Minjia Shi

Xiaoxiao Li
email: ahulixiaoxiao@163.com

Zahra Sepasdar
email: zsepas-dar@yahoo.com

Patrick Solé
email: patrick.sole@telecom-paristech.fr

Polycyclic codes as invariant subspaces

Keywords: Polycyclic codes, cyclic codes, BCH bound, invariant subspaces MSC (2010) : Primary 94B15, Secondary 11A15

whether they are published or not. The documents may come

Introduction

Polycyclic codes of length n over a finite field F can be described as ideals in the quotient ring of F [x]/(f ), where f is some polynomial of degree n. They reduce to cyclic codes when f = x n -1 and to constacyclic codes when f = x n -a, for some a ∈ F * . They have been known for a long time under the name of pseudo-cyclic codes [START_REF] Peterson | Error-Correcting Codes[END_REF]. They received a new name in [START_REF] Lopez-Permouth | Dual generalizations of the concept of cyclicity of codes[END_REF], and a renewed interest in [START_REF] Alahmadi | On the duality and the direction of polycyclic codes[END_REF], where their algebraic structure is studied in great detail. Replacing F by a finite chain ring is considered in [START_REF] Lopez-Permouth | Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes[END_REF][START_REF] Martinez-Moro | On polycyclic codes over a finite chain ring[END_REF]. In parallel, an algebraic approach to quasi-twisted and constacyclic codes was developed in [START_REF] Radkova | Cyclic codes and quasi-twisted codes: an algebraic approach[END_REF][START_REF] Radkova | Constacyclic codes as invariant subspaces[END_REF]. The idea is to consider the codes in the class under scrutiny as invariant subspaces under the action of an endomorphism; this allows the machinery from linear algebra to bear on the problem [START_REF] Hoffman | Linear Algebra[END_REF]. In the cases considered in [START_REF] Radkova | Cyclic codes and quasi-twisted codes: an algebraic approach[END_REF][START_REF] Radkova | Constacyclic codes as invariant subspaces[END_REF], the said endomorphism is the analogue of the cyclic shift of cyclic codes. For another aplication of this linear algebra method to Coding Theory see [START_REF] Gueneri | On the additive cyclic structure of quasi-cyclic codes[END_REF].

In the present paper, we apply the latter approach to the study of polycyclic codes. The transform mentioned before and called here the polyshift admits for matrix the companion matrix of the polynomial f (x). A notion of minimal invariant subspace is introduced, driven by the factorization of f (x) into irreducible factors over the base field. This allows to describe any invariant subspace as a direct sum of its intersections with these spaces. The first benefit of this decomposition is an analogue of the generator polynomial and of the check polynomial of cyclic codes. Similarly, an analogue of the idempotent of cyclic codes is developed. Last but not least, a bound on the minimum distance is derived that outperforms, in some cases, the BCH-like bound of [START_REF] Li | Pseudo-cyclic codes and the construction of quantum MDS codes[END_REF].

The material is arranged as follows. The next section develops the approach to polycyclic codes as invariant subspaces of the polyshift, and introduces the analogues of generator and check polynomials of cyclic codes. Section 3 derives a primitive idempotent decomposition of polycyclic codes. Section 4 is dedicated to a bound on the minimum distance which improves, in some cases the BCH-like bound of [START_REF] Li | Pseudo-cyclic codes and the construction of quantum MDS codes[END_REF]. Section 5 concludes the paper and points out some challenging open problems.

Linear polycyclic codes as invariant subspaces

Let F = F q be a finite field of order q and F n be the n-dimensional vector space over F with the standard basis e 1 = (1, 0, . . . , 0), e 2 = (0, 1, . . . , 0), . . . , e n = (0, 0, . . . , 1). E is the identify matrix. Let c = (c 0 , c 1 , ..., c n-1 ) be a codeword of C with c 0 = 0 and let T c denote the polyshift defined as:

   F n → F n , (x 1 , x 2 , . . . , x n ) → x n c + (0, x 1 , . . . , x n-1 ).
Then T c ∈ Hom(F n ) and it has the following matrix:

A(n, c) = A =            0 0 0 • • • 0 0 c 0 1 0 0 • • • 0 0 c 1 0 1 0 • • • 0 0 c 2 . . . . . . . . . . . . . . . . . . . . . 0 0 0 • • • 1 0 c n-2 0 0 0 • • • 0 1 c n-1            .
We observe the relations

A -1 (n, c) = A -1 =               -c 1 c 0 1 0 0 • • • 0 0 -c 2 c 0 0 1 0 • • • 0 0 -c 3 c 0 0 0 1 • • • 0 0 . . . . . . . . . . . . . . . . . . . . . -c n-2 c 0 0 0 0 • • • 1 0 -c n-1 c 0 0 0 0 • • • 0 1 -1 c 0 0 0 0 • • • 0 0               . The characteristic polynomial of A is f A (x) = f (x) =            -x 0 0 • • • 0 0 c 0 1 -x 0 • • • 0 0 c 1 0 1 -x • • • 0 0 c 2 . . . . . . . . . . . . . . . . . . . . . 0 0 0 • • • 1 -x c n-2 0 0 0 • • • 0 1 c n-1 -x            = (-1) n+1 (c 0 + c 1 x + c 2 x 2 + • • • + c n-2 x n-2 + c n-1 x n-1 -x n ).
Note that this polynomial is, up to sign, the same f as in the Introduction [START_REF] Alahmadi | On the duality and the direction of polycyclic codes[END_REF].

Let f (x) = (-1) n+1 f p 1 1 (x)f p 2 2 (x) • • • f pt t (x)
be the factorization of f (x) into irreducible factors over F . By the theorem of Cayley-Hamilton, we have

f (A) = (-1) n+1 f p 1 1 (A)f p 2 2 (A) • • • f pt t (A) = 0.
Next, we consider the homogeneous set of equations f p i i (A)x = 0, x ∈ F n for i = 1, . . . , t. If U i stands for the solution space of this homogeneous set of equations, then we may write U i = Ker(f p i i (T c )), and consider the homogeneous set of equations f i (A)x = 0, x ∈ F n for i = 1, . . . , t. If V i stands for the solution space of this homogeneous set of equations, then we may write V i = Ker(f i (T c )). Now, we need the following well-known fact [3, p.200

]. Proposition 2.1. Let V be a T c -invariant subspace of U . Then f Tc|U i (x) divides f Tc (x).
The general properties of minimal invariant subspaces can be summarized as follows.

Theorem 2.2. The subspaces U i and V i of F n satisfy the following conditions:

(1)

V i ⊆ U i , U i and V i are T c -invariant subspace of F n ; (2) if W is a T c -invariant subspace of F n and W i = W ∩ U i for i = 1, . . . , t, then W i is T c -invariant and W = W 1 ⊕ W 2 ⊕ • • • ⊕ W t ;
(3)

F n = U 1 ⊕ U 2 ⊕ • • • ⊕ U t ; (4) dim F U i = deg f p i i (x) = k i p i , where degf i (x) = k i ; (5) f Tc|U i (x) = (-1) k i p i f p i i (x); (6) V i is a minimal T c -invariant subspace of F n . Proof. (1) Let u ∈ U i , i.e. f p i i (A)u = 0, then f p i i (A)T c (u) = f p i i (A)Au = Af p i i (A)u = 0, so that T c (u) ∈ U i . (2) Let f p i i (x) = f (x) f p i i (x) for i = 1, . . . , t. Since ( f1 (x), f2 (x), . . . , ft (x)) = 1, there are poly- nomials a 1 (x), . . . , a t (x) ∈ F [x] such that a 1 (x) f1 (x)+a 2 (x) f2 (x)+• • •+a t (x) ft (x) = 1. Then for any w ∈ W , w = a 1 (A) f1 (A)w + a 2 (A) f2 (A)w + • • • + a t (A) ft (A)w. Let w i = a i (A) fi (A)w, then f i (A) p i w i = a i (A)f (A)w = 0 and w i ∈ U i W . Hence W = W 1 + W 2 + • • • + W t . Assume that w ∈ W i j =i W j , then f p i i (A)w = 0, f p i i (A)w = 0. Since (f p i i (x), f p i i (x)) = 1, there are polynomials a(x), b(x) ∈ F [x], such that a(x)f p i i (x) + b(x) f p i i (x) = 1. Hence a(A)f p i i (A)w + b(A) f p i i (A)w = w = 0, so that w ∈ W i j =i W j = {0}. Therefore W = W 1 W 2 ... W t .
(3) This follows from 2) with W = F n .

(4) Let k ≥ 1 be the smallest positive integer with the property that the vectors u, T c (u), . . . , T k c (u) are linearly dependent for all u ∈ U i . Then there exist elements a 0 , a 1 , . . . , a k-1 ∈ F such that

T k c (u) = a 0 u + a 1 T (u) + • • • + a k-1 T k-1 c (u). (1)
Consider the polynomial t

(x) = x k -a k-1 x k-1 -• • • -a 0 x 0 ∈ F [x]
. Thus by Eq. ( 1)

t(T c )(u) = T k c (u) -a k-1 T k-1 c (u) -• • • -a 0 T 0 c (u) = 0.
On the other hand, since u ∈ U i , we have

f p i i (T c )(u) = 0. Hence t(T c )(u) = f p i i (T c )(u) = 0. It follows that: [(t(x), f p i i (x))(T c )](u) = 0. (2)
Since

f i (x) is an irreducible polynomial, gcd(t(x), f p i i (x)) = 1 or f j i (x) for j ≤ p i . By Eq. (2), gcd(t(x), f p i i (x)) = f j i (x) for j ≤ p i . Thus f j i (T c )(u) = 0 for j ≤ p i . First, we show that j = p i . Let W = Ker(f j i (T c )). It is clear that W ⊆ U i . On the other hand, from u ∈ U i , we get that f j i (T c )(u) = 0, u ∈ W . Because u is an arbitrary element of U i , we conclude that U i ⊆ W and so W = U i . Therefore (t(x), f p i i (x)) = f p i i (x). So f p i i (x) divides t(x) and this means that k i p i = deg(f p i i ) ≤ deg(t(x)) = k. Now, from f p i i (T c )(u) = 0 we get that u, T c (u), . . . , T k i p i c (u) are linearly dependent. By minimality of k, we get k ≤ k i p i . Hence k = k i p i . It is clear that dim U i ≥ k = k i p i , then we have t i=1 dim U i = dim F F n = n = deg(f ) = t i=1 deg(f p i i ) = t i=1 k i p i . Therefore dim F U i = k i p i .
(5) Suppose that g (i) = (g

(i) 1 , g (i)
2 , . . . , g

k i p i ) is a basis of U i over F for i = 1, . . . , t. Consider A i as the matrix of T c | U i with respect to that basis. Let f i = f Tc| U i . First, we show that gcd(f p i i , f i ) = 1. Suppose that gcd(f p i i , f i ) = 1, then there exist polynomials a(x), b(x) ∈ F [x] such that a(x)f p i i (x) + b(x)f i (x) = 1. Then a(A i )f p i i (A i ) + b(A i )f i (A i ) = E. By the theorem of Cayley-Hamilton, f i (A i ) = 0. So a(A i )f p i i (A i ) = E (i)
. Now, we want to show that f p i i (A i ) = 0, and we get the contradiction. By (3) in this Theorem, g = (g

(i) 1 , g (i) 2 , . . . , g (1)
k 1 p 1 , g (t) 1 , g (t) 2 , . . . , g (t)
ktpt ) is a basis of F n and T c is represented by

A =          A 1 A 2 A 3 . . . A t         
, with respect to that basis. Besides A = T -1 AT , where T is the transformation matrix from the standard basis of F n to the basis g. Then we have

f p i i (A ) =          f p i i (A 1 ) f p i i (A 2 ) f p i i (A 3 ) . . . f p i i (A t )          . Hence f p i i (A ) = f p i i (T -1 AT ) = T -1 f p i i (A)T . Let g (i) j = λ (i) j 1 e 1 + λ (i) j 2 e 2 + • • • + λ (i) jn e n , j = 1, . . . , k i p i . Since g (i) j ∈ U i , we get that f p i i (A )             0 . . . 1 0 . . . 0             = T -1 f p i i (A)T             0 . . . 1 0 . . . 0             = T -1 f p i i (A)     λ (i) j 1 . . . λ (i) jn     = 0,
where 1 is on the (k

1 + k 2 + • • • + k i-1 + j)-th position.
According to the last equation,

f p i i (A i ) = 0. Therefore gcd(f p i i , f i ) = 1.
By the proof of (3) in this theorem, it is easy to see that gcd(f

p i i , f i ) = f p i i .
On the other hand, deg(f

p i i ) = deg(f i ). Therefore f Tc|U i (x) = (-1) k i p i f p i i (x). (6) Suppose that V i is T c -invariant subspace of F n such that {0} = V i ⊆ V i . Then by Proposition 2.1, we know that f Tc|V i divides f i . Since f i is an irreducible polynomial, f i = f Tc|V i . So we have dim V i = deg f Tc|V i = deg f i = dim V i . Then we have V i = V i .
This completes the proof.

Proposition 2.3. Let U be a T c -invariant subspace of F n . Then U is a direct sum of some T c -invariant subspaces U i of F n .
Proof. This follows immediately from (2) of Theorem 2.2.

Definition 2.4. A linear code of length n and dimension k is a linear subspace C of dimension k of the vector space

F n . Definition 2.5. A linear code C of length n over F is called polycyclic, if whenever x = (x 1 , x 2 , . . . , x n ) ∈ C, then so is y = x n c + (0, x 1 , . . . , x n-1 ), where c = (c 0 , c 1 , . . . , c n-1 ) ∈ F n . Proposition 2.6. A linear code C of length n over F is polycyclic if and only if C is a T c -invariant subspace of F n .
Proof. The proof of this proposition is immediate from Definition 2.5.

The decomposition of a polycyclic code into minimal invariant subspaces enjoys the following properties. This is the main result of this section.

Theorem 2.7. Let C be a linear polycyclic code of length n over F . Then the following results hold.

(1)

C = U i 1 ⊕U i 2 ⊕• • •⊕U is for some T c -invariant subspaces U ir of F n and k := dim F C = k i 1 p i 1 + • • • + k is p is , where k ir p ir is the dimension of U ir ; (2) f Tc|C (x) = (-1) k i 1 p i 1 +k i 2 p i 2 +•••+k is p is f p i 1 i 1 (x)f p i 2 i 2 (x) • • • f p is is (x) = g(x); (3) c ∈ C iff g(A)c = 0;
(4) the polynomial g(x) has the smallest degree with respect to (3) in this theorem;

(5) rank(g(A)) = n -k and the matrix H the rows of which constitute an arbitrary set of n -k linearly independent rows of g(A) is a parity check matrix of C.

Proof.

(1) This follows from 2) of Theorem 2.1.

(2) Let (g

(ir) 1 , . . . , g (ir)
k ir p ir ) be a basis of U ir over F , where r = 1, . . . , s, and let A ir be the matrix of T c |U ir with respect to that basis. Then (g

(i 1 ) 1 , . . . , g (i 1 ) k i 1 p i 1 , . . . , g (is) 1 , . . . , g (is)
k is p is ) is a basis of C over F and T c |C is represented by the following matrix:

         A i 1 A i 2 A i 3 . . . A is         
with respect to that basis. Hence,

f Tc|C = (-1) k i 1 p i 1 +k i 2 p i 2 +•••+k is p is f p i 1 i 1 (x)f p i 2 i 2 (x) • • • f p is is (x).
(

) Let c ∈ C, then c = u i 1 + • • • + u is for some u ir ∈ U ir 3
, where r = 1, . . . , s, and

g(A)c = (-1) k i 1 p i 1 +•••+k is p is [(f p i 1 i 1 f p i 2 i 2 • • • f p is is )(A)u i 1 + (f p i 1 i 1 f p i 2 i 2 • • • f p is is )(A)u is ] = 0.
Conversely, suppose that g(A)c = 0 for some c ∈ F n . According to Theorem 2.2 part 3,

c = u 1 + u 2 + • • • + u t . Then g(A)c = (-1) k i 1 p i 1 +•••+k is p is [(f p i 1 i 1 f p i 2 i 2 • • • f p is is )(A)u 1 +• • •+(f p i 1 i 1 f p i 2 i 2 • • • f p is is )(A)u t ] = 0, so that g(A)(u j 1 + • • • + u j l ) = 0, where {j 1 , . . . , j l } = {1, . . . , t}\{i 1 , . . . , i s }. Let v = u j 1 + • • • + u j l and h(x) = f (x) g(x) . Since (h(x), g(x)) = 1, there are polynomials a(x), b(x) ∈ F [x] such that a(x)h(x) + b(x)g(x) = 1. Therefore, v = a(A)h(A)v + b(A)g(A)v = 0 and so c ∈ C.
k i 1 p i 1 +k i 2 p i 2 +•••+k im p im f p i 1 i 1 (x)f p i 2 i 2 (x) • • • f p im im (x) and m < s. Let us consider the code C = U i 1 U i 2 • • • U im ⊂ C. Then b(x) = f Tc|C (x)
and by the equation g(A)c = 0 for all c ∈ C, we obtain that C ⊆ C . This contradiction proves the statement.

(5

) dim F C = k = n -rank(g(A)), so rank(g(A)) = n -k = dim F C ⊥ .

Idempotent matrices

Let C = U i be a linear polycyclic code of length n over F ,

g(x) = (-1) k i p i f p i i (x) and h(x) = (-1) n-k i p i f p i i (x),
where

k i p i = dim F U i . Since gcd(g(x), h(x)) = 1, there are unique polynomials u i (x), v i (x) ∈ F [x] such that u i (x)g(x) + v i (x)h(x) = 1, deg(u i (x)) < deg(h(x)), deg(v i (x)) < deg(g(x)). It follows that v i (x)h(x)[u i (x)g(x) + v i (x)h(x)] = v i (x)h(x)
and hence

v i (A)h(A)[u i (A)g(A) + v i (A)h(A)] = v i (A)h(A).
If we let e i (A) = v i (A)h(A), then we have e 2 i (A) = e i (A) because of h(A)g(A) = f (A) = 0. The main properties of the idempotent matrices e i (A) are as follows.

Theorem 3.1. The matrices e i (A), i = 1, . . . , t, satisfy the following relations: (1) e 2 i (A) = e i (A); (2) e i (A)e j (A) = 0 for j = i;

(3) c ∈ U i iff e i (A)c = c ; (4) e i (A)c = 0 iff for all c ∈ U j , j = i;

(5) t i=1 e i (A) = E; (6) the columns of e i (A) generate U .

Proof. (1) e

i (A) = (-1) n-k i p i v i (A) f p i i (A) = v i (A)h(A)
, so e 2 i (A) = e i (A).

(

) e i (A)e j (A) = (-1) 2n-k i p i -k j p j v i (A)v j (A) f p i i (A) f p j j (A) = u(A)f (A) = 0 for a suitable polynomial u(x) ∈ F [x]. (3) Let c ∈ U i , then we have (-1) k i p i u i (A)f i (A) p i c + (-1) n-k i p i v i (A) fi (A) p i c = e i (A)c = c . 2
Conversely, let e i (A)c = c , then

f p i i (A)c = f p i i (A)e i (A)c = (-1) n-k i p i v i (A)f (A)c = 0, so c ∈ U i . (4) Let c ∈ U j , j = i. Then e i (A)c = (-1) n-k i p i v i (A) f p i i (A)c = u(A)f p j j (A)c = 0.
(5) For the proof of ( 5) we refer to Theorems 3 to 5 in [START_REF] Radkova | Constacyclic codes as invariant subspaces[END_REF].

(6) For the proof of ( 6) we refer to Theorems 3 to 6 in [START_REF] Radkova | Constacyclic codes as invariant subspaces[END_REF].

This completes the proof.

Because of the preceding theorem and the form of constructing polynomial, the proofs of the following two theorems are similar to those of Theorems 4 and 5 in [START_REF] Radkova | Constacyclic codes as invariant subspaces[END_REF]. Thus we omit them here. Theorem 3.2. e i (A) is the only idempotent matrix satisfying e i (A)c = c for all c ∈ U i and i (A)x = 0 for all x ∈ j =i U j .

Now let C = U i 1 U i 2 • • • U is be an arbitrary linear polycyclic code of length n over F . Then f Tc|C (x) = (-1) k i 1 p i 1 +•••+k is p is f p i 1 i 1 (x)f p i 2 i 2 (x) • • • f p is is (x) = g(x), and h(x) = f (x) g(x) = (-1) n-(k i 1 p i 1 +k i 2 p i 2 +•••+k is p is ) f p j 1 j 1 (x)f p j 2 j 2 (x) • • • f p j l j l (x),
where {j 1 , . . . , (2) the columns of e(A) generate C;

j l } = {1, . . . , t}\{i 1 , . . . , i s }. Let e(A) = v(A)h(A) for some polynomial v(A) ∈ F [x], e 2 (A) = e(A). Theorem 3.3. Let C = U i 1 U i 2 • • • U is be
(3) e(A) = e i 1 (A) + e i 2 (A) + • • • + e is (A); (4) the polycyclic code C = U j 1 U j 2 • • • U j l has the idempotent matrix E -e(A).

Bounds for polycyclic codes

Let K = F q m be the splitting field of the polynomial f (x) = (-1) n+1 ( n-1 i=0 c i x i -x n ) over F q , where 0 = c 0 ∈ F q . Let z be a root of f (x). Then z is an eigenvalue of A, where

A =            0 0 0 • • • 0 0 c 0 1 0 0 • • • 0 0 c 1 0 1 0 • • • 0 0 c 2 . . . . . . . . . . . . . . . . . . . . . 0 0 0 • • • 1 0 c n-2 0 0 0 • • • 0 1 c n-1            .
The corresponding eigenvector is given by

A t v(z) t = zv(z) t , v(z) = (1, z, . . . , z n-1 ),
where * t is transposition and v is an eigenvector of A t with associated eigenvalue z.

Now we study the case that f (x) has no repeated roots. Assume that f (x) has n distinct roots z j , 1 ≤ j ≤ n. Each root is an eigenvalue of A and A t .

A t v(z j ) = z j v(z j ), 1 ≤ j ≤ n.
These n statements can be expressed in just one matrix statement (each eigenvector being a column of the Vandermonde matrix V = V (z 1 , z 2 , . . . , z n )), where

V =            1 1 1 • • • 1 1 1 z 1 z 2 z 3 • • • z n-2 z n-1 z n z 2 1 z 2 2 z 2 3 • • • z 2 n-2 z 2 n-1 z 2 n . . . . . . . . . . . . . . . . . . . . . z n-2 1 z n-2 2 z n-2 3 • • • z n-2 n-2 z n-2 n-1 z n-2 n z n-1 1 z n-1 2 z n-1 3 • • • z n-1 n-2 z n-1 n-1 z n-1 n           
.

Let D = diag(z 1 , z 2 , . . . , z n ) be the diagonal matrix with the roots {z j |1 ≤ j ≤ n} on the main diagonal and with zeros everywhere else, we have

A t V = V D ⇔ V -1 A t V = D ⇔ V t A(V t ) -1 = D. Let T = (V t ) -1 , then T -1 AT = D, where
T -1 =            1 z 1 z 2 1 • • • z n-3 1 z n-2 1 z n-1 1 1 z 2 z 2 2 • • • z n-3 2 z n-2 2 z n-1 2 1 z 3 z 2 3 • • • z n-3 3 z n-2 3 z n-1 3 . . . . . . . . . . . . . . . . . . . . . 1 z n-1 z 2 n-1 • • • z n-3 n-1 z n-2 n-1 z n-1 n-1 1 z n z 2 n • • • z n-3 n z n-2 n z n-1 n            . Let u i = (1, z i , . . . , z n-1 i ), 1 ≤ i ≤ n, which is a row of T -1 .
Since D is a diagonal matrix, the matrices g(D) and h(D) are also diagonal.

Let deg(h(x)) = n -k = r, and let its r zeros be z i 1 , z i 2 , . . . , z ir and its k nonzeros be z j 1 , z j 2 , . . . , z j k . It is obvious that the zeros of g(x) are the nonzeros of h(x) and vice versa. We denote I = {i 1 , i 2 , . . . , i r } and J = [n] \ I = {j 1 , j 2 , . . . , j k }. Let K be any finite field and A = [a 1 , a 2 , . . . , a n ] any matrix over K with n columns a i , 1 ≤ i ≤ n. Let C A denote the linear code over K with A as parity check matrix. The minimum distance of C A will be denoted as d A .

For any m×n matrix X = [x 1 , x 2 , . . . , x n ] with nonzero columns x i ∈ K m for 1 ≤ i ≤ n, we define the matrix A(X) as

A(X) =       x 11 a 1 x 12 a 2 • • • x 1n a n x 21 a 1 x 22 a 2 • • • x 2n a n . . . . . . . . . . . . x m1 a 1 x m2 a 2 • • • x mn a n      
We recall the following lemma [START_REF] Radkova | Constacyclic codes as invariant subspaces[END_REF], which describes how the parity-check matrix A for a linear code can be extended with new rows in such a way that the minimum distance increases. Next, we recall the following BCH type lower bound that is a direct consequence of the observation that a shortened code has at least as great a minimum distance as the original code [8, p. 241]. Theorem 4.2. (BCH Type Lower Bound) [START_REF] Li | Pseudo-cyclic codes and the construction of quantum MDS codes[END_REF] Let C be a polycyclic code over F q with g(x) = f Tc|C (x) and h(x) = f (x) g(x) . If f (x) has no repeated roots and deg(f (x)) = n 1 ≥ 1. Suppose f (x)|(x n -1) for some positive integer n with gcd(n, q) = 1. Let β be a primitive element of order n in some finite extension of F q . Suppose there are integers a, b, d such that {β a+bi : 0 ≤ i ≤ d -2 and h(β a+bi ) = 0} and

n gcd(b,n) ≥ n 1 ≥ d -1.
Then the minimum distance of C is at least d. Definition 4.3. Let K = F q m . A set M = {α j 1 , α j 2 , . . . , α j l } is a consecutive set of n-th roots of unity if there is some primitive n-th root of unity β in K such that M consists of consecutive powers of β. Definition 4.4. Given n 1 such that n 1 ≤ n. If N = {α j 1 , α j 2 , . . . , α jt } is a set of zeros of n-th roots of unity, we denote by U N or by U (α j 1 ,α j 2 ,...,α j t ) the matrix of size t by n 1 over K that has (1, α js , . . . , α n 1 -1 js ) as its s-th row.

From the discussion above, it is clear that U N is a parity-check matrix for the polycyclic code C over F having N as a set of zeros of h(x). Let C N be the polycyclic code over K with U N as the parity-check matrix, and let this code have minimum distance d N . So the minimum distance of C is at least d N , because C is a subfield subcode of C N .

The proof of the following theorem is similar to that of Theorem 6 in [START_REF] Radkova | Constacyclic codes as invariant subspaces[END_REF], so we omit it here. According to Theorems 4.2 and 4.5, a systematic algorithm to compute the bound for a polycyclic code C in the multiplicity free case can be sketched as follows.

(i) For a polycyclic code C over F q with f (x) = f Tc (x), g(x) = f Tc|C (x) and h(x) = f (x) g(x) . Write n 1 = deg(f (x)). If f (x) has no repeated roots, we can find a minimal integer n such that f (x)|(x n -1), and gcd(n, q) = 1. Let β be a primitive element of order n in the splitting field F q m of x n -1 over F.

(ii) Compute the cyclotomic cosets of n mod q, denoted by C i for i = 0, 1, . . . , s.

(iii) Write the zeros of h(x) as β i , with i belonging to some union of the cyclotomic cosets.

If h(x) has a string of δ -1 consecutive zeros, then the BCH bound of C is δ.

(iv) Find two sets M, N satisfy the conditions in Theorem 4.5 such that M N is contained in the set of zeros of h(x).

(v) Find the matrix U N that has (1, α, . . . , α n 1 -1 ) as its row for all α ∈ N . Let U N be the parity-check matrix over F q m and compute the minimum distance d N . Then our bound for the minimum distance Similar to the discussion as above, we have (a) Let C be a polycyclic code over F 7 with f (x) = (x 4 + x 3 + x 2 + x + 1)(x 4 + 2x 3 + 4x 2 + 2x + 1)(x 4 + 4x 3 + 4x + 1)(x 4 + 4x 3 + 3x 2 + 4x + 1)(x 4 + 6x 3 + 5x 2 + 6x + 1) and g(x) = (x 4 + 6x 3 + 5x 2 + 6x + 1)(x 4 + 4x 3 + 4x + 1). Then the BCH bound is 5 and our bound is 6.

C is d ≥ d M N ≥ d N + |M | -1.
(b) Let C be a polycyclic code over F 3 with f (x) = (x + 2)(x 5 + x 4 + 2x 3 + x 2 + 2)(x 5 + 2x 3 + 2x 2 + 2x + 1) and g(x) = x + 2. Then the BCH bound is 4 and our bound is 5.

(c) Let C be a polycyclic code over F 5 with f (x) = (x+1)(x+2)(x+3)(x 2 +x+2)(x 2 +3x+3) and g(x) = (x 2 + 3x + 3)(x + 3). Then the BCH bound is 3 and our bound is 4.

Conclusion and open problems

In the present paper, we have developed an approach to polycyclic codes based on the theory of invariant subspaces under a fixed endomorphism. When the characteristic polynomial of this endomorphism is multiplicity free in its factorization, we have derived a lower bound on the minimum distance of the polycyclic codes. This mild hypothesis is equivalent, in the cyclic codes case, to the coprimality of the length and the alphabet size. The first open problem is to derive a similar bound for the repeated root case. Another open problem would be to generalize our results from finite fields alphabets to chain rings [START_REF] Martinez-Moro | On polycyclic codes over a finite chain ring[END_REF]. Finally, an algebraic decoding algorithm would be a natural continuation of the minimum distance bound.

( 4 )

 4 Suppose that b(x) ∈ F [x] is a nonzero polynomial of smallest degree such that b(A)c = 0 for all c ∈ C. By the division algorithm in F [x] there are polynomials q(x), r(x) such that g(x) = b(x)q(x)+r(x), where deg(b(x)) < deg(g(x)). Then for each vector c ∈ C, we have g(A)c = q(A)b(A)c + r(A)c and hence, r(A)c = 0. But this contradicts the choice of b(x) unless r(x) = 0. Thus b(x) divides g(x). If deg(b(x)) < deg(g(x)), then b(x) is a product of some irreducible factors of g(x), and without loss of generality we may assume that b(x) = (-1)

  an arbitrary linear polycyclic code of length n over F . Then the following facts hold:(1) c ∈ C iff e(A)c = c ;

  Assume that d = (d 1 , d 2 , . . . , d n ) ∈ F n q and let d = T -1 d. We know d ∈ C iff g(A)d = 0. The latter condition is equivalent to g(D)d = T -1 g(A)T T -1 d = T -1 g(A)d = 0, which, in its turn, is equivalent to d i = 0, i ∈ I. Hence, d ∈ C ⇔ u i d = 0, i ∈ I.

Lemma 4 . 1 .

 41 If d A ≥ 2 and every m × (m + d A -2) submatrix of X has full rank, then d A(X) ≥ d A + m -1.

Theorem 4 . 5 .

 45 If M, N are sets of n-th roots of unity such that |M | ≤ |M | + d N -2 for some consecutive set M containing M , then d M N ≥ d N + |M | -1, where M N = {αβ|α ∈ M, β ∈ N }.

Example 4 . 6 .

 46 A polycyclic code C over F 2 with f (x) = (x 2 + x + 1)(x 4 + x 3 + 1)(x 4 + x 3 + x 2 + x + 1) and g(x) = x 4 + x 3 + 1. We can find f |(x 15 -1) and (15, 2) = 1. Let β be a primitive 15-th root of unity. We determine the cyclotomic coset of 2 mod 15. These areC 0 = {0}, C 1 = {1, 2, 4, 8}, C 3 = {3, 6, 9, 12}, C 5 = {5, 10} and C 7 = {7, 11, 13, 14}.It is easy to check that the zeros of h(x) = f (x) g(x) are β i with i ∈ C 3 ∪ C 5 . Since h(x) has a string of two consecutive zeros, the linear polycyclic code C defined by h(x) has a minimum distance d ≥ 3 according to Theorem 4.2.Now take {βi |i = 5, 9} and M = {β i |i = 0, 1}. Then the elements of M N are zeros of h(x). Since d N = 3 and |M | = 2 ≤ |M | + d N -2 = 3, Theorem 4.5 implies d ≥ d M N ≥ |M | + d N -1 = 4.

Acknowledgement:

This research is supported by National Natural Science Foundation of China (61672036), the Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20), the Academic Fund for Outstanding Talents in Universities (gxbjZD03)


A C C García-Wong

D Pilloud

S Bruyère

S Mathieu

S Migot

J F Pierson

F Capon

Oxidation of sputter-deposited vanadium nitride as a new precursor

Keywords: Vanadium dioxide, Vanadium Nitride, Thermochromism, Optical and electrical properties

Introduction

In 1959, Morin demonstrated that vanadium dioxide (VO2) exhibits a reversible metal-insulator transition (MIT) approximately at Tc = 68°C [START_REF] Morin | Oxides Which Show a Metal-to-Insulator Transition at the Neel Temperature[END_REF]. At low temperature, VO2 behaves as a semiconductor with a monoclinic structure, high transmittance in the infrared region and high electrical resistivity, while at temperatures higher than Tc a transition to a metallic tetragonal rutile structure occurs and VO2 exhibits low transmittance and low electrical resistivity. The change of these properties during the phase transition of VO2 makes this material convenient for several applications, such as infrared modulation in smart windows [START_REF] Ji | Al-doped VO2 films as smart window coatings: Reduced phase transition temperature and improved thermochromic performance[END_REF][START_REF] Li | Thermochromic undoped and Mg-doped VO2 thin films and nanoparticles: Optical properties and performance limits for energy efficient windows[END_REF][START_REF] Liang | Active and passive modulation of solar light transmittance in a hybrid thermochromic soft-matter system for energy-saving smart window applications[END_REF][START_REF] Ji | High thermochromic performance of Fe/Mg co-doped VO2 thin films for smart window applications[END_REF][START_REF] Ji | Inkjet printing of vanadium dioxide nanoparticles for smart windows[END_REF][START_REF] Kim | High-throughput roll-toroll fabrication of flexible thermochromic coatings for smart windows with VO2 nanoparticles[END_REF][START_REF] Vu | Physical vapour deposition of vanadium dioxide for thermochromic smart window applications[END_REF], thermal solar collectors [START_REF] Mercs | Innovative Smart Selective Coating to Avoid Overheating in Highly Efficient Thermal Solar Collectors[END_REF][START_REF] Paone | Thermal solar collector with VO2 absorber coating and V1-xWxO2 thermochromic glazing -Temperature matching and triggering[END_REF], dynamic thermal control of spacecraft [START_REF] Benkahoul | Thermochromic VO2 film deposited on Al with tunable thermal emissivity for space applications[END_REF][START_REF] Hendaoui | Enhancement of the positive emittanceswitching performance of thermochromic VO2 films deposited on Al substrate for an efficient passive thermal control of spacecrafts[END_REF][START_REF] Hendaoui | VO2-based smart coatings with improved emittance-switching properties for an energy-efficient near room-temperature thermal control of spacecrafts[END_REF], ultrafast electronic devices [START_REF] Pergament | Oxide Electronics and Vanadium Dioxide Perspective: A Review[END_REF][START_REF] Qiu | Electrically-driven metalinsulator transition of vanadium dioxide thin films in a metal-oxide-insulator-metal device structure[END_REF] and many others. Although the research on this material has been carried on for several decades, the synthesis of VO2 films with good thermochromic properties on large surfaces still remains a challenge. The reason behind this defiance is the imperative need to avoid several other phases in the vanadiumoxygen system [START_REF] Kang | Critical evaluation and thermodynamic optimization of the VO-VO2.5 system[END_REF].

Magnetron sputtering is one of the most used methods to produced VO2. Among the different approaches to obtain VO2, reactive sputtering in a mixture of Ar and O2 without a post-annealing process is one of the most used [START_REF] Benkahoul | Thermochromic VO2 film deposited on Al with tunable thermal emissivity for space applications[END_REF][START_REF] Yuce | Investigation of electron beam lithography effects on metal-insulator transition behavior of vanadium dioxide[END_REF][START_REF] Zhao | Tuning the properties of VO2 thin films through growth temperature for infrared and terahertz modulation applications[END_REF][START_REF] Zhu | Thermochromism of vanadium dioxide films controlled by the thickness of ZnO buffer layer under low substrate temperature[END_REF][START_REF] Brassard | Grain size effect on the semiconductor-metal phase transition characteristics of magnetron-sputtered VO2 thin films[END_REF][START_REF] Kana | Thermochromic VO2 thin films synthesized by rf-inverted cylindrical magnetron sputtering[END_REF]. However, the need to properly control the O2 flow rate makes this method not suitable on large surfaces. On the other hand, a post-annealing process in air or oxygen is commonly used to improve the quality of the deposited films [START_REF] Melnik | Low-temperature method for thermochromic high ordered VO2 phase formation[END_REF][START_REF] Yu | Oxygen pressure manipulations on the metal-insulator transition characteristics of highly (0 1 1)-oriented vanadium dioxide films grown by magnetron sputtering[END_REF][START_REF] Dou | Oxidizing annealing effects on VO2 films with different microstructures[END_REF][START_REF] Yoon | Correlation between surface morphology and electrical properties of VO2 films grown by direct thermal oxidation method[END_REF]. Another interesting and simple proposal is the Sputtering Oxidation Coupling (SOC) method. It is a twostage procedure, in which a vanadium film is deposited and then annealed in a furnace outside of the deposition chamber [START_REF] Xu | A novel sputtering oxidation coupling (SOC) method to fabricate VO2 thin film[END_REF]. The possibility to regulate the temperature and duration during the oxidation process makes this method suitable and attractive for the industrial process.

In this study, we chose to modify the SOC method using another precursor: vanadium nitride (VN). At room temperature, the NaCl-like structure of the VN phase is more resistant to oxidation than the body-cubic centered of the V one [START_REF] Kubel | Structural phase transition at 205 K in stoichiometric vanadium nitride[END_REF][START_REF] Mei | Dynamic and structural stability of cubic vanadium nitride[END_REF]. Besides, several studies have shown that the oxidation of VN at elevated temperatures resulted in the formation of a series of vanadium oxides, which significantly improved the tribological properties of the film [START_REF] Guo | Comparison of microstructures and properties of VN and VN/Ag nanocomposite films fabricated by pulsed laser deposition[END_REF][START_REF] Fateh | The Beneficial Effect of High-Temperature Oxidation on the Tribological Behaviour of V and VN Coatings[END_REF]. Nevertheless, to the best of our knowledge, controlled oxidation of VN has never been performed to produce thermochromic VO2.

In this paper, VN thin films are reactively sputter-deposited on Si substrates. The air-oxidation conditions of these films have been optimized. The results indicate that VN is an interesting new precursor to synthesize high-quality thermochromic VO2 films.

Experimental details

VN thin films were deposited on undoped (100) silicon substrates by reactive magnetron sputtering of a metallic vanadium target (330 mm length, 50 mm width, 6 mm thick and 99.95% purity) in an Ar-N2 mixture with flow rates of 30 and 5 standard cubic centimeters per minute (sccm), respectively. The deposition pressure was fixed to 1 Pa and no intentional heating was used during growth. A pulsed-DC power supply (Pinnacle + Advanced Energy), working on a set current of 1.5 A, provided power to the V target. The frequency of DC-pulses and their off-time were kept at 50 kHz and 4 µs, respectively. The distance between target and substrate was 5 cm. The volume of the semi-industrial chamber is around 350 L. The Si wafer (4 inches diameter) fixed to the substrate holder moved back and forth in front of the V target (to ensure films homogeneity) at a speed of 50 cm/min during 1 h. The 175 nm thickness of the VN film was measured by a GBS Smart WLI 3D optical profilometer.

The second step consisted of air-oxidation of the VN films to form vanadium oxide films.

The as-deposited VN films were annealed in a modified Carbolite HRF 7/45 furnace equipped with a quick sample tray loader to prevent any drop in the temperature when the sample is introduced.

The furnace worked under atmospheric pressure at a preheated temperature of 450°C during different durations [START_REF] Kim | High-throughput roll-toroll fabrication of flexible thermochromic coatings for smart windows with VO2 nanoparticles[END_REF][START_REF] Paone | Thermal solar collector with VO2 absorber coating and V1-xWxO2 thermochromic glazing -Temperature matching and triggering[END_REF][START_REF] Qiu | Electrically-driven metalinsulator transition of vanadium dioxide thin films in a metal-oxide-insulator-metal device structure[END_REF][START_REF] Brassard | Grain size effect on the semiconductor-metal phase transition characteristics of magnetron-sputtered VO2 thin films[END_REF][START_REF] Yoon | Correlation between surface morphology and electrical properties of VO2 films grown by direct thermal oxidation method[END_REF][START_REF] Fateh | The Beneficial Effect of High-Temperature Oxidation on the Tribological Behaviour of V and VN Coatings[END_REF][START_REF] Schilbe | Raman scattering in VO2[END_REF][START_REF] Jin | Design, formation and characterization of a novel multifunctional window with VO2 and TiO2 coatings[END_REF] and 50 minutes. The films were labeled "VN" for as-deposited vanadium nitride films and "VN_time" for the oxidized films for various annealing durations.

X-rays diffractograms (XRD) were collected by using a Bruker D8 Advanced diffractometer with Cu Kα1 radiation (λ = 1.5406 Å) in the Bragg-Brentano configuration. Raman spectra of samples were acquired on a Horiba Jobin-Yvon LabRAM spectrometer. Samples were excited by a Nd:YAG laser (532 nm) with a power of approximately 10 mW. This laser beam power was low enough to avoid any heating effect (such as oxidation from VO2 to V2O5). The morphology of the films was studied by scanning electron microscopy (SEM) in a ZEISS GeminiSEM 500. All images were taken by using an Inlens secondary electron detector and under the same conditions: magnification of x100000 and a low acceleration voltage of 1 kV.

Transmission electron microscopy (TEM) was carried out in the sample oxidized for 25 minutes with a JEOL ARM 200-Cold FEG (point resolution 0.19 nm). The cross-section TEM sample was prepared by a focused ion beam (FIB)-SEM dual beam system (FEI Helios Nanolab 600i). Electron energy loss spectroscopy (EELS) experiments were carried out in scanning electron transmission microscopy mode (STEM). The spectrometer was set to an energy dispersion of 0.1 eV/channel and the resolution of the full width at half maximum (FWHM) of the zero-loss peak was 0.7 eV.

The study of the sheet resistance as a function of the temperature of the samples was performed by the four-point probe method (Jandel) equipped with a Linkam ® HFS600E heater system in the 30-120°C temperature range.

The dependence of optical properties of the films with temperature (30-120°C range) was investigated by using an infrared camera (FLIR A300) that supplies a cumulative response sensitive in the 7.5-13 µm spectral range. The emissivity of the oxidized VN films was calculated from the slope of the temperature measured using the IR camera (TIR) as a function of the real temperature (TS) of the sample. Measurements were carried out by considering = 1 irrespective to the sample.

Besides, TS was measured using a Prazitherm heater system. The emissivity switch, ∆ = -( corresponds to the emissivity at high temperature and corresponds to the emissivity at low temperature) was determined by the difference between the slopes corresponding to each region of the MIT according to the procedure employed by Benkahoul et al [START_REF] Benkahoul | Thermochromic VO2 film deposited on Al with tunable thermal emissivity for space applications[END_REF].

Results

Microstructure and morphology

The X-ray diffractograms for as-deposited and air annealed VN films deposited on silicon are shown in Fig. 1. VN films (175 nm thick) exhibit two peaks located at approx. 37.9° and 80.3° corresponding to (111) and ( 222) planes of the face-centered cubic VN structure (PDF no.00-035-0768), respectively. No other peak is detected except those of silicon substrate. Fig. 1 also presents diffractograms for the films after air-annealing at 450°C during different durations in the 7-50 min range. For the lowest duration, the monoclinic VO2 phase is not detected. However, the intensity of the VN peaks diminishes, which indicates that the oxidation process has effectively begun. In addition, these peaks are shifted to higher angles because of stress relaxation during the oxidation process. As the annealing time increases, the VN peaks intensity decreases even more and the main showing that the oxide layer corresponds to an equiaxed structure (i.e. that it does not grow with a strong preferred orientation). The most intense peak of VO2 appears at 30 min of oxidation and the VN peak completely disappears at the same time. In this sample (VN_30), two new peaks with a low intensity corresponding to orthorhombic V2O5 phase are detected (PDF no. 00-041-1426). For oxidation times greater than 30 min, the peaks corresponding to VO2 decrease in intensity while the opposite occurs with the peaks of V2O5. For the VN_50 sample, the VO2 vanishes, leading to the V2O5 being the only crystalline phase present. These results show oxidation from V 4+ state to V 5+ state as the oxidation time increases and are consistent with the findings of several authors [START_REF] Dou | Oxidizing annealing effects on VO2 films with different microstructures[END_REF][START_REF] Rampelberg | In situ X-ray diffraction study of the controlled oxidation and reduction in the V-O system for the synthesis of VO2 and V2O3 thin films[END_REF].

Due to the overlapping of the diffraction peaks of the various vanadium oxide phases, Xray diffraction is not a powerful method to evince the formation of a pure VO2 phase. To verify the formation of VO2, cross-examination with Raman spectrometry is carried out for all oxidized films.

Figure 2 presents room-temperature Raman spectra of the oxidized VN films, deposited on silicon substrates.

The Si substrate signal (520 cm -1 ) is barely visible until 10 min of oxidation, thanks to the presence of the VN film that exhibits metallic behavior. As the oxidation time increases the Si signal becomes more perceptible indicating a decrease of the VN layer thickness. From sample VN_25, the peak at 520 cm -1 is intense; suggesting that the sample is almost completely oxidized, as shown by XRD. Thus, the evolution of the Si Raman band is a powerful tool to monitor the VN oxidation.

Oxidized samples in the range of 10 to 25 min show relatively similar vibration bands, with most of the peaks (located at approx. 192, 223, 264, 296, 331, 389, 499 and 622 cm -1 ) assigned to m-VO2 phase, in accordance with literature data [START_REF] Zhang | Characterization of vanadium oxide thin films with different stoichiometry using Raman spectroscopy[END_REF][START_REF] Ureña-Begara | Raman and XPS characterization of vanadium oxide thin films with temperature[END_REF][START_REF] Parker | Raman scattering from VO2 single crystals: A study of the effects of surface oxidation[END_REF][START_REF] Schilbe | Raman scattering in VO2[END_REF].

In the case of the oxidized films during 40 and 50 minutes, the shape of the spectra is different from the other samples. The vibration bands detected at approx. 195, 284, 303, 405, 528, 701 and 992 are readily assigned to the V2O5 phase [START_REF] Zhang | Characterization of vanadium oxide thin films with different stoichiometry using Raman spectroscopy[END_REF][START_REF] Baddour-Hadjean | Raman Microspectrometry Study of Electrochemical Lithium Intercalation into Sputtered Crystalline V2O5 Thin Films[END_REF].

The simultaneous presence of VO2 and V2O5 is detected by X-ray diffraction results in oxidized samples between 30 and 40 min. Although several bands may overlap, it is difficult to notice a big change in the spectrum of these samples. However, one can notice a slight shift of the band located at 389 cm -1 (VN_20 sample) until reaching 405 cm -1 (VN_40 sample). The Raman results confirm the formation of VO2 and V2O5 as shown by the X-ray results.

Optical and electrical properties

Near the MIT transition of VO2, some physical properties present a hysteresis between the heating and cooling cycles. Within these properties, changes in electrical resistivity and emissivity at low and high temperatures are considered as indicators of VO2 quality [START_REF] Yoon | Correlation between surface morphology and electrical properties of VO2 films grown by direct thermal oxidation method[END_REF][START_REF] Xu | The study of optimal oxidation time and different temperatures for high quality VO2 thin film based on the sputtering oxidation coupling method[END_REF][START_REF] Mlyuka | Thermochromic VO2-based multilayer films with enhanced luminous transmittance and solar modulation[END_REF]. The higher the difference, the better the thermochromic properties.

For all oxidized samples, the optical modulation properties in the infrared (IR) domain are studied using an infrared camera. Fig. 6 represents apparent infrared temperature (TIR) of samples monitored versus the sample temperature (TS) during both, heating and cooling for the studied films. All the films of Fig. 6 exhibit hysteresis loops near the MIT region. Samples oxidized for 10 minutes or less show no representative thermochromic behavior (∆ = -0.03). This is due to the presence of a considerable residual layer of VN (see Fig. S1 in the supplementary information), higher oxidation time is needed to observe thermochromic behavior. A thermochromic effect is not observed for samples oxidized for a duration longer or equal to 40 min, ∆ _ = -0.03. In this case, this is attributed to the increased formation of V2O5 phase with oxidation time. On the contrary, for samples oxidized in the range of 15 and 35 minutes, a thermochromic behavior with values of ∆ between -0.14 and -0.20 is evidenced, confirming the presence of m-VO2 consistent with XRD and Raman results. Nevertheless, the emissivity contrast obtains from our samples is lower than that reported in the literature. For example, Benkahoul et al. found ∆ = -0.36 in 300 nm thick VO2 film deposited on Si [START_REF] Benkahoul | Thermochromic VO2 film deposited on Al with tunable thermal emissivity for space applications[END_REF]. Emissivity switching properties can be improved by modifying the temperature and duration of the air-annealing process (see Fig. S2 in the supplementary information), thickness changes [START_REF] Brassard | Grain size effect on the semiconductor-metal phase transition characteristics of magnetron-sputtered VO2 thin films[END_REF][START_REF] Voti | Optimization of thermochromic VO2 based structures with tunable thermal emissivity[END_REF] and doping [START_REF] Ji | Al-doped VO2 films as smart window coatings: Reduced phase transition temperature and improved thermochromic performance[END_REF][START_REF] Mercs | Innovative Smart Selective Coating to Avoid Overheating in Highly Efficient Thermal Solar Collectors[END_REF][START_REF] Hendaoui | VO2-based smart coatings with improved emittance-switching properties for an energy-efficient near room-temperature thermal control of spacecrafts[END_REF].

Fig. 7 shows the evolution of the sheet resistance (SR) as a function of the temperature for all oxidized samples during both heating and cooling. In this study, sheet resistance is calculated rather than the resistivity, due to the presence of the remaining metallic VN layer for films oxidized for less than 25 minutes. Up and down triangles in Fig. 7 The thermochromic behavior of oxidized samples will be affected if they remain exposed to air open contact because the oxidation process continues. Infrared camera measurement of the oxidized sample 25 minutes was repeated 10 months later, showing a significant decrease (50%) in the absolute value of the emissivity switch (Fig. S3 supporting information). To protect the films for further and undesired oxidation anti-reflective layers such as TiO2 [START_REF] Jin | Design, formation and characterization of a novel multifunctional window with VO2 and TiO2 coatings[END_REF][START_REF] Zheng | TiO2(R)/ VO2 (M)/TiO2(A) multilayer film as smart window: Combination of energy-saving, antifogging and self-cleaning functions[END_REF], SiO2 [START_REF] Yu | Enhanced Visible Transmittance of Thermochromic VO2 Thin Films by SiO2 Passivation Layer and Their Optical Characterization[END_REF], AZO [START_REF] Kang | Thermochromic properties and low emissivity of ZnO:Al/VO2 double-layered films with a lowered phase transition temperature[END_REF] and SiNx [START_REF] Long | Application-oriented VO2 thermochromic coatings with composite structures: Optimized optical performance and robust fatigue properties[END_REF] are used. rises. This reduction of thermochromic behavior is evident at 40 min of oxidation. Finally, as oxidation time continues to increase, there is a region where V2O5 is the only compound because of the over-oxidation of the samples.

Conclusions

This study reveals that VN is a new efficient precursor to achieve a high-quality monoclinic VO2 using a sputtering oxidation coupling method. In correspondence with previous studies of oxidation of vanadium films, the importance of optimizing the annealing duration to obtain thermochromic vanadium dioxide is demonstrated. During the oxidation process at 450 °C, there are only two phases corresponding to the V-O system, VO2 and V2O5. The first oxide formed is VO2, it is present in several samples oxidized between 10 and 40 min. On the other hand, V2O5 is only detected after the total oxidation of VN. The best thermochromic behavior is accomplished in the oxidized samples in the range between 25-35 min. However, it is important to point out that in these samples two phases coexist:

1-VN and VO2 in the sample VN_25, 2-VO2 and V2O5 in samples VN_30 and VN_35.

In both cases, VN and V2O5 represent a small fraction compared to the VO2 content. Therefore, we could expect to obtain a sample with only VO2 in the range between 25-30 minutes of oxidation, which could present better thermochromic behavior.

The TEM analysis performed on the oxidized sample for 25 minutes show that there is an abrupt interface between VN and VO2.

(

  111) peak is noticed up to 25 min, indicating the presence of a remaining layer of vanadium nitride. Monoclinic VO2 is detected for the first time after 10 min of air oxidation. The most significant peaks corresponding to m-VO2 are located at 2θ = 37.1, 42.3 and 72.1° (PDF no. 04-003-2035),

Fig. 3

 3 Fig.3shows the top view SEM images of six relevant samples: the as-deposited VN films

Fig. 4 a

 4 Fig. 4 a) shows transmission electron microscopy micrograph for the VN_25 sample. The

  represent the profiles for temperature ramping up and down, respectively. The samples oxidized during 7 and 50 min do not evidence any hysteresis. For the low time oxidation duration (7 minutes) there is no remarkable change between low and high temperatures sheet resistance values, this is due to the presence of a remaining layer of VN. On the other hand, the sample oxidized for a long duration (50 minutes) displays no sign of VO2, only V2O5. All other studied samples exhibit hysteresis loops in the near MIT region with different magnitudes: from a feeble smoothed hysteresis for VN_10, that rises to maximum marked hysteresis curves for VN_25 and then decreases again until an almost imperceptible hysteresis for VN_50. The enlarging presence of V2O5 in the samples oxidized for more than 25 minutes, is the reason for the reduction of the difference between the values of sheet resistance at low and high temperatures. These hysteresis loops are an intrinsic feature of thermochromic VO2 MIT and, again confirm the XRD, Raman and IR thermometry results for the series of studied samples.

3. 2

 2 Phase diagram of VN oxidation at 450 °C Gathering the results of XRD and Raman, as well as the study of optical and electrical properties, it is possible to obtain a phase diagram of VN oxidation. In Fig. 8 is displayed on the left axis (blue squares) the logarithm of the ratio between the sheet resistance at low (SRLT, measured at 30°C), and at high temperatures (SRHT, measured at 120°C) as a function of the oxidation time for all the oxidized samples. The right axis (red circles) of the same figure shows the dependence of the emissivity switch with the oxidation time for all studied samples.A color gradient is used in Fig.8to illustrate the five different regions that exist in the oxidation of VN at 450 °C. The first region is a VN predominant zone. As the oxidation time progresses, a region with the coexistence of VN and VO2 appears. By increasing the annealing duration in this region, the ratio between SRLT and SRHT (the emissivity switch) increases (decreases) due to the competing effect of VO2 formation with simultaneous VN degradation. During the interval between 25 and 30 min, there are the best thermochromic properties due to the almost complete oxidation of VN, where VO2 is the predominant compound. Beyond 30 min, the amount of VO2 declines and begins the formation of V2O5. Continuing with the oxidation process in this region, thermochromic performance decays as the amount of VO2 diminishes while the presence of V2O5

Figure 1 :

 1 Figure 1: X-ray diffractograms of as-deposited VN film on silicon substrate and the resulting oxide

Figure 2 :

 2 Figure 2: Raman spectra of air-oxidized VN films at 450°C for various durations.

Figure 3 :

 3 Figure 3: Top-view SEM images corresponding to a) the as-deposited VN films and sample

Figure 4 :

 4 Figure 4: a) TEM cross-sectional image of sample VN_25. b) HRTEM image of the highlighted

Figure 5 :

 5 Figure 5: a) High angle annular dark field image of the sample VN_25, EELS characterization was

Figure 1 -

 1 Figure 1

Figure 2 -

 2 Figure 2

Figure 3 -

 3 Figure 3

Figure 4 -

 4 Figure 4

Figure 5 -

 5 Figure 5

Figure 6 :

 6 Figure 6:

Figure 7 :

 7 Figure 7:

Figure 8 :

 8 Figure 8:

Acknowledgments

This work was supported by the French PIA project "Lorraine Université d'Excellence", reference ANR-15-IDEX-04-LUE. The authors acknowledge the financial support of the Contrat de Plan Etat-Région MatDS. They are also grateful to the Centre de Competence Daνm (cc-Daνm) of the Institut Jean Lamour for the semi-industrial chamber provision.

Resulting VO2 films exhibit thermochromic properties that are suitable for applications of VO2 as smart windows or as thermal controllers.