Liqin Qian

Xiwang Cao
email: xwcao@nuaa.edu.cn

Wei Lu

Patrick Solé
email: sole@enst.fr

A new method for constructing linear codes with small hulls *

Keywords: Linear codes, hull of a code, LCD codes, Gauss sums, quasi-abelian codes, double circulant codes MSC(2010): 94B05, 11T24, 11T71

The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key. In this paper, we develop a method to construct linear codes with trivial hull ( LCD codes) and one-dimensional hull by employing the positive characteristic analogues of Gauss sums. These codes are quasiabelian, and sometimes doubly circulant. Some sufficient conditions for a linear code to be an LCD code (resp. a linear code with one-dimensional hull) are presented. It is worth mentioning that we present a lower bound on the minimum distances of the constructed linear codes. As an application, using these conditions, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) with respect to the online Database of Grassl.

Introduction

The hull of a linear code C over a finite field is defined to be

Hull(C) := C ∩ C ⊥ .
It is clear that Hull(C) is also linear. It is easy to see that a linear code C is self-orthogonal if and only if the dimension of Hull(C) is the dimension of C, i.e., Hull(C) = C, and it is Linear Complementary Dual (LCD) if and only if the dimension of Hull(C) is zero, i.e., Hull(C) = {0}. Specifically, a linear code C is self-dual if and only if the dimension of Hull(C) is n 2 for even n, where n is the length of C. Hulls of linear codes have been introduced to classify finite projective planes in [START_REF] Assmus | Affine and projective planes[END_REF]. Later, it turned out that hulls of linear codes play a vital role in determining the complexity of some algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code [START_REF] Leon | Computing automorphism groups of error-correcting codes[END_REF][START_REF] Leon | Permutation group algorithms based on partition, I: theory and algorithms[END_REF][START_REF] Sendrier | Finding the permutation between equivalent codes: the support splitting algorithm[END_REF][START_REF] Sendrier | On the computation of the automorphism group of a linear code[END_REF]. It has been shown that these algorithms are always effective when the dimension of the hull is small. Due to their wide applications, some families of linear codes with special hulls such as LCD codes and linear codes with one-dimensional hull have been of interest and extensively studied [START_REF] Carlet | Linear codes with small hulls in semi-primitive case[END_REF][START_REF] Liu | LCD codes over finite chian rings[END_REF][START_REF] Liu | Galois LCD codes over finite fields[END_REF][START_REF] Mesnager | Complementary dual algebraic geometry codes[END_REF][START_REF] Li | Constrctions of linear codes with one-dimensional hull[END_REF][START_REF] Shi | Double circulant LCD codes over Z 4[END_REF][START_REF] Sok | Constructions of optimal LCD codes over large finite fields[END_REF]. It is worth noting that the equivalence of many types of codes with LCD codes has been extensively studied. Jin and Xing [START_REF] Jin | Algebraic geometry codes with complementary duals exceed the asymptotic Gilbert-Varshamov bound[END_REF] showed that an algebraic geometry code over F 2 m (m ≥ 7) is equivalent to an LCD code. Moreover, a celebrated result was presented in [START_REF] Carlet | Linear codes over F q are equivalent to LCD codes for q > 3[END_REF], which proved that any linear code over F q (q > 3) is equivalent to an LCD code. These codes are practically useful in communications systems, various applications, and link with other objects as shown in [START_REF] Carlet | Complementary dual codes for counter-measures to side-channel attacks[END_REF][START_REF] Carlet | Linear codes over F q are equivalent to LCD codes for q > 3[END_REF][START_REF] Carlet | Euclidean and Hermitian LCD MDS codes[END_REF][START_REF] Carlet | New characterization and parametrization of LCD codes[END_REF][START_REF] Jin | Construction of MDS codes with complementary duals[END_REF] and references therein. Consequently, it is of interest to study hulls, families of linear codes with small hulls. What needs to be emphasized is that Li and Zeng [START_REF] Li | Constrctions of linear codes with one-dimensional hull[END_REF] constructed linear codes with one-dimensional hull by utilizing quadratic Gaussian sums from quadratic number fields and Carlet, Li and Mesnager [START_REF] Carlet | Linear codes with small hulls in semi-primitive case[END_REF] constructed LCD codes and linear codes with one-dimensional hull by employing character sums in semi-primitive case from cyclotomic fields and multiplicative subgroups of finite fields. They have made a lot of contributions in this regard.

Inspired by the above research work, we construct LCD codes and codes with onedimensional hull dimension, by using an analogue of Gauss sums where both the corresponding additive and multiplicative character take their take values in a finite field instead of the complex numbers. This method generalizes previous work [START_REF] Li | Constrctions of linear codes with one-dimensional hull[END_REF][START_REF] Qian | Linear codes with one-dimensional hull associated with Gaussian sums[END_REF][START_REF] Qian | Linear complementary dual codes constructed by general Gaussian sums over finite fields[END_REF]. Moreover, we consider the order N ≥ 2 of the homomorphism, while [START_REF] Li | Constrctions of linear codes with one-dimensional hull[END_REF] only considers N = 2. It turns out that our constructions are more general and direct than previous work on small hulls of linear codes. It is worth observing that we obtain some optimal or almost optimal LCD codes and linear codes with one-dimensional hull from our constructions. Compared with [START_REF] Li | Constrctions of linear codes with one-dimensional hull[END_REF], the linear codes we constructed may be new when N > 2 in the sense. Furthermore, we also present a lower bound on the minimum distances of the codes presented in this paper. These codes have a lot of built-in symmetry: they are quasi-abelian of index 2 in general [START_REF] Jitman | Quasi-abelian codes[END_REF], and double circulant in many cases.

The rest of this paper is organized as follows. Section 2 gives the preliminaries. In Section 3, we give two concrete homomorphisms from a finite field into a finite field and present the idea of constructing linear codes determined by a special generator matrix. In Sections 4 and 5, we investigate LCD codes and linear codes with one-dimensional hull by employing these two homomorphisms from a finite field to a finite field, respectively. In addition, we present some examples of optimal or almost optimal LCD codes and linear codes with one-dimensional hull. In Section 6, we present a lower bound on the minimum distances of the constructed linear codes. Section 7 concludes the article.

Preliminaries

In this section, we introduce some notation and results in order for the exposition in this paper to be self-contained, which will be useful later.

Codes

Let q be a power of a prime p and F q denote the finite field with q elements. For a positive integer n, a linear code of length n over F q is defined to be a subspace of the F q -vector spaces

F n q . A linear code C of length n over F q is called an [n, k, d] q code if its F q -dimension is k and the minimum Hamming distance of C is d. If C is an [n, k, d]
code, then from the Singleton bound, its minimum distance is bounded above by d ≤ n -k + 1. A code meeting the above bound is called Maximum Distance Separable (MDS). A code is called almost MDS if its minimum distance is one less than the MDS case. For u :

= (u 1 , u 2 , • • • , u n ) and v := (v 1 , v 2 , • • • , v n ) in F n
q , the inner product of u and v is defined to be u, v :=

n i=1 u i v i .
The dual C ⊥ of a linear code C of length n over F q is defined to be the set

C ⊥ = {v ∈ F n q | c, v = 0 for all c ∈ C}. A linear code C is said to be self-orthogonal if C ⊆ C ⊥ and it is said to be self-dual if C = C ⊥ . A linear code C is said to be linear complementary dual (LCD) code if C ∩ C ⊥ = {0}.

Homomorphisms

Starting from this subsection till the end of this paper, we let F r m denote the finite field of order r m , where r is a prime number and m is a positive integer. Let F * r m = F r m \{0}. Let F q be the algebraic closure of the finite field F q .

Let ϕ be a homomorphism from F * r m into F * q , that is, a mapping from F * r m into F * q with ϕ(xy) = ϕ(x)ϕ(y) for all x, y ∈ F * r m . Define ϕ(x) := ϕ(x -1 ). Let ϕ 0 be the trivial homomorphism, which is defined by ϕ 0 (x) = 1 for all x ∈ F * r m . The following lemma gives the orthogonality relations of the homomorphism ϕ. Lemma 2.1. Let ϕ be defined as above. Then we have

x∈F * r m ϕ(x) =    r m -1, if ϕ = ϕ 0 ; 0, if ϕ = ϕ 0 .
Proof. The proof is similar to that of [START_REF] Lidl | Finite fields[END_REF]Theorem 5.4] and omitted here.

Let χ be a homomorphism from F r m into F * q , that is, a mapping from F r m into F * q with χ(x + y) = χ(x)χ(y) for all x, y ∈ F r m . Define χ(x) := χ(-x). Let χ 0 be the trivial homomorphism, which is defined by χ 0 (x) = 1 for all x ∈ F r m .

We also have the following lemma, which presents the orthogonality relations of the homomorphism χ. Lemma 2.2. Let χ be defined as above. Then we have

x∈F r m χ(x) =    r m , if χ = χ 0 ; 0, if χ = χ 0 .
Proof. The proof is similar to that of [START_REF] Lidl | Finite fields[END_REF]Theorem 5.4] and omitted here.

Some results for the sum g(ϕ, χ)

Let ϕ and χ be defined as Subsection 2.1. Then we define the sums

g(ϕ, χ) = x∈F * r m ϕ(x)χ(x) and g(ϕ, χ) = g(ϕ, χ) = x∈F * r m ϕ(x -1 )χ(-x).
The following results show the value of the sum g(ϕ, χ).

Lemma 2.3. Let ϕ and χ be defined as Subsection 2.1. Then the sum g(ϕ, χ) satisfies

g(ϕ, χ) =          r m -1, if ϕ = ϕ 0 and χ = χ 0 ; -1, if ϕ = ϕ 0 and χ = χ 0 ; 0, if ϕ = ϕ 0 and χ = χ 0 .
Proof. The conclusion follows directly from Lemmas 2.1 and 2.2.

Lemma 2.4. Let ϕ and χ be defined as Subsection 2.1. If ϕ = ϕ 0 and χ = χ 0 , then

g(ϕ, χ)g(ϕ, χ) = r m ∈ F p .
Proof. For ϕ = ϕ 0 and χ = χ 0 , we get

g(ϕ, χ)g(ϕ, χ) = x∈F * r m ϕ(x)χ(x) y∈F * r m ϕ(y -1 )χ(-y) = x,y∈F * r m ϕ(xy -1 )χ(x -y) x -→ xy x,y∈F * r m ϕ(x)χ(y(x -1)) = x∈F * r m ϕ(x) y∈F * r m χ(y(x -1)) = ϕ(1) y∈F * r m χ(0) + x∈F * r m \{1} ϕ(x) y∈F * r m χ(y(x -1)) = r m -1 - x∈F * r m \{1} ϕ(x) = r m - x∈F * r m ϕ(x) = r m .
This completes the proof of this lemma.

The study of the behavior of the sum g(ϕ, χ) under various transformations of the ϕ or χ leads to a number of useful identities. Lemma 2.5. Let ϕ and χ be defined as Subsection 2.1. Then we have the following results.

(1) g(ϕ, χ) = ϕ(-1)g(ϕ, χ);

(2) g(ϕ, χ) = ϕ(-1)g(ϕ, χ);

(3) g(ϕ, χ)g(ϕ, χ) = ϕ(-1)r m for ϕ = ϕ 0 and χ = χ 0 ;

(4) (g(ϕ, χ)) p s = g(ϕ p s , χ p s ), where p is the characteristic of F q and s is a positive integer.

Proof. The results of (1)-(3) are obvious by the definition g(ϕ, χ) and Lemma 2.4. Next, we prove the result of (4). Combined with the definitions of ϕ and χ, we have

(g(ϕ, χ)) p s = x∈F * r m ϕ(x)χ(x) p s = x∈F * r m (ϕ(x)) p s (χ(x)) p s = x∈F * r m ϕ p s (x)χ p s (x) = g(ϕ p s , χ p s ).
Remark 2.6. The ϕ and χ in Section 2.1 are not the usual multiplicative and additive characters, respectively. Moreover, the g(ϕ, χ) is also not the usual Gaussian sums. However, we can prove that the sum g(ϕ, χ) has similar properties to Gaussian sums (see Lemmas 2.3 and 2.5 [START_REF] Assmus | Affine and projective planes[END_REF][START_REF] Bosma | Hand-book of Magma functions[END_REF][START_REF] Carlet | Complementary dual codes for counter-measures to side-channel attacks[END_REF]). The definition of the sum g(ϕ, χ) may have been studied before, but we haven't found any relevant references.

On characterizations of LCD codes and codes having one-dimensional hull

In this paper, we consider the constructions of linear codes with small hull, mainly refer to LCD codes and linear codes with one-dimensional hull. We will characterize when a linear code is an LCD code or a linear code with one-dimensional hull. We next present two lemmas for this purpose.

A complete characterization of LCD codes via the nonsingularity of their generator matrices was employed in [START_REF] Carlet | Complementary dual codes for counter-measures to side-channel attacks[END_REF][START_REF] Massey | Linear codes with complementary duals[END_REF], which provides a sufficient and necessary condition for a linear code to be an LCD code.

Lemma 2.7. [START_REF] Carlet | Complementary dual codes for counter-measures to side-channel attacks[END_REF][START_REF] Massey | Linear codes with complementary duals[END_REF] Let C be an [n, k] linear code over F q with generator matrix G = [I k , P ]. Then the code C is LCD if and only if I k +P P T is nonsingular, i.e., -1 is not an eigenvalue of the matrix P P T , where P T denotes the transpose of P.

We also have the following lemma on a linear code having one-dimensional hull, which provides an idea to construct linear codes with one-dimensional hull by using the eigenvalues of the generator matrices.

Lemma 2.8. [START_REF] Carlet | Linear codes with small hulls in semi-primitive case[END_REF][START_REF] Li | Constrctions of linear codes with one-dimensional hull[END_REF] Let C be an [n, k] linear code over F q with generator matrix G = [I k , P ]. Then the code C has one-dimensional hull if the matrix P P T has an eigenvalue -1 with (algebraic) multiplicity 1.

Linear codes associated with homomorphisms

In this section, we construct the linear codes by using the two homomorphisms in Section 2.1.

Let r be a prime number and m a positive integer. F r m denotes the finite field of order r m . Let F * r m = F r m \{0} and F * r m = α , where α is a fixed primitive element of F * r m . Assume that N > 1 is a positive integer and N |(r m -1). Let q be a power of p, where p is a prime number. Assume that N |(q -1). Let F * q = β , where β is a fixed primitive element of F * q . For the sake of convenience, we let u = β q-1 N . Define the function

ϕ : F * r m -→ F * q , ϕ(α k ) = u k , (1)
where 0 ≤ k ≤ r m -2. It is easy to know that ϕ is a homomorphism of order N . Define the kernel of the homomorphism ϕ is the set

ker(ϕ) := {α k , 0 ≤ k ≤ r m -2 : ϕ(α k ) = 1} = α N .
Assume that (p, r) = 1. Then there exists a positive integer t such that r|(q t -1). Let

F * q t = γ and ζ = γ q t -1
r , where γ is a fixed primitive element of F * q t . For any a ∈ F r m , we define

χ a : F r m -→ F * q , χ a (x) = ζ Tr r m r (ax) , x ∈ F r m , (2)
where Tr r m r denotes the trace function from F r m onto F r . It is easy to know that χ a is a homomorphism. It follows from the definition of χ a that

g(ϕ, χ ab ) = ϕ(b)g(ϕ, χ a ) (3) for a ∈ F r m and b ∈ F * r m . Fix v ∈ F q . Let F r m = {x i : 1 ≤ i ≤ r m }. Define the r m ×r m matrix P = (p ij ) ∈ M r m (F q ) by p ij = ρ(x j -x i ), where ρ(x j -x i ) =    ϕ(x j -x i ), if i = j; v, if i = j. (4)
For any a ∈ F r m , set η a := (χ a (x 1 ), χ a (x 2 ), • • • , χ a (x r m )) T , where "T " denotes the transpose operator. Then the ith component of P η a is

r m j=1 ρ(x j -x i )χ a (x j ) = x∈F r m ρ(x-x i )χ a (x)y := x -x i y∈F r m ρ(y)χ a (y+x i ) = y∈F r m ρ(y)χ a (y)χ a (x i ).
Hence, P η a = y∈F r m ρ(y)χ a (y) η a and η a is an eigenvector of P .

Similarly, the ith component of

P T η a is r m j=1 ρ(x i -x j )χ a (x j ) = x∈F r m ρ(x i -x)χ(x)y := x i -x y∈F r m ρ(y)χ a (x i -y) = y∈F r m ρ(y)χ a (-y)χ a (x i ).
Hence, P T η a = y∈F r m ρ(y)χ a (-y) η a and η a is also an eigenvector of P T .

Next, we will prove that the r m vectors {η a := (χ a (x 1 ),

χ a (x 2 ), • • • , χ a (x r m )) T : a ∈ F r m } are linearly independent over F q .
Suppose that a∈F r m k a η a = 0, where k a ∈ F q . Then we have

a∈F r m k a (χ a (x 1 ), χ a (x 2 ), • • • , χ a (x r m )) T = 0, =⇒   a∈F r m k a χ a (x 1 ), a∈F r m k a χ a (x 2 ), • • • , a∈F r m k a χ a (x r m )   T = 0. Hence, a∈F r m k a χ a (x i ) = 0 for any 1 ≤ i ≤ r m .
Given an element a 0 ∈ F r m , we have

a∈F r m k a χ a (x i )χ a 0 (-x i ) = 0 for any 1 ≤ i ≤ r m , =⇒ a∈F r m k a χ 1 ((a -a 0 )x i ) = 0 for any 1 ≤ i ≤ r m , =⇒ x∈F r m a∈F r m k a χ 1 ((a -a 0 )x) = 0, =⇒ a∈F r m k a x∈F r m χ 1 ((a -a 0 )x) = 0.
By Lemma 2.2, we obtain k a 0 r m = 0 and then k a 0 = 0 by (r, p) = 1. Because a 0 is arbitrary, we have k a = 0 for any a ∈ F r m . Hence, the r m vectors {η a := (χ a (x 1 ), χ a (x 2 ), • • • , χ a (x r m )) T : a ∈ F r m } are linearly independent over F q .

Therefore, the multisets Let the symbols be the same as above. According to Lemma 2.5(1), we obtain

λ a = y∈F r m ρ(y)χ a (y) y∈F r m ρ(y)χ a (-y) =   v + y∈F * r m ϕ(y)χ a (y)     v + y∈F * r m ϕ(y)χ a (-y)   = (v + g(ϕ, χ a ))(v + g(ϕ, χ a )) = v 2 + vg(ϕ, χ a ) + vg(ϕ, χ a ) + g(ϕ, χ a )g(ϕ, χ a ) = v 2 + vg(ϕ, χ a ) + ϕ(-1)vg(ϕ, χ a ) + ϕ(-1)(g(ϕ, χ a )) 2 = v 2 + (1 + ϕ(-1))vg(ϕ, χ a ) + ϕ(-1)(g(ϕ, χ a )) 2 .
Hence, all eigenvalues of P P T are given by the multiset

{λ a := v 2 + (1 + ϕ(-1))vg(ϕ, χ a ) + ϕ(-1)(g(ϕ, χ a )) 2 : a ∈ F r m }. (5)
Let C := C (ϕ,v) be a linear code over F q with generator matrix G = [I r m , P ]. Then C is a [2r m , r m ] linear code over F q . In Section 4, we construct LCD codes according to Lemma 2.7. In Section 5, we construct linear codes with one-dimensional hull by Lemma 2.8.

The constructions of LCD codes

In this section, we present two simple constructions of LCD codes by using the two homomorphisms (1) and [START_REF] Bosma | Hand-book of Magma functions[END_REF]. When m = 1 these codes are double circulant. In general, they are quasi-abelian of index 2 as F q [H]-submodules of F q [H] 2 with H the additive group of F r m [START_REF] Jitman | Quasi-abelian codes[END_REF]. Construction A. Define ρ(0) = v = 0. We then obtain a r m × r m matrix P = (p ij ) by

p ij = ρ(x j -x i ),
which is defined as (4). It follows from ( 5) that all eigenvalues of P P T are given by

λ a =    0, if a = 0; ϕ(-1)(g(ϕ, χ a )) 2 , if a ∈ F * r m . (6)
The following theorem gives the sufficient conditions for linear codes to be LCD codes by Construction A.

Theorem 4.1. Let r be a prime number and m be a positive integer. Assume that N > 1 is a positive integer and N |(r m -1). Let q be a power of prime p and (p, r) = 1. Assume that N |(q -1). Let C := C (ϕ,0) be the linear code over F q with generator matrix [I r m , P ]. Then we have the following.

(1) If there exists a positive integer s such that N |(p s -1) and ϕ(p 2s ) = 1, then C is a [2r m , r m ] LCD code over F q . In particular, if ϕ(q 2 ) = 1, then C is an [2r m , r m ] LCD code over F q .

(2) If there exists a positive integer s such that N |(p s + 1) and ϕ(p -2s ) = r 2m , then C is an [2r m , r m ] LCD code over F q .

Proof. It follows from ( 6) that all eigenvalues of P P T are 0 when a = 0 and ϕ(-1)(g(ϕ, χ a )) 2 when a ∈ F * r m . According to Lemma 2.7, we just have to prove that ϕ(-1)(g(ϕ, χ a )) 2 = -1 for any a ∈ F * r m . Assume on the contrary that there exists a ∈ F * r m such that ϕ(-1)(g(ϕ, χ a )) 2 = -1.

(1) If there exists a positive integer s such that N |(p s -1), we get ϕ p s = ϕ. Then (ϕ(-1)(g(ϕ, χ a )) 2 ) p s = (-1)

p s = -1 =⇒ (ϕ(-1)) p s (g(ϕ, χ a )) p s ) 2 = -1 =⇒ ϕ p s (-1)(g(ϕ p s , χ ap s )) 2 = -1 =⇒ ϕ(-1)(g(ϕ, χ ap s )) 2 = -1 =⇒ ϕ(-1)(ϕ(p s )g(ϕ, χ a )) 2 = -1 =⇒ ϕ(-1)(g(ϕ, χ a )) 2 = -(ϕ(p s )) -2 = -ϕ(p 2s ).
Combined ϕ(-1)(g(ϕ, χ a )) 2 = -1 with ϕ(-1)(g(ϕ, χ a )) 2 = -ϕ(p 2s ), we get ϕ(p 2s ) = 1 which is a contradiction. Hence, ϕ(-1)(g(ϕ, χ a )) 2 = -1 for any a ∈ F * r m . (2) If there exists a positive integer s such that N |(p s + 1), we get ϕ p s = ϕ -1 . Then (ϕ(-1)(g(ϕ, χ a )) 2 ) p s = (-1)

p s = -1 =⇒ ϕ p s (-1)(g(ϕ p s , χ ap s )) 2 = -1 =⇒ ϕ -1 (-1)(g(ϕ -1 , χ ap s )) 2 = -1 =⇒ ϕ -1 (-1)(ϕ -1 (p s )ϕ -1 (-1)g(ϕ -1 , χ -a )) 2 = -1 =⇒ ϕ -1 (-1)(g(ϕ -1 , χ -a )) 2 = -(ϕ -1 (p -s )ϕ -1 (-1)) -2 = -ϕ(p -2s ) =⇒ ϕ -1 (-1)(g(ϕ, χ a )) 2 = -ϕ(p -2s ). Combined ϕ(-1)(g(ϕ, χ a )) 2 = -1 with ϕ -1 (-1)(g(ϕ, χ a )) 2 = -ϕ(p -2s ), we get ϕ(-1)(g(ϕ, χ a )) 2 ϕ -1 (-1)(g(ϕ, χ a )) 2 = ϕ(p -2s ) =⇒ (g(ϕ, χ a )g(ϕ, χ a )) 2 = ϕ(p -2s ) =⇒ r 2m = ϕ(p -2s ) by Lemma 2.4,
which is a contradiction. Therefore, ϕ(-1)(g(ϕ, χ a )) 2 = -1 for any a ∈ F * r m .

To sum up, -1 is not an eigenvalue of the matrix P P T . By Lemma 2.7, C is an [2r m , r m ] LCD code over F q . This finishes the proof of the theorem.

Two concrete examples with respect to Theorem 4.1 are given as follows.

Example 4.2. Let r = 7, m = 1, N = 3, p = 2 and q = 4. Let F * 4 = β , where β is a fixed primitive element of F * 4 . It is easy to check that q, r, N satisfy the conditions in Theorem 4.1(1)(2). Then C is a quaternary [14, 7, 5] LCD code with generator matrix [I 7 , P ], where

P =              0 1 β 2 β β β 2 1 1 0 1 β 2 β β β 2 β 2 1 0 1 β 2 β β β β 2 1 0 1 β 2 β β β β 2 1 0 1 β 2 β 2 β β β 2 1 0 1 1 β 2 β β β 2 1 0             
, which is almost optimal in the sense that the minimum distance of the optimal quaternary linear code with the length 14 and the dimension 7 is 6 by the online Database [START_REF] Grassl | Bounds on the minimum distance of linear codes and quantum codes[END_REF]. Moreover, the dual code of C has parameters [START_REF] Liu | LCD codes over finite chian rings[END_REF][START_REF] Carlet | New characterization and parametrization of LCD codes[END_REF][START_REF] Carlet | Linear codes over F q are equivalent to LCD codes for q > 3[END_REF], which is also almost optimal.

Example 4.3. Let r = 3, m = 1, N = 2 and q = p = 5. It is easy to check that q, r, N satisfy the conditions in Theorem 4.1 [START_REF] Bosma | Hand-book of Magma functions[END_REF]. Then C is an [6, 3, 3] LCD code over F 5 with generator matrix [I 3 , P ], where

P =    0 1 -1 -1 0 1 1 -1 0    ,
which is almost optimal in the sense that the minimum distance of the optimal 5-ary linear code with the length 6 and the dimension 3 is 4 by the online Database [START_REF] Grassl | Bounds on the minimum distance of linear codes and quantum codes[END_REF]. Moreover, the dual code of C has parameters [START_REF] Carlet | Euclidean and Hermitian LCD MDS codes[END_REF][START_REF] Carlet | Complementary dual codes for counter-measures to side-channel attacks[END_REF][START_REF] Carlet | Complementary dual codes for counter-measures to side-channel attacks[END_REF], which is also almost optimal.

In view of Theorem 4.1, since the sufficient condition is abstract for a linear code to be an LCD code, we present a concrete result as corollary in the following.

Corollary 4.4. Let r be a prime number and m be a positive integer. Assume that N > 1 is a positive integer and N |(r m -1). Let q be a power of prime p and (p, r) = 1. Assume that N |(q -1). Let C := C (ϕ,0) be the linear code over F q with generator matrix [I r m , P ]. Then we have the following.

(1) If there exists a positive integer s such that N |(p s -1) and p 2s(r m -1) N ≡ 1 (mod r), then C is an [2r m , r m ] LCD code over F q .

(2) If there exists a positive integer s such that N |(p s + 1) and r 2mN ≡ 1 (mod p), then

C is an [2r m , r m ] LCD code over F q .

Proof. Compared with the conditions of Theorem 4.1, we just have to prove that (1) if p 2s(r m -1) N ≡ 1 (mod r), then ϕ(p 2s ) = 1 and (2) if r 2mN ≡ 1 (mod p), then ϕ(p -2s ) = r 2m , respectively.

(1) Assume on the contrary that ϕ(p 2s ) = 1, then p 2s ∈ ker(ϕ). Hence,

p 2s ∈ α N . Since ord(α N ) = r m -1 N , we have p 2s• r m -1 N ≡ 1 (mod r), it is a contradiction.
(2) Assume on the contrary that ϕ(p -2s ) = r 2m , then (ϕ(p -2s )) N = r 2mN . Hence, r 2mN ≡ 1 (mod p), it is a contradiction. This completes the proof.

Construction B. Define ρ(0) = v.
We then obtain a r m × r m matrix P = (p ij ) by

p ij = ρ(x j -x i ),
which is defined as (4). For any a ∈ F r m , we define f a : F q -→ F q by

f a (x) =    x 2 ,
if a = 0;

x 2 + (1 + ϕ(-1))ϕ(a)g(ϕ, χ 1 )x + ϕ(-1)(ϕ(a)g(ϕ, χ 1 )) 2 , if a ∈ F * r m . (7)
It follows from ( 5) that all eigenvalues of P P T are given by

λ a := f a (v)
for all a ∈ F r m .

In order to construct LCD codes over F q , we hope that there exists v ∈ F q satisfying f a (v) = -1 for any a ∈ F r m . Hence, we present a lemma as follows.

Lemma 4.5. Let the symbols be the same as above. If q > 2(N + 1), then there exists v ∈ F q satisfying λ a := f a (v) = -1 for any a ∈ F r m .

Proof. Since the order of ϕ is N , the set {f a (x) : a ∈ F r m } has at most N + 1 distinct polynomials. For any a ∈ F r m , f a (x) = -1 has at most two solutions in F q . Theorefore, all these equations in {f a (x) = -1 : a ∈ F r m } have at most 2(N + 1) solutions over F q . Since q > 2(N + 1), there exists an element v ∈ F q such that v is not a solution of any equation f a (x) = -1, i.e., there exists v ∈ F q satisfying λ a = f a (v) = -1 for any a ∈ F r m .

Based on the discussion above, we can easily get the following theorem.

Theorem 4.6. Let r be a prime number and m be a positive integer. Assume that N > 1 is a positive integer and N |(r m -1). Let q be a power of prime p and (p, r) = 1. Assume that N |(q -1). Let C := C (ϕ,v) be the linear code over F q with generator matrix [I r m , P ]. If q > 2(N + 1), then there exists v ∈ F q such that C is an [2r m , r m ] LCD code over F q .

Proof. By Lemmas 2.7 and 4.5, we can easily obtain the desired results. So we omit the detail here.

Next, we present an example to explain the result of Theorem 4.6. It is easy to check that q, r, N satisfy the conditions in Theorem 4.6. Then C is an [START_REF] Grassl | Bounds on the minimum distance of linear codes and quantum codes[END_REF][START_REF] Carlet | Linear codes with small hulls in semi-primitive case[END_REF][START_REF] Carlet | Linear codes with small hulls in semi-primitive case[END_REF] LCD code over F 25 with generator matrix [I 4 , P ], where

P =       β 2 β 16 β 8 1 β 16 β 2 1 β 8 β 8 1 β 2 β 16 1 β 8 β 16 β 2      
, which is an almost MDS code. Moreover, the dual code of C has parameters [START_REF] Grassl | Bounds on the minimum distance of linear codes and quantum codes[END_REF][START_REF] Carlet | Linear codes with small hulls in semi-primitive case[END_REF][START_REF] Carlet | Linear codes with small hulls in semi-primitive case[END_REF], which is also an almost MDS code.

The constructions of linear codes with one-dimensional hull

In this section, we present the constructions of linear codes with one-dimensional hull by using the two homomorphisms (1) and [START_REF] Bosma | Hand-book of Magma functions[END_REF]. In order to construct linear codes with onedimensional hull over F q , we hope that there exists v ∈ F q satisfying λ 0 = v 2 = -1 and λ a = -1 for any a ∈ F r m by Lemma 2.8. Let q be a power of a prime p. In what follows, we shall consider the construction dividing into two cases p = 2 and p ≥ 3.

The case p = 2

Define ρ(0

) = v = 1. Then v 2 = 1 = -1.
We then obtain a r m × r m matrix P = (p ij ) by

p ij = ρ(x j -x i ),
which is defined as (4). It follows from ( 5) that all eigenvalues of P P T are given by

λ a =    -1, if a = 0; -1 + (g(ϕ, χ a )) 2 , if a ∈ F * r m . (8)
Therefore, we present the following theorem.

Theorem 5.1. Let r be an odd prime number and m be a positive integer. Assume that N > 1 is a positive integer and N |(r m -1). Let q be a power of p = 2 and N |(q -1). Let C := C (ϕ,1) be the linear code over F q with generator matrix [I r m , P ]. Then C is a [2r m , r m ] linear code over F q with one-dimensional hull.

Proof. It follows from ( 8) that all eigenvalues of P P T are -1 when a = 0 and -1+(g(ϕ, χ a )) 2 when a ∈ F * r m . By using Lemma 2.8, we just have to prove that -1 + (g(ϕ, χ a )) 2 = -1 for any a ∈ F * r m . Note that the result g(ϕ, χ a )g(ϕ, χ a ) = r m for any a ∈ F * r m from Lemma 2.4. Then g(ϕ, χ a ) = 0 and g(ϕ, χ a ) = 0 for any a ∈ F * r m . Hence, (g(ϕ, χ a )) 2 = 0 and -1 + (g(ϕ, χ a )) 2 = -1 for any a ∈ F * r m . The desired conclusion then follows.

Here, we give a concrete example as follows.

Example 5.2. Let r = 13, m = 1, N = 3, p = 2 and q = 4. Let F * 4 = β , where β is a fixed primitive element of F * which is defined as (4). In addition, ϕ(-1) = ϕ(α

r m -1 2 ) = u r m -1 2 = (β q-1 N ) r m -1 2 = (β q-1 2 ) r m -1 N = (-1) r m -1 N . When r m -1 N is odd, ϕ(-1) = -1; when r m -1 N is even, ϕ(-1) = 1.
Combined with (5), when r m -1 N is odd, we have

λ a =    -1, if a = 0; -1 -(g(ϕ, χ a )) 2 , if a ∈ F * r m ; (9)
when r m -1

N is even, we get

λ a =    -1, if a = 0; -1 + (2v + g(ϕ, χ a ))g(ϕ, χ a ), if a ∈ F * r m . (10)
Collecting all discussions above, we first present the sufficient conditions for constructing linear codes with one-dimensional hull when r m -1 N is odd.

Theorem 5.3. Let r be a prime number and m be a positive integer. Assume that N > 1 is a positive integer and N |(r m -1). Let q be a power of prime p and (p, r) = 1. Assume that N |(q -1) and 4|(q -1).

Let C := C (ϕ,β q-1 4 )
be the linear code over F q with generator matrix

[I r m , P ]. When r m -1 N is odd, C is a [2r m , r m ] linear code over F q with one-dimensional hull.
Proof. The proof is similar to that of Theorem 5.1 and omitted here. It is easy to check that q, r, N satisfy the conditions in Theorem 5.3. Then C is a [START_REF] Li | Constrctions of linear codes with one-dimensional hull[END_REF][START_REF] Jin | Construction of MDS codes with complementary duals[END_REF][START_REF] Grassl | Bounds on the minimum distance of linear codes and quantum codes[END_REF] linear code over F 49 with one-dimensional hull and its generator matrix [I 9 , P ], where

P =                   β 12 β 42 β 6 β
                  .
Moreover, the hull of C is a [START_REF] Li | Constrctions of linear codes with one-dimensional hull[END_REF][START_REF] Assmus | Affine and projective planes[END_REF][START_REF] Li | Constrctions of linear codes with one-dimensional hull[END_REF] quasi-cyclic code of index 2 over F 49 with generator matrix

1 1 1 1 1 1 1 1 1 β 12 β 12 β 12 β 12 β 12 β 12 β 12 β 12 β 12 .
Compared with Example 2(2) in [START_REF] Li | Constrctions of linear codes with one-dimensional hull[END_REF], the linear code C over F 49 with one-dimensional hull we obtained has better parameters than its parameters. In other words, the linear code C of the length 18 with the dimension 9 has the minimal distance 8, while the linear code C of the length 18 with the dimension 9 in [18, Example 2(2)] has the minimal distance 7.

That is to say, the linear code C over F 49 with one-dimensional hull we obtained is also considered new. It is easy to check that q, r, N satisfy the conditions in Theorem 5.3. Then C is a [START_REF] Liu | LCD codes over finite chian rings[END_REF][START_REF] Carlet | New characterization and parametrization of LCD codes[END_REF][START_REF] Carlet | New characterization and parametrization of LCD codes[END_REF] linear code over F 25 with one-dimensional hull and its generator matrix [I 7 , P ], where

P =              2
            
, which is an almost MDS code. Moreover, the hull of C is a [14, 1, 14] quasi-cyclic code of index 2 over F 25 with generator matrix

1 1 1 1 1 1 1 2 2 2 2 2 2 2 .
Next, we turn to the sufficient conditions for constructing linear codes with one-dimensional hull when r m -1 N is even.

Theorem 5.6. Let r be a prime number and m be a positive integer. Assume that N > 1 is a positive integer and N |(r m -1). Let q be a power of prime p and (p, r) = 1. Assume that N |(q -1) and 4|(q -1).

Let C := C (ϕ,β q-1 4 )
be the linear code over F q with generator matrix [I r m , P ]. When r m -1 N is even and 2v + g(ϕ, χ a ) = 0 for all a ∈ F * r m , C is a [2r m , r m ] linear code over F q with one-dimensional hull.

Proof. It follows from [START_REF] Jin | Algebraic geometry codes with complementary duals exceed the asymptotic Gilbert-Varshamov bound[END_REF] that all eigenvalues of P P T are -1 when a = 0 and -1 + (2v + g(ϕ, χ a ))g(ϕ, χ a ) when a ∈ F * r m . According to Lemma 2.8, we just have to prove that -1 + (2v + g(ϕ, χ a ))g(ϕ, χ a ) = -1 for all a ∈ F * r m . By utilizing Lemma 2.4 and the proof of Theorem 5.1, we obtain that g(ϕ, χ a ) = 0 for any a ∈ F * r m . When 2v + g(ϕ, χ a ) = 0 for all a ∈ F * r m , we have -1 + (2v + g(ϕ, χ a ))g(ϕ, χ a ) = -1 for all a ∈ F * r m .

Therefore, the matrix P P T has an eigenvalue -1 with multiplicity 1. It then follows from Lemma 2.8 that C is a [2r m , r m ] linear code over F q with one-dimensional hull.

In Theorem 5.6, the condition "2v + g(ϕ, χ a ) = 0 for all a ∈ F * r m " is not very straightforward. Hence, we will present the following corollary as a concrete result.

Corollary 5.7. Let r be a prime number and m be a positive integer. Assume that N > 1 is a positive integer and N |(r m -1). Let q be a power of odd prime p and (p, r) = 1. Assume that N |(q -1) and 4|(q -1).

Let C := C (ϕ,β q-1 4 )
be the linear code over F q with generator matrix [I r m , P ]. Let r m -1 N be even. If ϕ(q) = 1, then C is a [2r m , r m ] linear code over F q with one-dimensional hull.

Proof. Since r m -1 N is even and it follows from (10) that all eigenvalues of P P T are -1 when a = 0 and -1 + (2v + g(ϕ, χ a ))g(ϕ, χ a ) when a ∈ F * r m . Suppose that 2v + g(ϕ, χ a ) = 0 for some a ∈ F * r m . Then g(ϕ, χ a ) = -2v ∈ F p when p ≡ 1 (mod 4) and g(ϕ, χ a ) = -2v ∈ F p 2 when p ≡ 3 (mod 4). In addition,

(g(ϕ, χ a )) q =   x∈F * r m ϕ(x)χ a (x)   q = g(ϕ q , χ aq ) = g(ϕ, χ aq ) = ϕ(q -1 )g(ϕ, χ a ) = ϕ(q) -1 g(ϕ, χ a )
by N |(q -1) and Section 3(3). If ϕ(q) = 1, then (g(ϕ, χ a )) q = g(ϕ, χ a ), i.e., g(ϕ, χ a ) / ∈ F q .

When p ≡ 1 (mod 4), F p ⊆ F q , which implies that g(ϕ, χ a ) / ∈ F p . It is a contradiction.

When p ≡ 3 (mod 4), F p 2 ⊆ F q by 4|(q -1), which implies that g(ϕ, χ a ) / ∈ F p 2 . It is a contradiction.

Hence, 2v + g(ϕ, χ a ) = 0. By using Lemma 2.4, we obtain that g(ϕ, χ a ) = 0. Then -1 + (2v + g(ϕ, χ a ))g(ϕ, χ a ) = -1 for all a ∈ F * r m . Thus the matrix P P T has an eigenvalue -1 with multiplicity 1. It then follows from Lemma 2.8 that the desired result then follows.

We now employ Corollary 5.7 to present a example as follows. It is easy to check that q, r, N satisfy the conditions in Corollary 5.7. Then C is a [START_REF] Liu | LCD codes over finite chian rings[END_REF][START_REF] Carlet | New characterization and parametrization of LCD codes[END_REF][START_REF] Carlet | Euclidean and Hermitian LCD MDS codes[END_REF] linear code over F 25 with one-dimensional hull and its generator matrix [I 7 , P ], where

P =              2
             .
Moreover, the hull of C is a [14, 1, 14] quasi-cyclic code of index 2 over F 25 with generator matrix

1 1 1 1 1 1 1 2 2 2 2 2 2 2 .
Furthermore, some optimal or almost optimal LCD codes (resp. linear codes with onedimensional hull) derived from Theorems 4.1 and 4.6 (resp. Theorems 5.1, 5.3 and Corollary 5.7) are listed in Table 1 by Magma [START_REF] Bosma | Hand-book of Magma functions[END_REF]. Remark 5.9. In Table 1, optimal linear codes with one-dimensional hull in [18, Section A] can also be obtained by our construction methods when N = 2 (see the first row, fourth row and fifth row of Table 1), which implies that our results contain partial results in [18, Section A]. When N > 2, the linear codes are different from those in [18, Section A]. In addition, although the second and third low parameters of Table 1 are the same, we verified by Magma that the two codes are not equivalent.

6 The minimum distance of the linear code C (ϕ,v)

In this section, we discuss the lower bound on the minimum distance of linear code C := C (ϕ,v) with generator matrix G = [I r m , P ] defined in Section 3.

Assume that q is a power of odd prime p and N

= 2. Let F r m = {x i : 1 ≤ i ≤ r m }, where x 1 , • • • , x r m -1 2 are non-zero squares in F r m , x r m +1 2 , • • • , x r m -1 are non-squares in F r m and
x r m = 0. From Section 3, we have P η a = θ a η a for any a ∈ F r m , where θ a := Note that when v = 0,

θ a =    0, if a = 0;
ϕ(a -1 )g(ϕ, χ 1 ), if a ∈ F * r m .

Let's just say g := g(ϕ, χ 1 ) for convenience.

It is easy to know that Λ = g(ϕ, χ 1 )

              1 . . . 1 -1 . . . -1 0               . (11)
It follows the definition of the linear code C that C can be expressed in the following form:

C = {c(k) = kG, k ∈ F r m q }, where k = (k 1 , k 2 , • • • , k r m ).
From [START_REF] Leon | Computing automorphism groups of error-correcting codes[END_REF], we obtain

x > r m -A -1. ( 13
)
Let u i = l i -gk i and w i = l i + gk i , where 1 ≤ i ≤ r m . Then

  y∈F r m ρ(y)χ a (y) : a ∈ F r m and y∈F r m ρ(y)χ a (-y) : a ∈ F r m present all eigenvalues of the matrix P and P T , respectively. To sum up, P P T η a = P y∈F r m ρ(y)χ a (-y) η a = y∈F r m ρ(y)χ a (y) y∈F r m ρ(y)χ a (-y) η a . Then the multiset λ a := y∈F r m ρ(y)χ a (y) y∈F r m ρ(y)χ a (-y) : a ∈ F r m presents all eigenvalues of the matrix P P T and {η a : a ∈ F r m } presents all eigenvectors of P P T .

Example 4 . 7 .

 47 Let r = 2, m = 2, N = 3, p = 5 and q = 25. Let F * 25 = β , where β is a fixed primitive element of F * 25 . Taking v = β 2 .

Example 5 . 4 .

 54 Let r = 3, m = 2, N = 8, p = 7 and q = 49. Let F * 49 = β , where β is a fixed primitive element of F * 49 .

Example 5 . 5 .

 55 Let r = 7, m = 1, N = 6, p = 5 and q = 25. Let F * 25 = β , where β is a fixed primitive element of F * 25 .

Example 5 . 8 .

 58 Let r = 7, m = 1, N = 3, p = 5 and q = 25. Let F * 25 = β , where β is a fixed primitive element of F * 25 .

  y∈F r m ρ(y)χ a (y) and η a:= (χ a (x 1 ), χ a (x 2 ), • • • , χ a (x r m )) T . Let Q := (η x 1 , η x 2 , • • • , η x r m ). Then P Q = (P η x 1 , P η x 2 , • • • , P η x r m ) = (θ x 1 η x 1 , θ x 2 η x 2 , • • • , θ x r m η x r m ) = (η x 1 , η x 2 , • • • , η x r m )Λ = QΛ,

For(l 1 - 2 ) = 0 ;(l 1 + 2 ,x r m +1 2 ,

 120122 any codeword c(k) in C, we have c(k) = kG = k(I r m , P ) = (k, kP ) = (k, l), where l := l(k) = kP= (k 1 , k 2 , • • • , k r m , l 1 , l 2 , • • • , l r m ).Multiply both sides of the equation l = kP by the matrix Q, we obtainlQ = kP Q = kQΛ.Based on the above discussion and combined with Eq. (11), we have the following three equations:gk 1 , • • • , l r m -gk r m )(η x 1 , • • • , η x r m -1 gk 1 , • • • , l r m + gk r m )(η x r m +1 • • • , η x r m -1 ) = 0; l 1 + • • • + l r m = 0.In view of the above three equations, we present the following theorem. Before we do that, let's give some definitions. We define (ηx 1 , • • • , η x r m -1 • • • , η x r m -1 ).Theorem 6.1. Let C := C (ϕ,0) be a linear code over F q with generator matrix G = [I r m , P ] defined by Section 3. Let A be a positive integer. Assume that any A vectors in {µ x 1 , • • • , µ x r m } are linearly independent and any A vectors in{ν x 1 , • • • , ν x r m } are also linearly independent. Then d min (C) ≥ A + 1. Proof. Suppose that c(k) is any codeword in C which satisfies that wt(c(k)) < A + 1. Note that c(k) = (k, kP ) = (k, l) = (k 1 , • • • , k r m , l 1 , • • • , l r m ). Set Ω := {(l 1 , k 1 ), • • • , (l r m , k r m )}. Let x = #{(l i , k i ) ∈ Ω | (l i , k i ) = (0, 0) and 1 ≤ i ≤ r m }, y = #{(l i , k i ) ∈ Ω |Only one of l i and k i is 0 and 1 ≤ i ≤ r m } and z = #{(l i , k i ) ∈ Ω | l i = 0 and k i = 0, and 1 ≤ i ≤ r m }. Then we have    x + y + z = r m 2x + y > 2r m -A -1.

(u 1 ,= u 1 µ x 1 +

 11 • • • , u r m ) • • • + u r m µ x r m = 0 (14) and (w 1 , • • • , w r m ) w 1 ν x 1 + • • • + w r m ν x r m = 0.

  30 1 β 36 β 18 β 12 6 β 18 β 12 1 β 12 β 36 β 6 β 42 6 β 30 β 30 6 β 12 β 6 β 18 β 42 β 12 1 β 36 β 6 β 36 β 30 β 12 β 12 6 1 β 18 β 42 6 β 12 β 42 β 36 β 12 β 18 β 30 β 6 1 β 12 β 30 β 18 1 β 42 β 12 6 β 36 β 6 β 42 β 18 β 36 6 β 6 1 β 12 β 30 β 12 β 36 1 6 β 42 β 30 β 12 β 6 β 12 β 18 1 β 6 β 12 β 18 6 β 30 β 36 β 42 β 12

* This research is supported by the National Natural Science Foundation of China under Grant 11771007 and Grant 61572027.

The case p ≥ 3

In this subsection, we let F * q = β , where β is a fixed primitive element of F * q . Assume that 4|(q -1).

Define ρ(0

We then obtain a r m × r m matrix P = (p ij ) by p ij = ρ(x j -x i ),

According to [START_REF] Leon | Permutation group algorithms based on partition, I: theory and algorithms[END_REF], it is easy to know that there are at least r m -A zeros in u 1 , • • • , u r m . Similarly, there are also at least r m -A zeros in w 1 , • • • , w r m . Without loss of generality, let's assume that u A+1 = • • • = u r m = 0. Combined Eq. ( 14) and µ x 1 , • • • , µ x A are linearly independent, then we have

Similarly, we also deduce Conjecture 6.3. Let p > 3, r be two distinct prime numbers and m a positive integer. Assume that N = 2. Let C := C (ϕ,v) be a linear code over F q with generator matrix G = [I r m , P ], where P is defined by Section 3. Then (1) When v = 0, we have

(2) When v = 0 and r m ≡ 1 (mod 4), we have

Remark 6.4. Example 4.3 and some examples in Table 1 can illustrate the validity of the above results. In fact, we have tried a lot of examples by Magma, the conjecture is also correct. But we fail to prove it. Thus we would like to put it here as a conjecture.

Conclusion

In this paper, we propose a general method to construct LCD codes and linear codes with one-dimensional hull through the homomorphisms from finite fields into finite fields. Based on the eigenvalues of the matrix P P T , some sufficient conditions for a linear code to be an LCD code (resp. a linear code with one-dimensional hull) have been presented in this paper. With these conditions, we obtain some optimal and almost optimal LCD codes (resp. linear codes with one-dimensional hull) by Magma [START_REF] Bosma | Hand-book of Magma functions[END_REF], which are exhibited in Table 1. Additionally, we also obtain several almost MDS LCD codes (resp. almost MDS codes with one-dimensional hull) (see Examples 4.7 and 5.5).

Compared with [START_REF] Li | Constrctions of linear codes with one-dimensional hull[END_REF], their construction methods are specific and special, while our methods are more general and direct. It is mainly reflected in three aspects:

1. In [START_REF] Li | Constrctions of linear codes with one-dimensional hull[END_REF], the matrix P studied by the authors satisfies the symmetry property, while the matrix P we employed in this paper is a general matrix whose eigenvalues are completely determined;

2. In [START_REF] Li | Constrctions of linear codes with one-dimensional hull[END_REF], the authors constructed linear codes with one-dimensional hull over finite fields by using the generator matrix over quadratic number fields, while we construct them directly by utilizing the generator matrix over finite fields;

3. Taking N = 2, we obtain that [18, Theorem 5] is a special of our results in Theorem 5.3 by comparing the constraints. The results of Theorem 5.6 contain [18, Theorems 3 and 4]. In some sense, some of linear codes with one-dimensional hull we constructed may be new when N > 2 by comparing with [START_REF] Li | Constrctions of linear codes with one-dimensional hull[END_REF] (see Example 5.4). In addition, we present a lower bound on the minimum distance of linear code C over F q with generator matrix G = [I r m , P ] when N = 2.

We should emphasize that our results apply to (p, r) = 1. It would be interesting to extend the results of the present work to p = r. The main open problem is Conjecture 6.3. In addition, although there are many LCD codes and linear codes with one-dimensional hull, it seems to be difficult to determine the minimum distances of the codes presented in this paper when N > 2. It will be of interest to find other constructions such that the minimum distances of these codes can be determined.

Table 1: The list of optimal or almost optimal linear codes over small fields r, m, N The codes with asterisk ( * ) have the property that linear codes are best known q-ary linear codes in [START_REF] Bosma | Hand-book of Magma functions[END_REF], which is optimal. The codes with asterisk ( ) have the property that linear codes have better parameters according to the Database [START_REF] Bosma | Hand-book of Magma functions[END_REF], which is almost optimal in the sense. For example, the linear code over F 9 of the length n = 10 with the dimension k = 5 has the minimum distance 5, while the code in the Database [START_REF] Bosma | Hand-book of Magma functions[END_REF] has the minimum distance 6.


J Kuhlmann
email: kuhlmann@tfd.mw.tum.de

S Marragou

I Boxx

T Schuller

W Polifke

LES based prediction of technically premixed flame dynamics and comparison with perfectly premixed mode

The present study combines Large Eddy Simulation (LES) with System Identification (SI) to determine the Flame Transfer Functions (FTF) of technically premixed flames, which respond to fluctuations of upstream velocity as well as equivalence ratio. Two variants to obtain the corresponding FTFs from numerically determined time series data are reported and compared with experimental results. The experiment does not measure heat release rate directly, but instead CH* chemiluminescence. This is insufficient for FTF identification of technically premixed flames, but can be used for validation of the simulation. We implemented a CH* post-processor in the simulation and validate with experiment. After validation the simulation is used to identify the contributions of velocity and equivalence ratio to the FTF of technically premixed flame dynamics. We propose and compare two approaches for the identification of FTFs. The direct approach via Multiple Input Single Output system identification requires one simulation with simultaneous excitation of fuel and air inlets and carefully chosen input signals. The second approach reconstructs the FTF decomposition from two separate simulations, one perfectly premixed and one technically premixed, with reduced requirements on signal quality. We compare both approaches and discuss the FTFs of perfectly and technically premixed flames. Overall the LES/SI approach proved to be flexible and reliable for technically premixed flames.

I. INTRODUCTION

Thermoacoustic instabilities are a major problem in almost all lean premixed or high power combustion systems, e.g. gas turbines or rocket engines [START_REF]Combustion Instabilities in Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling[END_REF] . Directly assessing the stability of a given system either experimentally or numerically 2 is possible, but may be prohibitively expensive if several operating conditions have to be considered. The network model approach for thermoacoustic stability analysis reduces the complexity significantly [START_REF] Polifke | Low-order analysis tools for aero-and thermoacoustic instabilities[END_REF] . This approach requires an accurate, robust and efficient prediction of the Flame Transfer Function 4 (FTF), which couples acoustics with flame dynamics and vice versa. The flame dynamics of perfectly premixed flames has been investigated extensively. The FTF can be determined by experiment [START_REF] Palies | The combined dynamics of swirler and turbulent premixed swirling flames[END_REF][START_REF] Cuquel | Theoretical and experimental determination of the flame transfer function of confined premixed conical flames[END_REF][START_REF] Ćosić | Response of a swirl-stabilized flame to simultaneous perturbations in equivalence ratio and velocity at high oscillation amplitudes[END_REF] or simulation [START_REF] Polifke | Black-box system identification for reduced order model construction[END_REF][START_REF] Han | Prediction of combustion instability limit cycle oscillations by combining flame describing function simulations with a thermoacoustic network model[END_REF] . Experiments are expensive and time consuming to build, but can produce long and extensive time series data for FTF identification. Numerical simulation, on the other hand, can be set up quickly and may even be automatized, but requires significant computational resources. A multi-fidelity approach for LES/SI can reduce return time and at the same time increase accuracy significantly [START_REF] Kuhlmann | A top level parallelization and data fusion approach for identification of flame transfer functions with increased reliability, accuracy and efficiency[END_REF] . In high power technical systems, perfect premixing cannot be achieved. Insufficient mixing and modulation of the mixing process by acoustic fluctuations cause mixture inhomogeneities, which lead to additional flame and flow perturbations [START_REF] Ćosić | Response of a swirl-stabilized flame to simultaneous perturbations in equivalence ratio and velocity at high oscillation amplitudes[END_REF][START_REF] Lieuwen | The role of equivalence ratio oscillations in driving combustion instabilities in low nox gas turbines[END_REF] . The interference of several mechanisms of acoustic/flow/flame interaction may provoke or dampen instability [START_REF] Huber | Impact of Fuel Supply Impedance on Combustion Stability of Gas Turbines[END_REF][START_REF] Huber | Dynamics of practical premixed flames, part ii: Identification and interpretation of cfd data[END_REF] . In order to make accurate stability predictions, all interaction mechanisms must be identified [START_REF] Ducruix | Combustion dynamics and instabilities: Elementary coupling and driving mechanisms[END_REF] . For both experiment and simulation, the choice of the excitation scheme must be selected carefully. A high quality signal generation is critical to correctly distinguish between the mechanisms. Experiments are often restricted due to insufficient diagnostics and construction constraints, limiting access for measurements and the excitation approach. Simulation, on the other hand, faciliates excitation of the system in a versatile manner and optimal identification of the system dynamics. Numerical simulation depends on an adequate choice of the combustion model, turbulence model, numerical schemes, mesh base and quality, boundary conditions and much more to ensure a correct reproduction of flame dynamics [START_REF] Han | Prediction of combustion instability limit cycle oscillations by combining flame describing function simulations with a thermoacoustic network model[END_REF][START_REF] Tay-Wo-Chong | Comparative Validation Study on Identification of Premixed Flame Transfer Function[END_REF][START_REF] Wysocki | Effect of fuel mixture fraction and velocity perturbations on the flame transfer function of swirl stabilized flames[END_REF][START_REF] Tay-Wo-Chong | Les combustion model with stretch and heat loss effects for prediction of premix flame characteristics and dynamics[END_REF][START_REF] Chatelier | Experimental and numerical investigation of the response of a swirled flame to flow moddulations in a nonadiabatic combustor[END_REF][START_REF] Kuhlmann | Assessing accuracy, reliability and efficiency of combustion models for prediction of flame dynamics with large eddy simulation[END_REF][START_REF] Dupuy | Combining analytical models and les data to determine the transfer function from swirled premixed flames[END_REF] . Obviously, a valid estimate of the FTF can only be identified if the simulation as well as the identification approach are correct. It can be summarized that simulation may offer more flexibility, reliability and shorter study durations, but may be difficult to set up and therefore needs experiments for validation. The PRECCINSTA gas turbine model combustor developed by Turbomeca and investigated at DLR over the past 15 years has helped deepen our understanding of the intriguing dynamics of swirling flows and swirling flames interacting with acoustic waves (see for examples [START_REF] Meier | Detailed characterization of the dynamics of thermoacoustic pulsations in a lean premixed swirl flame[END_REF][START_REF] Steinberg | Flow-flame interactions causing acoustically coupled heat release fluctuations in a thermo-acoustically unstable gas turbine model combustor[END_REF][START_REF] Boxx | Temporally resolved planar measurements of transient phenomena in a partially pre-mixed swirl flame in a gas turbine model combustor[END_REF] ). Unfortunately, until now the flame transfer function, which describes the response of heat release rate to acoustic perturbations, has not been characterized for this burner. Knowledge of the flame response is necessary to fully assess and interpret the combustor stability with respect to self-sustained thermoacoustic instabilities [START_REF] Dowling | Acoustic analysis of gas turbine combustors[END_REF][START_REF] Sattelmayer | Assessment of methods for the computation of the linear stability of combustors[END_REF][START_REF] Poinsot | Prediction and control of combustion instabilities in real engines[END_REF] . The PRECCINSTA combustor can be operated in fully premixed as well as technically premixed (also: "partially premixed") conditions. In the latter mode of operation, the fuel is injected through small holes located within the radial swirl channel, resulting in a fuel/air mixture at the burner outlet that is not fully homogeneous. Furthermore, in the technically premixed case acoustic oscillations in the fuel injection and mixing sections will in general lead to modulation of the equivalence ratio [START_REF] Agostinelli | On the impact of h2 -enrichment on flame structure and combustion dynamics of a lean partially-premixed turbulent swirling flame[END_REF] . Indeed, it has been shown that self-sustained combustion oscillations in PRECCINSTA combustor are accompanied by large fluctuations of the mixture composition at the burner outlet [START_REF] Meier | Detailed characterization of the dynamics of thermoacoustic pulsations in a lean premixed swirl flame[END_REF][START_REF] Stoehr | Interaction between velocity fluctuations and equivalence ratio fluctuations during thermoacoustic oscillations in a partially premixed swirl combustor[END_REF] . The objective of the present study is to determine the FTF of methane/air flames in the PRECCINSTA burner. The FTF is identified not only for perfectly premixed, but also for technically premixed conditions. In the experiment, the transfer function from velocity to CH * chemiluminescence intensity is determined. These data are used to validate the numerical simulation for technically premixed conditions. The LES is used to generate multiple data sets for identification of the velocity and equivalence ratio contribution to the FTF. Two identification approaches are tested and evaluated. The Multiple-Input, Single-Output (MISO) identification method [START_REF] Huber | Dynamics of practical premixed flames, part ii: Identification and interpretation of cfd data[END_REF][START_REF] Huber | Dynamics of practical premixed flames, part i: Model structure and identification[END_REF] is tested for the first time in an LES/SI based study of thermoacoustics. The paper is structured as follows. The next section contains a detailed discussion of the test case, experimental setup and simulation model and then validates the simulation results for the CH * transfer function (CTF) against the experimental data. In section III the identification approaches are discussed and evaluated in detail.

II. EXPERIMENTAL AND NUMERICAL SETUP

The PRECCINSTA burner is a swirler stabilized burner operating in perfectly or technically premixed conditions. When it operates in perfectly premixed conditions, the air and the fuel mixture is injected at the bottom of the plenum, crosses in a radial swirler and flows into the combustion chamber. In technically premixed configuration, only air is injected at the bottom of the plenum and the fuel is injected through small holes as a jet-in-cross-flow inside the swirler vane leading to partial mixing. It is operated with methane and air under atmospheric conditions. The reference case for our study has an equivalence ratio of 0.8 and a fixed thermal power of 10 kW.

A. Experiment

The original PRECCINSTA combustor has been slightly modified for the FTF measurements by replacing the bottom plate of the plenum by a transition component to fix a loudspeaker as shown in Figure 1.

In this version, air is injected from two lateral holes at the bottom of the plenum. To allow the FTF estimation based on CH* chemiluminescence, a photomultiplier is installed in front of the flame on one side of the transparent combustion chamber. The photomultiplier is equipped with a CH * narrow bandpass filter (430 ± 5 nm). The reference velocity is measured in cold flow conditions (i.e. without combustion) with a hot-wire probe installed 1 mm above the injector outlet. In order to have physical access with the hot-wire probe to the outlet of the injector, a hole was drilled in one of the quartz windows of the combustion chamber. Several tests were made to determine the radial position of the hot-wire probe providing a representative signal of the oscillations of the bulk flow at the injector outlet. Experiments were only conducted for technically premixed injection conditions in which case the relation between the CH * flame luminosity and heat release rate is questionable. The CTF is determined by imposing a fixed modulation level of 15% on the flow velocity at the hot wire location when the flow is submitted to harmonic excitation with the loudspeaker:

Ĩ = F CH * ũ, (1)
where Ĩ and ũ denote the Fourier transforms -normalized by their mean respective values -of the time series recorded by the hot wire and the photomultiplier.

Since CH * chemiluminescence intensity depends also on equivalence ratio and equivalence ratio can not be measured with the experimental setup, the FTF can not be reconstructed from experimental data only and numerical simulation is needed. In order to validate the numerical results, an experimentally determined CH * predictor is required for the simulation.

A relation between the CH * chemiluminescence intensity I with the thermal power Q and the equivalence ratio φ is adapted from Higgins [START_REF] Higgins | An experimental study on the effect of pressure and strain rate on ch chemiluminescence of premixed fuel-lean methane/air flames[END_REF] and written as

I = k Qφ β ,
where k and β are two constants that are determined experimentally. First, the global equivalence ratio φ is set to 0.8 and the thermal power Q is varied. The voltage provided by the photomultiplier is plotted as a function of the thermal power Q in Figure 2 (top axis). The experiments are then repeated at a thermal power fixed to 10 kW and the equivalence ratio is varied (bottom axis).

The linearity of I with the thermal power Q31 is globally verified and the constant k is estimated to 0.98. The small non-linearity visible can be attributed to the effect of strain on the chemiluminescence 32 . The non-linearity of the CH * chemiluminescence with the equivalence ratio φ is verified and the exponent β is estimated to 3.81.

In both cases, the flame with a thermal power of 10 kW and a global equivalence ratio of 0.8 is considered as the reference case and the gain of the photomultiplier was set in order to obtain a mean output voltage I = 4 V for this operating condition.

FIG. 2: CH * chemiluminescence intensity as a function of thermal power Q for a fixed equivalence ratio φ = 0.8 (top axis, red) and as a function of equivalence ratio φ for fixed thermal power of 10 kW (bottom axis, blue)

For the selected operating point of Q = 10 kW and φ = 0.8, the thermoacoustic state is considered to be globally stable. The Power Spectral Density (PSD) detected by the photomultiplier is plotted in Figure 3 for cases when the loudspeaker is turned off (i.e. without external forcing) and when the loudspeaker is submitted to an harmonic excitation at 180 Hz, respectively. Although the setup is considered to be globally stable, there is some coherent activity characterized by small bumps around f ≈ 51 Hz, 146 Hz and 253 Hz clearly emerging from the background noise. Figure 4 shows a modal analysis of the setup using an external loudspeaker and recording the acoustic signal inside the plenum. The signal is normalized by the pressure signal p ref recorded by a microphone placed in front of the external loudspeaker. These experiments were made without combustion and without flow, the loudspeaker at the bottom was also turned off. Acoustic modes at lower frequencies were not detected in the plenum or the combustion chamber, implying that the peaks detected at f ≈ 51 Hz and 146 Hz have a different origin. The tiny peak around 50 Hz may be due to an interference with the electrical network. The peak at 146 Hz may be associated with a hydrodynamic instability of the swirl flow or result from resonance with the intrinsic thermoacoustic feedback loop.

B. Numerics

We employ a generic meshing, modelling, solving and analysis process as applied in our previous publications 10, [START_REF] Kuhlmann | Assessing accuracy, reliability and efficiency of combustion models for prediction of flame dynamics with large eddy simulation[END_REF] . The highly automated approach allows fast adaption to We concentrate on a predictive evaluation and do not tune the LES. In contrast to our prior studies, the combustion model has been extended for inhomogeneous fuel-air mixtures instead of perfectly premixed gas only. The extension of the Artificial Thickened Flame (ATF) model for inhomogeneous mixture is trivial. Thanks to its generality, replacing the model constants with prior calculated function over equivalence ratio from fully resolved laminar 1-D simulations is enough.

Turbulent Combustion LES

The pressure based reactingFoam solver (OpenFOAM) was modified to weakly-compressible form and ex-tended by a mixture-ratio parameterized version of the ATF model with local sensor and BFER chemistry mechanism 2 . The weakly compressible formulation (ρ is only a function of T ) has several advantages compared to the compressible one. It lifts the requirement of complex acoustic boundary conditions with unknown acoustic parameters, allows an increased time step size, prevents unstable thermoacoustic modes and thereby increases signal to noise ratio.

In the ATF model [START_REF] Colin | A thickened flame model for large eddy simulations of turbulent premixed combustion[END_REF] , discrete species are transported and the reaction rates are calculated by chemistry mechanism integration. Artificial diffusivity is introduced in the flame region, which spreads the flame over a given number of cells and allows direct computation of reaction rates via Arrhenius law. Since the thickened flame is less sensitive to wrinkling, a correction must be employed [START_REF] Charlette | A power-law flame wrinkling model for les of premixed turbulent combustion part i: non-dynamic formulation and initial tests[END_REF] .

For technically premixed injection the model parameters, laminar flame speed, flame thickness and flame tracking function, are functions of equivalence ratio obtained from 1-D laminar fully resolved simulations. One of the biggest advantages of the model is its generality. Fuel mixtures, varying degrees of premixing, heat loss, strain, geometric influences are directly resolved to a great extent and do not require further modeling. The Wall-Adapting Local Eddy-viscosity (WALE) turbulence model adapts consistently towards the wall, does not require wall functions or full wall resolution, and therefore reduces the conditions on the mesh and allows to simplify the mesh generation process. We checked the resolution of the sub-grid scale with Pope criterion and estimation of the Taylor scale. Using the values provided by Moureau et al. [START_REF] Moureau | From large-eddy simulation to direct numerical simulation of a lean premixed swirl flame: Filtered laminar flame-pdf modeling[END_REF] , the Taylor length scale can be estimated to 0.7 mm for the 30 kW case. Due to the reduced Reynolds number for the 10 kW case, this value will even rise. The grid with 0.5 mm in the flame zone and further refinement at the walls and near the injectors results in fine LES resolution. The time step size is set at 1 × 10 -6 s for the technically premixed mode, which corresponds to a maximum CFL number of 0.4 at the injectors and typical one in the flame zone of 0.1. For the perfectly premixed mode with zero mass flow through the fuel injectors, the time step size is less restricted and can be increased to 2 × 10 -6 s.

Actually, only a few cells at the injector edges are badly cut with a smaller volume and CFL numbers above 0.3, all other cells have CFL numbers well below 0.3. This ensures correct time resolution. All equations are solved in a segregated approach with PISO pressure-velocity coupling.

Discretization schemes are all second order. We chose implicit in time (backward), central for gradients with cellbased limiter (cellLimited linear 1.0), and central gradient schemes with minimal limiting to upwind gradient in regions of highly varying gradients (limitedLinear 0.1, V for velocity, 01 for scalar, linear else) for divergence terms.

Boundary Conditions

For this case the swirler produces most of the turbulence and a turbulent inflow model is not required. All walls are assumed isothermal and are set to mean values from Conjugate Heat Transfer (CHT) simulation [START_REF] Agostinelli | Impact of wall heat transfer in large eddy simulation of flame dynamics in a swirled combustion chamber[END_REF] . Applying CHT would be the more robust and general approach but we assume the heat loss modeling to be good enough considering the other uncertainties. No-slip condition is imposed at walls and velocity signals are imposed at air and fuel inlets using "uniformFixedValue" specifying a "tableFile" . The outlet imposes a total pressure and zero gradient else. We employ broad-band signals with fixed amplitude of 15%. The broad-band excitation signal must exhibit minimal crest factor and auto-correlation, i.e. maximum amplitude utilization and signal content uniqueness, in order to investigate the system in an optimal manner and reduce required time series lengths. Furthermore, it must FIG. 5: Details of computational domain and mesh topology.

be smooth enough regarding the simulation time step to generate a stable and properly resolved simulation. We applied a Daubechies wavelets based signal [START_REF] Föller | Advances in identification techniques for aero-acoustic scattering coefficients from large eddy simulation[END_REF] offering the best compromise between all requirements.

Mesh

The mesh was built semi-automatically from an uniform background mesh and the geometry's stl file with Open-FOAM's built-in mesher snappyHexMesh. Only a few refinement regions and surfaces were specified to obtain the mesh used. Within the meshing process in snap-pyHexMesh the mesh starts with a hex cell base and is refined, trimmed and locally deformed to conform to the geometry and quality criteria. The resulting mesh may comprise arbitrary polyhedra but consists mostly of hex cells. It has around 5 million cells, very low nonorthogonality < 45, and skewness < 2. In the flame region the cells are of uniform size of approximately 0.5 mm, roughly equal to the flame thickness for this case, leading to a maximum thickening factor of 9. The cells close to the injector and within their jets, swirler and bluff body walls are refined one additional level in order to accurately reproduce the geometry and the corresponding flow field.

Figure 5 shows a sketch of the computational domain and mesh topology. Our mesh has comparable resolution as the optimized one from Agostinelli et al. [START_REF] Agostinelli | Static mesh adaptation for reliable large eddy simulation of turbulent reacting flows[END_REF] used at the same case showing very good reproduction [START_REF] Agostinelli | Impact of wall heat transfer in large eddy simulation of flame dynamics in a swirled combustion chamber[END_REF] of the experimental data. A further refinement of up to 0.25 mm in the flame zone could not improve the overall flame shape, heat release and dynamic behavior noticeably.

CH* Predictor

A predictor for the measured CH * is included in the simulation. It relates the local heat release rate Q and the equivalence ratio φ to the CH * intensity signal I as

I = k Qφ β (2)
The coefficients k = 0.98 and β = 3.81 are determined from two steady state experiments, one with constant power and varying mixture ratio, and vice versa, see Figure 2. The global experimental correlations are locally applied in the LES and integrated over the volume.

C. Validation of LES

The validation process is twofold, first we compare steady state results in terms of mean flame shape and second we compare flame dynamics in terms of FTF.

FIG. 6: Mean heat release rate for different simulations: current simulation with fixed temperature BC (T) vs. reference from Agostinelli et al. [START_REF] Agostinelli | Impact of wall heat transfer in large eddy simulation of flame dynamics in a swirled combustion chamber[END_REF] without CHT (R).

Figure 6 shows the numerically predicted mean heat release rate from our simulation with fixed temperature BC vs. the reference simulation from Agostinelli et al. [START_REF] Agostinelli | Impact of wall heat transfer in large eddy simulation of flame dynamics in a swirled combustion chamber[END_REF] without CHT. It is important to note that our simulation employs a weakly-compressible formulation while the reference employs a fully-compressible one. The reference simulation exhibits a thermoacoustic limit cycle, which is by design not present in our simulation. Flame length and angle agree very well. The current simulation shows a more compact flame branch and higher activity towards the flame foot. This can be explained by the coarser mesh and therefore increased thickening applied in this simulation. For similar cases, a finer mesh results in increased liftoff height and a thicker flame branch [START_REF] Benard | Large-eddy simulation of the lean-premixed preccinsta burner with wall heat loss[END_REF] . Nonetheless the mean flame shape is qualitatively well recovered. Since equivalence ratio measurement is unavailable in the current experiment, the FTF can not be reconstructed and the CTF is used for validation of the flame dynamics. We therefore compare the transfer functions from velocity to numerically predicted and experimentally measured CH * shown in Figure 7. The numerically predicted phase of the CH * transfer function -see Figure 7 (bottom) -matches the experimentally measured one with good accuracy over the complete range of frequencies considered. The gain, on the other hand, shows discrepancies at low frequencies and in the vicinity of 260 Hz. The latter may be explained by the fact that the experiment suffers from a resonance around 260 Hz, see Figure 4, which results in poor signal-to-noise ratio in this frequency range. The discrepancies at low frequency result to a certain extent from vibrations of the chamber walls, which were observed in experiment but were of course absent in LES. More important is, however, the acoustic impedance of the fuel injection system. If a stiff injector is assumed, as set in the simulation, the low frequency limit of the CH * predictor can be approximated analytically, inspired by 40 .

In the low frequency limit and for stiff fuel injection, an increase in air mass flow rate δu results in zero change in heat release rate δ Q. For a stiff fuel injector, equivalence ratio perturbations are directly anti-proportional to velocity perturbations at the injector:

δφ φ = - δu u (3)
The low frequency limit of the CH * predictor can be expressed as

δI I = β δφ φ = -β δu u (4)
In the limit of zero frequency, this gives values of β = 3.81 and π for gain and phase of the CTF, respectively. The CH * transfer function determined with LES and shown in Figure 7 exhibits this behavior in agreement with the assumptions made.

In the experiments, on the other hand, acoustic activity has been detected in the fuel supply line during selfsustained oscillations as well as in forced flow conditions. These observations suggest that the fuel line also responds to low frequency perturbations. Assuming a quasi-steady response of the fuel line to external pressure perturbations, a link may be established between the heat release rate and CH * luminosity. A perturbation analysis yields

δφ φ = δ ṁf ṁf - δ ṁa ṁa ≈ δ Q Q - δu u (5)
Combining this expression with Eq. ( 2) one obtains

δ Q Q = 1 1 + β δI I + β δu u , (6)
which accounts for the qualitative discrepancies between experimental and numerical CTF seen in Figure 7 at low frequencies.

The origin of other discrepancies are more difficult to identify. The bump in the experimental CTF around 250 Hz could probably be attributed to the acoustic mode of the plenum at around 220 Hz. Furthermore, the loudspeaker calibration used to fix the amplitude of the velocity perturbations in the experiment was set in cold flow conditions, while the simulation used direct measurement of the velocity in hot conditions. Taking the experimental uncertainties and the assumptions for the numerical CH * predictor into account, the agreement between numerical and experimental results are reasonable for the gain and very good for the phase.

Since the numerical simulation is now verified, we want to take a closer look on the decomposition of the FTF into its contributions by velocity and equivalence ratio.

III. IDENTIFICATION APPROACH

In perfectly premixed mode the flame responds to velocity and swirl perturbations only. In technically premixed mode the flame additionally responds to equivalence ratio perturbation. This can be expressed in terms of a combined FTF [START_REF] Ćosić | Response of a swirl-stabilized flame to simultaneous perturbations in equivalence ratio and velocity at high oscillation amplitudes[END_REF][START_REF] Huber | Dynamics of practical premixed flames, part ii: Identification and interpretation of cfd data[END_REF][START_REF] Huber | Dynamics of practical premixed flames, part i: Model structure and identification[END_REF] . The MISO approach identifies the contributions of velocity and equivalence ratio directly from input-output data of one single simulation with simultaneous excitation of the input channels. Therein uncorrelated signals must be generated to distinguish the contributions otherwise the contributions may be wrongly attributed. In experiment the excitation of the input lines is limited. Gaseous fuel injectors may already be choked or the equipment used for excitation may be insufficient. The generated signals are often correlated and do not satisfy the high requirements for MISO identification.

The double Single-Input, Single-Output (SISO) identification approach [START_REF] Ćosić | Response of a swirl-stabilized flame to simultaneous perturbations in equivalence ratio and velocity at high oscillation amplitudes[END_REF] does not require uncorrelated input channels. In fact two separate simulations are performed, one with perfectly premixing and one with technically premixing. Their results are used to reconstruct the equivalence ratio transfer function. Therefore the equivalence ratio must be known at the reference point, either by measurement or by transport model 41 prediction. We want to compare both identification approaches qualitatively in a numerical study.

A. Double SISO Identification

In the perfectly premixed case fuel and air mixture is injected in the air inlet only. In the technically premixed case fuel is injected through injectors in the swirler vanes and mixed with the air cross flow. The technically premixed FTF F tp consists of velocity F u and equivalence ratio F φ FTFs [START_REF] Huber | Dynamics of practical premixed flames, part ii: Identification and interpretation of cfd data[END_REF][START_REF] Huber | Dynamics of practical premixed flames, part i: Model structure and identification[END_REF] , with Ũ and φ being the Fourier transformed and normalized by mean velocity and equivalence ratio at reference position, according to:

F tp Ũ = F u Ũ + F φ φ (7)
The underlying assumption of our decomposition approach is that we can compute the velocity FTF contribution F u of the technically premixed case from the perfectly premixed case F pp . This can only hold if the flow field is similar enough and all changes in the FTFs from F tp to F pp can be attributed to the missing equivalence ratio modulation. In order to evaluate the sensitivity of F pp to changes in the flow field caused by the injector jets, we have injected perfectly premixed gas through the injectors and air inlet, and investigated several injector mass flows to resemble the flow field of the technically premixed case. For our test case, the differences between zero mass flow injectors, same impulse or same mass flow injectors were negligible, due to the small fuel injector to air inlet mass ratio. Inserting the assumption and sorting for the equivalence ratio contribution gives 7 :

F φ φ = (F tp -F pp ) Ũ (8)
Even though we can identify both FTFs on the right hand side from the two separate cases, both sides of the equation are referenced to different quantities Ũ and φ. They must be of the same kind in order to allow computations. This can be done by inserting the velocity to equivalence ratio transfer function H with

φ = H Ũ (9)
into the equation and transforming the left hand side to Ũ space and solving after equivalence ratio FTF

F φ = (F tp -F pp ) H -1 (10)
The velocity to equivalence ratio transfer function is determined as input-output model from velocity and equivalence ratio signals at the reference position at the burner mouth. The choice of reference position is arbitrary but has implications on the modelling and required instrumentation. Our choice reduces the instrumentation complexity significantly but complicates the evaluation of H. This method suffers from possible uncertainty due to the inversion and multiple identifications, i.e amplification of their noise contributions.

FIG. 8: Decomposed FTF from SISO LES/SI: velocity vs. equivalence ratio Figure 8 shows the identified FTFs from technically premixed case F tp (techPremix) and from the perfectly premixed case F pp (fullyPremix), the velocity to equivalence ratio transfer function H and the reconstructed equivalence ratio contribution F φ . F pp looks as expected, i.e it has unity gain for steady state limit, overall low-pass behavior and significant peak at lower frequencies. F tp shows one additional peak at 140 Hz and zero gain at steady state. At steady state velocity and equivalence ratio contributions cancel out due to a similar gain and phase in F u and F φ and a π phase delay between Ũ and φ caused by H. H behavior can be explained by two major contributions, first the mode conversion from velocity perturbations to equivalence ratio perturbations at the injectors with constant phase delay of π and constant gain of 1 and second a transport part with linear phase delay and dispersive decrease in gain. The broad excess of gain around mid frequencies can be attributed to the recirculating flow with agglomeration of fuel in the swirler vanes during the low velocity phases of the oscillations. This mechanism was observed to couple with the acoustics and cause the limit cycle oscillation [START_REF] Agostinelli | Impact of wall heat transfer in large eddy simulation of flame dynamics in a swirled combustion chamber[END_REF] . The reconstructed F φ has almost unity gain for steady state limit and overall low-pass behavior without pronounced peak.

B. MISO Identification

In Multiple-Input, Single-Output (MISO) identification multiple inputs are related to one output. Individual transfer functions from each input to the common output are estimated in a combined optimization process. In order to attribute the origin of a response to the correct input, the inputs must exhibit minimal cross-correlation [START_REF] Tangirala | Principles of System Identification: Theory and Practice[END_REF] . This must be considered for the data generation process.

The excitation signals must be carefully chosen for the LES to generate appropriate time series data captured at the reference point [START_REF] Huber | Dynamics of practical premixed flames, part ii: Identification and interpretation of cfd data[END_REF][START_REF] Huber | Dynamics of practical premixed flames, part i: Model structure and identification[END_REF] . In the technically premixed case of the double SISO approach the fuel injectors were assumed to be stiff, i.e acoustic perturbations do not alter the velocity and mass flow of the fuel injectors and mode conversion, i.e. conversion from velocity perturbations to equivalence ratio perturbations, at the injector could be assumed. The input-output data triplet with velocity and equivalence ratio as inputs and heat release rate as output was measured for the double SISO approach. Using this data set for MISO identification does not give satisfying results, since the velocity and equivalence ratio signals at reference position exhibit a high level of cross-correlation making their contributions to the FTF hardly distinguishable. Within MISO each input channel must be excited simultaneously and independently to reduce cross-correlation significantly and allow a correct model identification. In our numerical simulation this can be done by imposing velocity signals at the air and fuel inlets. In addition to the single broad-band signal properties, the signal pairs must exhibit minimal cross-correlation. A pair of Daubechies 37 wavelets based signals is an optimal choice. In MISO identification both contributions, F u and F φ , are directly identified. The amount of model coefficients is doubled, which may increase the required single time series length compared to SISO, but only one time series is required instead of two. The overall required time series length should be shorter for MISO. Figure 9 shows the F u and F φ from MISO identification.

They look very similar to the ones obtained from double SISO approach, both satisfy the steady state limit and qualitative behavior. Only slight changes are present.

Overall the gain curves are smoother exhibiting less oscillation and the F φ gain now shows a distinctive peak. The MISO model seems to be more reasonable but a further evaluation needs experimental reference.

FIG. 9: FTF from MISO model identification on heat release rate signal: velocity vs. equivalence ratio

IV. CONCLUSION

In this study the LES/SI approach was extended and tested for a technically premixed flame test case. The straight forward CFD setup approach proved to be generally applicable. In consideration of numerical model and experimental uncertainties, the LES reproduced the measurements of CH * intensity fluctuations with very satisfactory accuracy, thereby validating the LES/SI approach. Two variants for system identification of a technically premixed flame were tested, i.e. the double SISO and the MISO approach. Both gave comparable results for the velocity and equivalence based FTF components. If all required data are available, the MISO identification is preferable due to a slightly better qualitative prediction of the FTFs and a reduced computational effort. The double SISO identification extends the applicability to cases, where the fuel line can not be excited in a suitable manner. For the future, a correct modelling of the velocityto-equivalence-ratio transfer function would allow the use of the double SISO approach in experiments, where a measurement of equivalence ratio is not possible. This would reduce the complexity and cost of experimental designs significantly.

FIG. 1 :

 1 FIG.1: Experimental sketch of the PRECCINSTA combustor with a loudspeaker at the base, a hot-wire probe and a photomultiplier.

FIG. 3 :

 3 FIG. 3: Spectral Power Density (PSD) of the CH * signal from photomultiplier in unforced and forced flow conditions

FIG. 4 :

 4 FIG. 4: Modal analysis of the plenum in cold conditionswithout flow

( a )

 a Sketch of computational domain with half cut of fluid domain and edges shown (grey), inlet and outlet boundary patches (green), and isosurfaces of mean fuel (yellow) at φ = 0.86 and mean heat release rate (red) at 10% of its maximum.(b) Mesh topology showing a cut through the swirler with injector surface and a cut through plenum, swirler, mixing tube and combustion chamber. Color represents refinement levels at characteristic lengths 1000, 500, 250, 125 µm from red to blue.

FIG. 7 :

 7 FIG. 7: CH * transfer function: Experiment and LES/SI

V. ACKNOWLEDGMENTS

The authors gratefully acknowledge financial support by the German Research Foundation DFG project PO 710/20-1 and the provided computing time on the Leibniz Supercomputing Centres Linux-Cluster. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.682383 and No.832248). We greatfully acknowledge that P. W. Agostinelli and T. Poinsot from CERFACS provided mean values for our thermal boundary conditions.


Cary Hufman

Jon-Lark Kim

Patrick Solé

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Guest Editorial

W. Cary Hufman, Jon-Lark Kim, Patrick Solé

The present special issue is dedicated to Vera Pless, our beloved collaborator, advisor, and friend. Accounts of her biography and bibliography have appeared in several places [START_REF] Dougherty | The life and work of Vera Stepen Pless, Notices of the AMS[END_REF]2,3,4]. The topics of the articles that comprize the issue reflect the themes and trends of Vera long and fertile researcher career. As explained in some detail in [2] these consist more specifically of To begin with, Pless is remembered, amongst other things, for a class of self-dual ternary codes related to Hadamard matrices, called Pless symmetry codes. The paper by V.D. Tonchev revisits this classic favorite. Self-dual codes over large finite fields are explored in a paper of Choi and Kim. Selforthogonal constacyclic codes are studied in Fu and Liu, which is also relevant to point 5. Self-orthogonal codes over a non-unitary ring, thus embracing both point 1 and point 4 is the theme of the article by Shi, Wang and Solé. The paper of Chakraborthy and Meziaki is dedicated to a special kind oweight enumerator, the Jacobi polynomial, which is especially relevant to self-dual codes and invariant theory. This is a panorama of point 1. Studies of codes 1 with a small hull can be distantly rattached to this theme. These include the paper by Sok, and that by Qian et al.

Point As we are writing this text, the status of some papers is still uncertain, so we apologize in advance for not including them in this census.

  2 is represented in a philosophical sense, by a deep paper of Dysko and Wood on the MacWilliams extension theorem. Point 4 and 5 are both covered by a nice paper of Shi et al. on polycyclic codes over a mixed alphabet. Similarly the paper by Patel and Prakash on constacyclic codes over a special non-chain ring is relevant to both points. A pure point 5 paper would be a contribution by Borello et al. on quasi-abelian codes, a natural generalization of quasi-cyclic codes. Point 6 figures in the paper by Tagahashi and Hagiwara. Point 7 is represented by Shi, Wang, Helleseth, Solé, and also Qin, Zhao, Liu. We could even connect the paper by Duursma et al. which studies codes in the Johnson graph to that track. Point 8 is covered by Ayebie and Souidi. As for point 3 it is implicit in almost all papers. We have included some papers of good quality that do not belong to these 8 themes: the paper on Chudnovsky algorithm by Ballet et al., a paper on permutation codes by Jafari et al., a paper on spherical designs by Martinez, and a paper on character sum bounds by Wu et al.