2[CdBr4] [16]. Hybrid compounds based on cadmium halide have constantly attracted much attention because the Cd 2+ ion with a d 10 configuration show a variety of coordination numbers, geometries and structural flexibility depending on crystal packing and ligands. Besides, the anionic sub-lattice of the crystal may consist of a simple discrete octahedral, one-dimensional chain, 2D or 3D anionic framework [17][18][19]. Although many organo-bromocadmate compounds have been synthesized and characterized, by single-crystal X-ray diffraction, only some of them have been studied theoretically, even though computational analysis has become one of the effective methods to gain information about the electronic, reactivity and thermodynamic properties of this type of compounds. Due to the hybrid structure, these materials have resulted in exceptional intermolecular interactions. The energy difference between the frontier orbitals HOMO and LUMO can be used to predict the strength and stability of metal complexes transition [20].

Considering the attractive characteristics and attributes of halogenocadmate (II) and the new promising opportunities they may open with regard to the development of useful organicinorganic hybrid materials, the present study reports the synthesis and structural characterization of the 3-ammoniumethyl-pyridinium hexabromocadmate (II) dihydrate with the general chemical formula (C6H10N2)2[CdBr6]•2H2O. The characterizations of the grown crystals were achieved throughout X-ray diffraction, spectroscopic techniques (infrared spectrum and optical properties) and thermal studies. The molecular geometry optimization was performed using DFT/B3LYP methods with the LanL2DZ basis set to assist the interpretation and the assignment of experimental IR spectrum. Good agreement has been found between the calculated results and the experimental data. The optical property reveals a noticeable band energy gap stating the stability of the resulting structure. In an attempt to rationalize the possible biological activity of the compound, its bioavailability, pharmacokinetics and molecular interactions with key receptors involved in cancer were also studied using computational modeling.

Experimental

Chemical materials and methods

In this experiment, the following ingredients were used in accordance with their analytical purity, purchased from Sigma Aldrich: CdBr2•4H2O (98 %), 3-picolylamine (C6H8N2; 3-(aminoethyl)pyridine, noted 3-amp) (99%) and hydrobromic acid HBr (48%). The synthesis was carried out using distilled water.

The IR spectrum was measured using a PerkinElmer BX FT-IR spectrometer, in the 400-4000 cm -1 region at ambient temperature.

In order to measure the optical properties at room temperature, a Cary 4000 UV-Visible spectrophotometer will be used which covers a spectrum from 200 nm to 800 nm and uses a Xenon lamp for the UV-Visible range.

Thermal analyses were performed on raw powders with a TGA/DTA 'SETSYS Evolution' (Pt crucibles, Al2O3 as a reference) under air flow (100 ml/min) of (3-ampH2)2CdBr6•2H2O. The thermograms were gathered from 11 mg samples in the RT-600°C range (heating rate: 5°C. min -1 ).

Synthesis of (3-ampH2)2CdBr6•2H2O

The title compound was synthesized by the reaction of 3-picolylamine (2 mmol: 0.216g), cadmium (II) bromide tetrahydrate (1 mmol: 0.344g) and 10 ml of distilled water. The mixture was acidified by concentrated hydrobromic acid (0.080g), under continuous stirring.

The solution was slowly evaporated. After few days some transparent single crystals were carefully isolated under a polarizing microscope for analysis by X-ray diffraction.

X-ray Data Collection

A suitable crystal of (C6H10N2)2[CdBr6]•2H2O was glued to a glass fiber on an APEX II area detector 4-circle diffractometer. Intensity data sets were collected until a resolution of 0.8Å because the intensity beyond was too low using Ag Kα radiation (λ = 0.56086Å) through the Bruker AXS APEX2 Software Suite. The crystal structure was solved in the triclinic system, space group P1 ̅ . Using Olex2 [21], the structure was solved with the olex2.solve [22] structure solution program using Charge Flipping and refined with the ShelXL [23] refinement package using Least Squares minimization with anisotropic thermal parameters for all non-H-atoms. H atoms bonded to C and N atoms were located in a difference map and refined with C-H distance of 0.93(2) Å and N-H distance of 0.91(2) Å. The drawings were made with DIAMOND program [START_REF] Brandenburg | Diamond Version 3.2i. Crystal Impact GbR[END_REF].

Computational details

Gaussian 09 software was used to perform optimizations and calculations of vibrational wavenumbers using molecular geometry and DFT [START_REF] Frisch | GAUSSIAN 09, Revision A.02[END_REF]. The geometry optimizations and frequency calculations of (3-ampH2)2CdBr6•2H2O were carried out using the Becke's threeparameter hybrid exchange functional (B3) [START_REF] Ad | Density-functional thermochemistry. IV. A new dynamical correlation functional and implications for exact-exchange mixing[END_REF] combined with the Lee-Yang-Parr correlation functional (LYP) [START_REF] Lee | Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density[END_REF] in conjunction with the LanL2DZ basis set [START_REF] Pj | Ab initio effective core potentials for molecular calculations. Potentials for K to Au including the outermost core orbitals[END_REF]. DFT is used to optimize a molecule's geometry by rearranging its atoms in order to obtain a structure that is most stable and with the lowest possible ground state energy. The calculated vibrational wavenumbers were assigned with the aid of the GaussView 05 program [29] employed to generate visual presentations and verify the normal mode assignments. These properties were determined as highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO).

Druggability, pharmacokinetics and anticancer biological activity

Prediction of druggability and pharmacokinetics of the synthesized and characterized compounds was performed based on the absorption, distribution, metabolism, excretion and toxicity (ADMET) properties as previously reported [START_REF] Badraoui | Chloroquine and hydroxychloroquine interact differently with ACE2 domains reported to bind with the coronavirus spike protein: mediation by ACE2 polymorphism[END_REF][START_REF] Hchicha | A novel sulfate-bridged binuclear copper (II) complex: Structure, optical, ADMET and in vivo approach in a murine model of bone metastasis[END_REF][START_REF] Mhadhbi | Physico-Chemical Properties, Pharmacokinetics, Molecular Docking and In-Vitro Pharmacological Study of a Cobalt (II) Complex Based on 2-Aminopyridine[END_REF] The anticancer potential was carried out by measuring both binding scores and molecular interactions of the synthesized compound using some key receptors. For this end, TNFα, COX-2, CDK6, VEGF and KRAS were targeted as receptors. Both chemical compound and receptors were processed then subjected to a CHARMm force field for the complex formations as previously published reports [START_REF] Hchicha | A novel sulfate-bridged binuclear copper (II) complex: Structure, optical, ADMET and in vivo approach in a murine model of bone metastasis[END_REF][START_REF] Zammel | Antioxi-dant and Anti-Inflammatory Effects of Zingiber officinale roscoe and Allium subhirsutum: In Silico, Biochemical and Histological Study[END_REF] (Hchicha et al., 2021; Zammel 2021).

Results and Discussion

X-ray diffraction and geometric optimization

The which confirms well that protonation also takes place on the pyridine ring N atom [START_REF] Hamdi | Design, synthesis and physico-chemical studies of a Co(II)/Co(III) mixed-valence complex: an experi-mental and DFT approach[END_REF][START_REF] Mhadhbi | Crystal structure, spectroscopy, DFT studies and thermal characterization of Cobalt (II) complex with 2-protonated aminopyridinium cation as ligand[END_REF][START_REF] Mhadhbi | Experimental and DFT characterization of the organic-inorganic monohydrated Co (II) complex with 2,6-di-aminopyridine ligand,(C5H8N3) 2 [CoBr4]•H2O[END_REF][START_REF] Mhadhbi | Structural characterization and physicochemical features of a new arsenate salt templated by mono and di-protonated 4-aminopyridine cations: (C5H7N2)(C5H8N2)[AsO4]•H2O[END_REF][START_REF] Hamdi | Experimental and theoretical investigations on structural-function relationship of new iron (III) complex with 2-(Ammoniomethyl)pyridinium cation as ligand: A promising material for green solar cells[END_REF].

The selected bond distances and angles are illustrated in Table 2. As observed, the calculated geometric parameters represent a good approximation and can be used as a base to calculate the other parameters for the title compound. The structure shows various hydrogen bonds, as given in Fig. 2. Indeed, the [CdBr6] 4-anionic species are strongly attached to organic molecules. The N-H•••Br hydrogen bonds link organic cations and inorganic anions to give alternatively arranged organic-inorganic layers parallel to the (ac) plane (Fig. 2). Therefore, the water molecules are located above and below the [CdBr6] 4-octahedral and tied between organic and inorganic molecules via hydrogen bonds. In addition, the presence of hydrogen bonds between organic and inorganic parts plays an important role in the stability of the structure. In these hydrogen-bonded interactions, the N•••Br distances vary from 3.362(3) to 3.607(3) Å and the angles range from 126° to 172° (Table 3). The optimized geometry of the title compound model is presented in (Fig. 1(b)). The calculated geometrical parameters of the title compound are reported and compared with the experimental ones in Table 2, in which the average relative error of almost 1.82% for the inorganic structural parameters while it does not exceed 0.9% for the organic parameters. These discrepancies can be explained by the fact that the calculation relates to the isolated molecule where the intermolecular Coulombic interaction with the neighboring molecules is absent, whereas the experimental result corresponds to interacting molecules in the crystal lattice. This result does not only support our cluster choice and calculation results, but also indicates that the calculation precision is satisfactory and the B3LYP/LanL2DZ level is suitable for the complex.

Infrared spectroscopy

Fig. 3 focuses on the calculated (blue) and experimental (red) IR spectra of the title compound.

The most intense IR absorption bands were dealt with in Table 4. Asymmetric and symmetric O-H (H2O) stretching are observed at 3424 and 3409 cm -1 , respectively. The observed band corresponding to the N-H bending vibration is observed at 3064 cm -1 . The corresponding calculated value appear at 3076 cm -1 . The bands appearing at 3043 and 2925 cm -1 are assignable to asymmetric and symmetric CH2 stretching modes, respectively. The corresponding theoretical value is situated at 3079 and 3019 cm -1 . The bands identified at 1729 cm -1 can be assigned to ring deformation and the H-N-H in-plane bending appeared at 1627 cm -1 . They are theoretically predictable at 1739 and 1651 cm -1 , respectively. The band appeared at 1549 cm -1 results from the deformation CH2. As it can be seen in Fig. 3, the bands observed at 1467 cm -1 and 1425 cm -1 are assigned to the in-plane bending of H-C-C and H-N-C as well as torsion of H-N-H-Cl, respectively. Their calculated frequencies are found in the range from 1481 to 1443 cm -1 . The H-C-C-N torsion is reported also experimentally, at 1390 cm -1 and theoretically, at 1406 cm -1 . The vibrations modes between 1379 and 1317 cm -1 in the experimental spectrum refer to N-C stretching, in plane bending H-C-N which are modes predicted at 1261 cm -1 in the calculated spectrum. The stretching vibrations C-C are also observed within the range of 1225 and 1070 cm -1 for the experimental spectra. The calculated frequencies between 1239 and 1087 cm -1 are assigned to the last mode. The H-C-C-H torsion appears in the experimental spectrum as a peak located at 1050 cm -1 . This mode was provided theoretically at 1063 cm -1 . The band identified at 793 cm -1 can be assigned to the torsion H-C-C-H. Its theoretical value is located at 805 cm -1 . The band appearing at 494 cm -1 is allocated to the H-C-C-C torsion. Finally, the band that appears at 406 cm -1 is attributed to the C-C-C-C torsion. The DFT calculation provides this mode at 417 cm -1 . A detailed assignment of the observed bands is displayed in Table 4.

Mulliken atomic charges

The charge distribution of the title compound (Fig. 4) shows that the cadmium atom attached with bromide atoms and the carbon atoms attached with nitrogen atoms are positive. The Bromide atoms have more negative charges whereas all hydrogen atoms have a positive charge. The maximum atomic charge is obtained for Cd when compared with other atoms. This is due to the attachment of negatively charged on bromide (Br) atoms. Negatively charged brome atoms shows that charge is transferred from Cd to Br. The calculated Mulliken charges for Cd (0.565) and Br (-0.289/ -0.010 / -0.508483 / -0.492544 / -0.443283 / -0.25847) taking part in the intramolecular charge transfer is revealed in the optical absorption. Nitrogen atoms are positively charged which shows the charge transfer from Br to N through the hydrogen bridge.

Optical characteristics

UV-visible spectra analysis

The measured and simulated optical UV-Vis absorption spectra of the (3-ampH2)2CdBr6•2H2O compound are shown in Fig. 5. The UV-Vis spectrum exhibits four distinct absorption bands centered at 242 𝑛𝑚 (5.12 𝑒𝑉), 227 𝑛𝑚 (5.46 𝑒𝑉), 215 𝑛𝑚 (5.76 𝑒𝑉)

and 200 𝑛𝑚 (6.2 𝑒𝑉). The strong absorption band centered at 242 𝑛𝑚 should be allocated to the 𝜋→𝜋 * transitions with the ring in the organic cation. This peak appears in the spectrum of the cation alone at 292 nm (4.24 eV). While, the average absorption band centered at 227 𝑛𝑚 should be assigned to the charge transferred from organic cation to inorganic anion, this band appears in the simulated spectrum at 237 nm (5.23 eV). The calculated electronic transition (TDDFT) is in very good agreement with the experimental findings. The absorption bands centered at 215 𝑛𝑚 and 200 nm should be accredited to the n→𝜋 * transitions with the ring in the organic cation.

Global reactivity descriptors and energy gap analysis

Analysis of density functional theory descriptors gives additional information about the characteristics of stability, electrophilicity, and nucleophilicity. Therefore, the global reactivity descriptors were calculated based on the DFT method with the B3LYP/LAN2DZ basis set. The quantum chemical parameters calculated using TDDFT with B3LYP/LAN2DZ functional, molecular orbital energies (HOMO and LUMO), energy gap (∆𝐸 𝐻-𝐿 ), Electron Affinity (𝐸 𝐴 ), Ionization Potential (𝐼 𝑃 ), global hardness (𝜂), chemical potential (𝜇), global softness (𝜎), electronegativity (𝜒) and electrophilicity index (𝜔) are listed in Table 5. All relationships connecting these parameters with HOMO and LUMO energies, termed as global reactivity descriptors are given by the following equations [40]:

∆𝐸 𝐻-𝐿 = 𝐸 𝐿𝑈𝑀𝑂 -𝐸 𝐻𝑂𝑀𝑂 (1)
𝐼 𝑃 = -𝐸 𝐻𝑂𝑀𝑂 ( 2)

𝐸 𝐴 = -𝐸 𝐿𝑈𝑀𝑂 (3) 𝜒 = - 𝐸 𝐿𝑈𝑀𝑂 + 𝐸 𝐻𝑂𝑀𝑂 2 (4) 𝜇 = 𝐸 𝐿𝑈𝑀𝑂 + 𝐸 𝐻𝑂𝑀𝑂 2 (5) 𝜂 = 𝐸 𝐿𝑈𝑀𝑂 -𝐸 𝐻𝑂𝑀𝑂 2 (6) 𝜎 = 1 𝜂 (7) 𝜔 = 𝜇 2 2𝜂 ( 8
)
The chemical hardness and softness is related to the stability of a given coumpound.

Molecules having low energy gap are called as soft molecules which are polarizable, whereas those having high energy gap are called as hard molecules which are less polarizable (nonpolar). From the Table 5, it is clear that (3-ampH2)2CdBr6•2H2O exhibits a larger energy gap

Hyperpolarizability calculations

The NLO effect is extremely important since it plays a crucial role in various signals conditioning processes such as frequency shifting, optical modulation, optical switching, and optical interactions [START_REF] Nalwa | Handbook "Organic Conductive Molecules and Polymers[END_REF][START_REF] Chemla | Nonlinear Optical Properties of Organic Molecules and Crystals[END_REF][START_REF] Shen | The Principles of Nonlinear Optics[END_REF][START_REF] Rodrigues | [END_REF]. The calculation of the second-order hyperpolarizability and its components are very informative in expressing the charge delocalization and a measure of nonlinear optical effects in a compound. In this study, the total dipole moment, average linear polarizability, and the second molecular hyperpolarizability are given by using the following equations [45][46][47][48]:

𝜇 𝑡𝑜𝑡 = √𝜇 𝑥 2 + 𝜇 𝑦 2 + 𝜇 𝑧 2 (9)
𝛼 𝑡𝑜𝑡 = 𝛼 𝑥𝑥 + 𝛼 𝑦𝑦 + 𝛼 𝑧𝑧 3 (10)

〈𝛾〉 = 1 5 (𝛾 𝑥𝑥𝑥𝑥 + 𝛾 𝑦𝑦𝑦𝑦 + 𝛾 𝑧𝑧𝑧𝑧 + 2(𝛾 𝑥𝑥𝑦𝑦 + 𝛾 𝑥𝑥𝑧𝑧 + 𝛾 𝑦𝑦𝑧𝑧 )) (11)
Since the values of the Polarizability 𝛼 𝑡𝑜𝑡 , and the second Hyperpolarizability 〈𝛾〉 of Gaussian 09 output are reported in atomic units (𝑎𝑢), the calculated values have been converted in to electrostatic units (𝑒𝑠𝑢): 𝛼 𝑡𝑜𝑡 : 1 𝑎𝑢 = 0.1482 × 10 -24 𝑒𝑠𝑢, 〈𝛾〉: 1 𝑎𝑢 = 0.50367 × 10 -39 𝑒𝑠𝑢. As can be see Based on the above facts, it could be proposed that this material can be better placed for nonlinear optical applications.

Thermal behavior of (3-ampH2)2[CdBr6]•2H2O

The thermal stability of the examined crystal was studied by means of a simultaneous thermogravimetric analysis (TGA) and differential thermal analysis (DTA), carried out with a heating rate of 5 °C.min -1 between 25 and 600 °C (Fig. 7). The first endotherm peak at 77 °C, starts at 35 °C and ends at 110 °C, and is accompanied by a weight loss of about 4.06%. The experimental loss in mass corresponds to a value which is calculated for the removal of two water molecules (theoretical 4.24%), thus leading to the anhydrous phase (3-ampH2)2[CdBr6].

The crystal seems to be stable up to about 250 °C and above this temperature a continuous decomposition of the sample takes place. The endothermic peak characterized by a weight loss of 69.77% observed on the TGA curve in the range 250-470 °C, which can be attributed to the destruction of the organic entities (weight loss: calculated 69.77%, experimental 71.14%). (3-ampH2)2[CdBr6] is not stable and decompose, probably, to CdBr2 and the bromine derivatives. The ultimate change starts at about 500 °C and corresponds to the formation of cadmium oxide CdO. and CDK6 with unit 1 and unit 2 of the compound, respectively (Fig. 9). Nevertheless, interaction types with COX-2 were much better. In fact, the latter established one electrostatic bond with Glu465 and four conventional H-bonds with Asn39, Cys41, Asn39 and Gln461, which were additionally, enforced by one carbon H-bond and three hydrophobic (alkyl) bonds. Furthermore, the was found to be deeply embedded within the pocket region of each selected receptor. Here again, the deeper embedding distance was found in the compound complexed to COX-2. Taken together, these molecular interactions contribute certainly to the stability of the complex and satisfy the possible anticancer effect of the (3-ampH2)2[CdBr6]•2H2O compound.

Druggability, pharmacokinetics and anticancer biological activity

Conclusion

In a nutshell, a new organic-inorganic hybrid compound (C6H10N2)2[CdBr6]•2H2O has been successfully synthesized at room temperature by slow evaporation. This compound belongs to the triclinic system with the P1 ̅ space group. The atomic arrangement can be described as an

  [START_REF] Badraoui | Chloroquine and hydroxychloroquine interact differently with ACE2 domains reported to bind with the coronavirus spike protein: mediation by ACE2 polymorphism[END_REF][START_REF] Hchicha | A novel sulfate-bridged binuclear copper (II) complex: Structure, optical, ADMET and in vivo approach in a murine model of bone metastasis[END_REF][START_REF] Mhadhbi | Physico-Chemical Properties, Pharmacokinetics, Molecular Docking and In-Vitro Pharmacological Study of a Cobalt (II) Complex Based on 2-Aminopyridine[END_REF]. Likewise, the oral bioavailibility was studied based on the physicochemical structure of the complex[START_REF] Hchicha | A novel sulfate-bridged binuclear copper (II) complex: Structure, optical, ADMET and in vivo approach in a murine model of bone metastasis[END_REF][START_REF] Mhadhbi | Physico-Chemical Properties, Pharmacokinetics, Molecular Docking and In-Vitro Pharmacological Study of a Cobalt (II) Complex Based on 2-Aminopyridine[END_REF][START_REF] Badraoui | Expression profilimg of selected immune genes and trabecular architecture in breast cancer skeletal metastases model: effect od -tocopherol acetate supplementation[END_REF] [START_REF] Badraoui | Chloroquine and hydroxychloroquine interact differently with ACE2 domains reported to bind with the coronavirus spike protein: mediation by ACE2 polymorphism[END_REF][START_REF] Badraoui | Expression profilimg of selected immune genes and trabecular architecture in breast cancer skeletal metastases model: effect od -tocopherol acetate supplementation[END_REF].

( 3 -

 3 ampH2)2CdBr6•2H2O compound crystallizes in the triclinic P1 ̅ space group with four formula units in unit cell (Z = 1) with the following lattice parameters: a = 7.993(3) Å, b = 8.139(3), c = 10.092(4) Å, α = 90.735 (14)°, β = 95.964(3)° and γ = 112.451 (12)°. Table 1 best summarizes the key crystallographic information for the title compound. The basic unit of the studied material, Fig.1(a), is made up of one inorganic [CdBr6] 4-anion, one diprotonated cation, 3-ammoniumethyl-pyridinium, [C6H10N2] 2+ or (3-ampH2) 2+ ) and one uncoordinated water molecule, all of which lie in general positions. The cohesion of this structure is established by N/C-H ••• Br hydrogen bonds, giving rise to a three-dimensional network (Fig.1(a)). The [CdBr6] 4-anions exhibit a slightly distorted octahedral arrangement around the Cd 2+ center, with Cd-Br bond lengths in the range 2.7464(9) -2.9002(9) Å. The Br-Cd-Br angles fall in the range 88.99(3) to 180° (Table 2). Within the deprotonated amine molecule, the C-C and N-C distances range from 1.358(4) to 1.494(5) Å and from 1.334(4) to 1.481(4) Å, respectively. The C-N-C angle is equal to 123.2 (3) °. Pyridinium cations typically possess an expanded C-N-C angle in comparison with the parent pyridine [119°],

( 3 .

 3 56 eV). Its value denoted as hard and less polarizable (non-polar) molecule. For atoms, electrons include atomic orbitals, but for molecules they occupy similar molecular orbitals which are surrounded by the molecule, called frontier molecular orbitals (FMOs). The chemical activity of the molecule is reflected through the energies of the FMOs. The electron donating nature is expressed by the highest occupied molecular orbital (HOMO) which has a physical significance, which means that it has an energy value equal to the first ionization potential and the electron withdrawing nature is expressed by the lowest unoccupied molecular orbitals (LUMO) with the energy (completely or partly vacant) of a molecular entity, and has a tendency to attract electrons. These orbitals play a fundamental role in determining electric and optical properties of the compound. The molecular orbitals mapped surface with charged electron densities of HOMO and LUMO are observed in the CdBr6 anion. The bandgap obtained from the frontier molecular orbital calculation is 3.56 𝑒𝑉, which is the distance between the HOMO (-0.12 𝑒𝑉) and LUMO (-3.68 𝑒𝑉), respectively. The HOMO and LUMO plots of (3-ampH2)2CdBr6•2H2O are shown in Fig.6. The transition of electrons from ground state to first excited state due to the absorption by means of one electron excitation from the HOMO to LUMO has been known from the wave function analysis. The DOS diagram of (3-ampH2)2CdBr6•2H2O is shown in Fig.6. It is well known that chemical stability of a molecule is largely affected by the frontier orbitals. The electron donating capability of a molecule can be explained by HOMO-LUMO energy. Thus, it is assumed that the energy gap between the HOMO and LUMO reveals the reactivity of the molecule. The NLO responses can be understood by examining the energies of frontier molecular orbitals. There is an inverse relationship between hyperpolarizability and HOMO-LUMO.

  alternation of organic/inorganic layers. The cohesion and the stability of the atomic arrangement of the title compound stem from the establishment of N-H•••Br and O-H•••Br hydrogen bonds between the [C6H10N2] 2+ cations, the water molecules and the [CdBr6] 4- anions. The vibrational FT-IR spectrum of (C6H10N2)2[CdBr6]•2H2O was recorded and the vibrational frequencies were precisely assigned with the aid of the theoretical calculations at B3LYP/LanL2DZ level. The experimental and theoretical results were found to support each other. The optical properties were investigated by optical absorption, showing four bands at 200, 215, 227 and 242 nm. The HOMO and LUMO energy gap value explain the eventual charge transfer interactions that took place in the molecule. The Mulliken atomic charges describe the charge distribution of every atom in the molecule. Moreover, the thermal analysis results confirm the presence of two water molecules per formula in addition to an absence of any phase transition in the studied range of temperature. The physico-chemical characteristics and the properties evolution of the compound may explain the potential anticancer effect, which may result from the bioavailability, pharmacokinetics and the established molecular interactions with some cancer targeted receptors.

Table 6

 6 , the calculated dipole moment is equal to 19.6 D (Debye).The highest value of dipole moment is observed for component µ 𝑦 . In this direction, this value is equal to 17.14 D. For direction x and z, these values are equal to 8.40 D and 1.58 D respectively. The calculated Polarizability 𝛼 𝑡𝑜𝑡 , is equal to 23.56 × 10 -24 𝑒𝑠𝑢. The calculated Polarizability 𝛼 𝑖𝑗 have non-zero values and was dominated by the diagonal components. The second Hyperpolarizability value 〈𝛾〉 of our compound is equal to 2.06 × 10 -36 𝑒𝑠𝑢. The second Hyperpolarizability is dominated by the diagonal components of 𝛾 𝑥𝑥𝑥𝑥 and 𝛾 𝑦𝑦𝑦𝑦 . The intensive presence of particular components reveals a substantial delocalization of charges in these directions. In directions of 𝛾 𝑧𝑧𝑧𝑧 , 𝛾 𝑥𝑥𝑦𝑦 , 𝛾 𝑥𝑥𝑧𝑧 and 𝛾 𝑦𝑦𝑧𝑧 , the values of components are relatively medium. In directions of 𝛾 𝑥𝑥𝑥𝑦 , 𝛾 𝑦𝑦𝑦𝑧 and 𝛾 𝑥𝑥𝑥𝑧 the components values are weak. In other directions, the particular components are practically low. The non-zero 〈𝛾〉 value is probably due to the charge transfer between the organic cation and the CdBr6 anion which facilitates the delocalization of the electronic charge density.

Table 7

 7 exhibits the major criteria, which are commonly used for screening new drugs to avoid any possible failure, particularly during the clinical phase[START_REF] Badraoui | Chloroquine and hydroxychloroquine interact differently with ACE2 domains reported to bind with the coronavirus spike protein: mediation by ACE2 polymorphism[END_REF][START_REF] Hchicha | A novel sulfate-bridged binuclear copper (II) complex: Structure, optical, ADMET and in vivo approach in a murine model of bone metastasis[END_REF] [START_REF] Badraoui | Chloroquine and hydroxychloroquine interact differently with ACE2 domains reported to bind with the coronavirus spike protein: mediation by ACE2 polymorphism[END_REF][START_REF] Hchicha | A novel sulfate-bridged binuclear copper (II) complex: Structure, optical, ADMET and in vivo approach in a murine model of bone metastasis[END_REF][START_REF] Zammel | Antioxi-dant and Anti-Inflammatory Effects of Zingiber officinale roscoe and Allium subhirsutum: In Silico, Biochemical and Histological Study[END_REF]. It showed the druggability and the pharmacokinetic properties of the synthesized compound. It was predicted that both compound units obey Lipinski rule of five as they showed no violations. While presenting low to moderate skin permeability (as assessed by Log Kp), they possessed good bioavailability scores (0.55) that was confirmed further by the bioavailability radars (Fig.8). previously reported by similar docking scores[START_REF] Hchicha | A novel sulfate-bridged binuclear copper (II) complex: Structure, optical, ADMET and in vivo approach in a murine model of bone metastasis[END_REF][START_REF] Badraoui | Expression profilimg of selected immune genes and trabecular architecture in breast cancer skeletal metastases model: effect od -tocopherol acetate supplementation[END_REF][START_REF] Zammel | Antioxi-dant and Anti-Inflammatory Effects of Zingiber officinale roscoe and Allium subhirsutum: In Silico, Biochemical and Histological Study[END_REF] (Zammel et al., 2021; Hchicha et al., 2021; Badraoui et al., 2022). The best docking score was predicted for COX-2

	None of the subunits was found to be blood-brain-barrier permeant or P-glycoprotein
	substrate (P-gp), which permit to deduce not only the absence of any disruptions regarding the
	physiological drug transportation but also the safety on nervous tissues, as confirmed by
	previous similar results [30,49] (Badraoui et al., 2022; Zammel et al., 2022). Regarding
	cytochrome P450 (CYP), the major isoforms were investigated to check the compound
	metabolism within the body. In this context, only CYP3A4 was found to be inhibited, but no
	inhibition was predicted to the CYP1A2, CYP2C9, CYP2C19 and CYP2D6, which have been
	reported to interfere with more than 80% of the drugs and toxicants [49-51] (Zammel et al.,
	2022; Allouche et al., 2022; Akacha et al., 2020).
	In this study, five major macromolecules, which are commonly involved in the pathogenesis
	of cancer, its development and metastatic process [33] (Badraoui et al., 2022), were targeted
	and simulated. Hence, the network of bonds of interactions were studied. As shown in table 8,
	the compound subunits exhibited negative binding energies with all the selected

macromolecules. The obtained energies ranged between ̶ 6.0 and ̶ 3.2 kcal.mol -1 . Thus, the anticancer effect of the (3-ampH2)2[CdBr6]•2H2O compound sounds thermodynamically possible, as

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through a research groups program under Grant Number R.G.P. 2/213/43.

Appendix

Supplementary crystallographic data for this article in CIF format are available as Electronic Supplementary Publication from Cambridge Crystallographic Data Centre (CCDC This data can be obtained free of charge via http://www.ccdc.cam.ac.uk/conts/retrieving.html, or from the Cambridge Crystallographic Data Centre, 12 Union Rood, Cambridge CB2 1EZ, UK (Fax: (international): +44 1223/336 033; e-mail: deposit@ccdc.cam.ac.uk).

Conflicts of Interest

The authors declare no competing financial interest.