Hans Havlicek

Boris Odehnal

Metod Saniga

On invariant notions of Segre varieties in binary projective spaces

Keywords: Mathematics Subject Classification (2010): 51E20, 05B25, 15A69 Segre variety, binary projective space

Invariant notions of a class of Segre varieties S (m) (2) of PG(2 m -1, 2) that are direct products of m copies of PG(1, 2), m being any positive integer, are established and studied. We first demonstrate that there exists a hyperbolic quadric that contains S (m) (2) and is invariant under its projective stabiliser group G S (m) (2) . By embedding PG(2 m -1, 2) into PG(2 m -1, 4), a basis of the latter space is constructed that is invariant under G S (m) (2) as well. Such a basis can be split into two subsets of an odd and even parity whose spans are either real or complex-conjugate subspaces according as m is even or odd. In the latter case, these spans can, in addition, be viewed as indicator sets of a G S (m) (2) -invariant geometric spread of lines of PG(2 m -1, 2). This spread is also related with a G S (m) (2) -invariant non-singular Hermitian variety.

The case m = 3 is examined in detail to illustrate the theory. Here, the lines of the invariant spread are found to fall into four distinct orbits under G S (3) (2) , while the points of PG(7, 2) form five orbits.

Introduction

The present note is concerned with invariant notions in the ambient space of certain Segre varieties over fields of characteristic two and, in particular, over the smallest Galois field F 2 . The attribute invariant always refers to the stabiliser of the Segre in the projective group of the ambient space.

Our text is organised as follows: In Section 2 we collect some background results about those Segre varieties S (m) (F) which are products of m projective lines over a field F. The next section presents an invariant quadric of a Segre S (m) (F) for m ≥ 2 and a ground field F of characteristic two (Theorem 1). This quadric is regular, of maximal Witt index, contains the given Segre, and its polarity is the fundamental polarity of this Segre. The following sections deal with Segre varieties S (m) (2) over F 2 . By extending the ground field of the ambient space from F 2 to F 4 we find an invariant basis (Theorem 2) and an invariant Hermitian variety (Section 5). The theory splits according as m is even or odd. In the latter case there is an invariant geometric line spread (Corollary 1) which gives also rise to a spread of the invariant quadric. We make use of our previous results in Section [START_REF] Brouwer | Near polygons and Fischer spaces[END_REF], where we describe certain line orbits and all point orbits of the stabiliser group of the Segre S (3) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] in terms of the invariant basis. This complements [14, p. 82], where a completely different description of these point orbits was given without proof.

There is a widespread literature on closely related topics, like notions of rank of multi-dimensional arrays [START_REF] Comon | Generic and typical ranks of multi-way arrays[END_REF], secant varieties of Segre varieties (mainly over the complex numbers) [START_REF] Abo | Induction for secant varieties of Segre varieties[END_REF], the very particular properties of certain Segre varieties over F 2 [START_REF] Shaw | The polynomial degrees of Grassmann and Segre varieties over GF(2)[END_REF], [START_REF] Shaw | The cubic Segre variety in PG(5, 2)[END_REF], quantum codes [START_REF] Glynn | The geometry of additive quantum codes[END_REF], and entanglement of quantum bits in physics [START_REF] Doković | On polynomial invariants of several qubits[END_REF], [START_REF] Heydari | Geometrical structure of entangled states and the secant variety[END_REF], [START_REF] Lévay | Three fermions with six single-particle states can be entangled in two inequivalent ways[END_REF]. The few sources which are cited here contain a wealth of further references.

Notation and background results

Let F be a commutative field and let V 1 , V 2 , . . . , V m be m ≥ 1 two-dimensional vector spaces over F. So P(V k ) = PG(1, F) are projective lines over F for k ∈ {1, 2, . . . , m}. We consider the tensor product m k=1 V k and the projective space P m k=1 V k = PG(2 m -1, F). The non-zero decomposable tensors of m k=1 V k determine the Segre variety (see [START_REF] Burau | Mehrdimensionale projektive und höhere Geometrie[END_REF], [START_REF] Hirschfeld | General Galois Geometries[END_REF])

S 1,1,...,1 m (F) = F a 1 ⊗ a 2 ⊗ • • • ⊗ a m | a k ∈ V k \ {0}
of P m k=1 V k . This Segre will also be denoted by S (m) (F). We recall some facts which are well known from the classical case [7, p. 143], where F is the field of complex numbers. They can immediately be carried over to our more general settings. Given a basis (e (k) 0 , e (k) 1 ) for each vector space V k , k ∈ {0, 1, . . . , m}, the tensors E i 1 ,i 2 ,...,i m := e (1) i 1 ⊗ e (2) i 2 ⊗ • • • ⊗ e (m) i m with (i 1 , i 2 , . . . , i m ) ∈ I m := {0, 1} m (1) constitute a basis of m k=1 V k . For any multi-index i = (i 1 , i 2 , . . . , i m ) ∈ I m the opposite multi-index, in symbols i ′ , is characterised by i k i ′ k for all k ∈ {1, 2, . . . , m}. In other words, two multi-indices are opposite if, and only if, their Hamming distance is maximal.

Let f k ∈ GL(V k ) for k ∈ {1, 2, . . . , m}. Then f 1 ⊗ f 2 ⊗ • • • ⊗ f k ∈ GL m k=1 V k (2)
denotes their Kronecker (tensor) product. Each permutation σ ∈ S m gives rise to linear bijections

V k → V σ(k) sending (e (k) 0 , e (k) 1 ) to (e (σ(k)) 0 , e (σ(k)) 1
). Also, the symmetric group S m acts on

I m via σ(i) = σ(i 1 , i 2 , . . . , i m ) = (i σ -1 (1) , i σ -1 (2) , . . . , i σ -1 (m) ).
There is a unique mapping

f σ ∈ GL m k=1 V k such that E i → E σ(i) for all i ∈ I m . (3)
Clearly, this f σ depends on the chosen bases. The subgroup of GL m k=1 V k preserving decomposable tensors is generated by all transformations of the form ( 2) and (3). It induces the stabiliser G S (m) (F) of the Segre S (m) (F) within the projective group PGL m k=1 V k . Each of the vector spaces V k admits a symplectic (i. e., non-degenerate and alternating) bilinear form 1

[•, •] : V k × V k → F. Consequently, m k=1 V k is equipped with a symplectic bilinear form, again denoted as [•, •], which is given by a 1 ⊗ a 2 ⊗ • • • ⊗ a m , b 1 ⊗ b 2 ⊗ • • • ⊗ b m := m k=1 [a k , b k ] for a k , b k ∈ V k , (4)
and extending bilinearly. Like the forms on V k this bilinear form on m k=1 V k is unique up to a non-zero factor in F. In projective terms the symplectic form [•, •] on m k=1 V k (or any proportional one) determines the fundamental polarity of S (m) (F), i. e., a polarity which sends S (m) (F) to its dual. This polarity is orthogonal when m is even and Char F 2, but null otherwise: Indeed, it suffices to consider the tensors of our basis [START_REF] Abo | Induction for secant varieties of Segre varieties[END_REF]. Given i, j ∈ I m we have

[E i , E i ′ ] = m k=1 [e (k) i k , e (k) i ′ k ] = (-1) m [E i ′ , E i ] 0, (5)
[E i , E j ] = 0 for all j i ′ . (6)
Hence [•, •] is symmetric when m is even and Char K 2, and it is alternating otherwise.

Let m be even and Char F 2. Then Q :

m k=1 V k → F : X → [X, X]
is a quadratic form having Witt index 2 m-1 and rank 2 m . So, the fundamental polarity of the Segre S (m) (F) is the polarity of a regular quadric. The Segre coincides with this quadric precisely when m = 2.

The invariant quadric

We now focus on the case when F has characteristic two. Here [•, •] is a symplectic bilinear form on m k=1 V k for any m ≥ 1, whence the fundamental polarity of the Segre S (m) (F) is always null. Furthermore, (5) simplifies to

[E i , E i ′ ] = m k=1 [e (k) 0 , e (k) 1 ] = [E i ′ , E i ] 0. ( 7
)
Proposition 1. Let m ≥ 2 and Char F = 2. Then there is a unique quadratic form Q : m k=1 V k → F satisfying the following two properties: 1. Q vanishes for all decomposable tensors.

The symplectic bilinear form

[•, •] : m k=1 V k × m k=1 V k → F is the polar form of Q.
Proof. (a) We denote by I m,0 the set of all multi-indices (i 1 , i 2 , . . . , i m ) ∈ I m with i 1 = 0. In terms of our basis (1) a quadratic form is given by

Q : m k=1 V k → F : X → i ∈ I m,0 [E i , X][E i ′ , X] [E i , E i ′ ] . (8)
Given an arbitrary decomposable tensor we have (1) 0 , e (1) 1 ] (1) 0 , e (1) 1 ]

Q(a 1 ⊗ • • • ⊗ a m ) = i ∈ I m,0 [E i , a 1 ⊗ • • • ⊗ a m ][E i ′ , a 1 ⊗ • • • ⊗ a m ] [E i , E i ′ ] = i ∈ I m,0 [e (1) 0 , a 1 ][e (1) 1 , a 1 ] • • • [e (m) 0 , a m ][e (m) 1 , a m ] [e
• • • [e (m) 0 , e (m) 1 ] = 2 m-1 [e (1) 0 , a 1 ][e (1) 1 , a 1 ] • • • [e (m) 0 , a m ][e (m) 1 , a m ] [e
• • • [e (m) 0 , e (m) 1 ] = 0,
where we used [START_REF] Burau | Mehrdimensionale projektive und höhere Geometrie[END_REF], #I m,0 = 2 m-1 , m ≥ 2, and Char F = 2. This verifies property 1.

(b) Let j, k ∈ I be arbitrary multi-indices. Polarising Q gives

Q(E j + E k ) + Q(E j ) + Q(E k ) = Q(E j + E k ) + 0 + 0 = i ∈ I m,0 [E i , E j + E k ][E i ′ , E j + E k ] [E i , E i ′ ] .
The numerator of a summand of the above sum can only be different from zero if i ∈ { j ′ , k ′ } and i ′ ∈ { j ′ , k ′ }. These conditions can only be met for k = j ′ , whence in fact at most one summand, namely for j ∈ I m,0 the one with i = j, and for j ′ ∈ I m,0 the one with i = j ′ , can be non-zero. So

Q(E j + E k ) + Q(E j ) + Q(E k ) = 0 = [E j , E k ] for k j ′
and, irrespective of whether i = j or i = j ′ , we have

Q(E j + E j ′ ) + Q(E j ) + Q(E j ′ ) = [E j , E j + E j ′ ][E j ′ , E j + E j ′ ] [E j , E j ′ ] = [E j , E j ′ ].
But this implies that the quadratic form Q polarises to [•, •], i. e., also the second property is satisfied. (c) Let Q be a quadratic form satisfying properties 1 and 2. Hence the polar form of Q-Q = Q+ Q is zero. We consider F as a vector space over its subfield F comprising all squares in F.

So (Q + Q) : m k=1 V k → F
is a semilinear mapping with respect to the field isomorphism F → F : x → x 2 ; see, e. g., [9, p. 33]. The kernel of Q + Q is a subspace of m k=1 V k which contains all decomposable tensors and, in particular, our basis [START_REF] Abo | Induction for secant varieties of Segre varieties[END_REF]. Hence Q + Q vanishes on m k=1 V k , and

Q = Q as required.
From ( 8) and [START_REF] Burau | Mehrdimensionale projektive und höhere Geometrie[END_REF], the quadratic form Q can be written in terms of tensor coordinates x j ∈ F as

Q j ∈ I m x j E j = i ∈ I m,0 [E i , E i ′ ]x i x i ′ = m k=1 [e (k) 0 , e (k) 1 ] • i ∈ I m,0 x i x i ′ . (9)
The previous results may be slightly simplified by taking symplectic bases, i. e., [e (k) 0 , e (k) 1 ] = 1 for all k ∈ {1, 2, . . . , m}, whence also [E i , E i ′ ] = 1 for all i ∈ I m . Observe also that Proposition 1 fails to hold for m = 1. A quadratic form Q vanishing for all decomposable tensors of V 1 is necessarily zero, since any element of V 1 is decomposable. Hence the polar form of such a Q cannot be non-degenerate. Theorem 1. Let m ≥ 2 and Char F = 2. There exists in the ambient space of the Segre S (m) (F) a regular quadric Q(F) with the following properties:

1. The projective index of Q(F) is 2 m-1 -1.

Q(F) is invariant under the group of projective collineations stabilising the

Segre S (m) (F).

Proof. Any f k ∈ GL(V k ), k ∈ {1, 2, . . . , m}, preserves the symplectic form [•, •] on V k to within a non-zero factor. Any linear bijection f σ as in (3) is a symplectic transformation of m k=1 V k .
Hence any transformation from the group G S (m) (F) preserves the symplectic form (4) up to a non-zero factor. Consequently, also Q is invariant up to a non-zero factor under the action of G S (m) (F) .

From ( 9) the linear span of the tensors E j with j ranging in I m,0 is a singular subspace with respect to Q. So the Witt index of Q equals 2 m-1 , because [•, •] being non-degenerate implies that a greater value is impossible.

We conclude that the quadric with equation Q(X) = 0 has all the required properties.

We henceforth call Q(F) the invariant quadric of the Segre S (m) (F). The case m = 2 deserves special mention, as the Segre S 1,1 (F) coincides with its invariant quadric Q(F) given by Q( j∈I 2 x j E j ) = x 00 x 11 + x 01 x 10 = 0. This result parallels the situation for Char F 2.

The invariant basis

In what follows F q will denote the Galois field with q elements. We adopt the notation and terminology from Section 2, but we restrict ourselves to the case F = F 2 . Indeed, we shall always identify F 2 with the prime field of F 4 = {0, 1, ω, ω 2 }, where ω 2 + ω + 1 = 0. For each k ∈ {1, 2, . . . , m} we fix a basis (e (k) 0 , e (k) 1 ) so that we obtain the tensor basis [START_REF] Abo | Induction for secant varieties of Segre varieties[END_REF].

Let V be any vector space over F 2 . Then V can be embedded in W := F 4 ⊗ F 2 V, which is a vector space over F 4 , by a → 1 ⊗ a; see, for example, [18, p. 263]. We shall not distinguish between a and 1 ⊗ a. Likewise, if f denotes a linear mapping between vector spaces over F 2 , then the unique linear extension of f to the corresponding vector spaces over F 4 will also be written as f rather than 1 ⊗ f . Similarly, we use the same symbol for a bilinear form on V and its extension to a bilinear form on W. After similar identifications, we have P(V) ⊂ P(W), PGL(V) ⊂ PGL(W), and so on. We make use of the usual terminology for real and complex spaces also in our setting. We address the vectors of V to be real, we speak of complex-conjugate vectors, points, and subspaces. In particular, a subspace is said to be real if it coincides with its complex-conjugate subspace.

Applying this extension to our vector spaces V k and their tensor product

m k=1 V k gives vector spaces W k and F 4 ⊗ F 2 m k=1 V k .
The last vector space can be identified with m k=1 W k in a natural way, so that the Segre S (m) (F 2 ) =: S (m) (2) can be viewed as as subset of S (m) (F 4 ) =: S (m) [START_REF] Bichara | Primes and order structure in the product spaces[END_REF]. Likewise, we have

Q(2) := Q(F 2 ) ⊂ Q(F 4 ) =: Q(4).
Theorem 2. For each k ∈ {1, 2, . . . , m} let F 4 u (k) 0 and F 4 u (k) 1 be the only two points of the projective line [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] of the Segre S (m) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF].

P(W k ) = PG(1, 4) that are not contained in P(V k ) = PG(1, 2). Then B m := F 4 u (1) i 1 ⊗ u (2) i 2 ⊗ • • • ⊗ u (m) i m | (i 1 , i 2 , . . . , i m ) ∈ I m is a basis of P m k=1 W k = PG(2 m -1, 4) which is invariant, as a set, under the stabiliser G S (m) (F 2 ) =: G S (m)
Proof. We may assume that

u (k) 0 = e (k) 0 + ωe (k) 1 and u (k) 1 = e (k) 0 + ω 2 e (k) 1 = (e (k) 0 + e (k) 1 ) + ωe (k) 1 . (10)
As u (k) 0 and u (k) 1 are linearly independent, the 2 m tensors [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] follows from the fact that the points F 4 u (k) 0 and F 4 u (k) 1 are determined uniquely up to relabelling.

U i 1 ,i 2 ,...,i m := u (1) i 1 ⊗ u (2) i 2 ⊗ • • • ⊗ u (m) i m with (i 1 , i 2 , . . . , i m ) ∈ I m (11) constitute a basis of m k=1 W k , whence B m is a projective basis. The invariance of B m under G S (m)
We shall refer to B m as the invariant basis of the Segre S (m) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF]. In order to describe the action of the stabiliser group G S (m) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] of the Segre S (m) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] on the invariant basis we need a few technical preparations:

First, from now on the set I m = {0, 1} m of multi-indices will be identified with the vector space F m 2 . Secondly, for any 2-dimensional vector space V over F 2 we can define the F 2 -valued sign function sgn 2 : GL(V) → F 2 to be 0 if f induces an even permutation of V \ {0} and 1 otherwise. Proposition 2. The stabiliser group G S (m) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] of the Segre S (m) (2) has the following properties:

1. Let f k ∈ GL(V k ) for k ∈ {1, 2, . .

. , m} and write

s := (sgn 2 f 1 , sgn 2 f 2 , . . . , sgn 2 f m ) ∈ F m 2 . ( 12
)
The collineation given by f

1 ⊗ f 2 ⊗ • • • ⊗ f m sends any point F 4 U i ∈ B m to the point F 4 U i+s ∈ B m .
2. G S (m) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] acts transitively on the invariant basis B m .

3. Let σ ∈ S m be a permutation and define f σ as in [START_REF] Benz | Cross ratios and a unifying treatment of von Staudt's notion of reeller Zug[END_REF]. Then f σ sends any point

F 4 U i ∈ B m to F 4 U σ(i) ∈ B m . Proof. (a) Each mapping f k ∈ GL(V k ) ⊂ GL(W k ) induces a projectivity of the projective line P(W k ) = PG(1, 4) which stabilises P(V k ) = PG(1, 2).
If sgn 2 f k = 0 then the restriction to P(V k ) is an even permutation, namely either a permutation without fixed points or the identity on P(V k ). In the first case the characteristic polynomial of f k has two distinct zeros over F 4 , whence each of the two points F 4 u (k) 0 and F 4 u (k) 1 remains fixed. In the second case all points of P(V k ) are fixed.

If sgn 2 f k = 1 then f k gives a permutation of P(V k ) with precisely one fixed point. Such an f k is an involution, whence the points F 4 u (k) 0 and F 4 u (k) 1 are interchanged.

We infer from the above results that

f 1 ⊗ f 2 ⊗ • • • ⊗ f m sends the point of B m with multi-index i ∈ F m 2 to the point of B m with multi-index (i + s) ∈ F m 2 . (b) Given i, j ∈ F m 2 we let s := i + j. In order find a collineation from G S (m) (2) taking F 4 U i to F 4 U j , it suffices to choose for all k ∈ {1, 2, . . . , m} some f k ∈ GL(V k ) with sgn 2 f k = s k . This can clearly be done, so that f 1 ⊗ f 2 ⊗ • • • ⊗ f m
yields a collineation with the required properties.

(c) According to the (basis-dependent) definition of f σ in (3), we have to consider the linear bijections V k → V σ(k) sending (e (k) 0 , e (k) 1 ) to (e (σ(k)) 0 , e (σ(k))

1

). By [START_REF] Doković | On polynomial invariants of several qubits[END_REF], any such map sends (u (k) 0 ,

u (k) 1 ) to (u (σ(k)) 0 , u (σ(k)) 1 ). Now f σ (U i ) = U σ(i) follows im- mediately.
The parity of a point F 4 U i ∈ B m can be defined as the parity of the multi-index i (i. e., it is even or odd according to the number of 1s among the entries of i). We write B + m and B - m for the set of base points with even and odd parity, respectively. Even though we can distinguish points of even and odd parity due to our fixed bases (e (k) 0 , e (k) 1 ), a change of bases in the vector spaces V k may alter the parity of a point. But having the same parity is an equivalence relation on B m with two equivalence classes, namely B + m and B - m , each of cardinality 2 m-1 . We define the Hamming distance between F 4 U i and F 4 U j as the Hamming distance of their multi-indices i and j. In particular, we speak of opposite points if i and j are opposite. For each point of B m there is a unique opposite point. By [START_REF] Doković | On polynomial invariants of several qubits[END_REF] and [START_REF] Dye | Maximal subgroups of finite orthogonal groups stabilizing spreads of lines[END_REF] opposite points of B m are complex-conjugate with respect to the Baer subspace P m k=1 V k of P m k=1 W k . The opposite point to F 4 U i can also be characterised as the only point F 4 U j ∈ B m such that [U i , U j ] 0. We remark that the Hamming distance on B m admits another description due to B m ⊂ S (m) [START_REF] Bichara | Primes and order structure in the product spaces[END_REF]. The Hamming distance of p, q ∈ B m equals the number of lines on a shortest polygonal path in S (m) (4) from p to q.

From the proof of Proposition 2 and the above remarks we immediately obtain: [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] , and the partition

B m = B + m ∪ B - m is a G S (m) (2) -invariant notion.
We now use the invariant basis for describing some other G S (m) (2) -invariant subsets. In the following theorem we also make use of a particular property of Segre varieties over F 2 . Recall that (for an arbitrary ground field F) there are precisely m generators through any point p of the Segre S (m) (F). They span the (m-dimensional) tangent space of S (m) (F) at p. The tangent lines at p are the lines through p which lie in its tangent space. For F = F 2 there are 2 m -1 tangents at p. Precisely one of them does not lie in any of the (m -1)-dimensional subspaces which are spanned by m -1 generators through p. This line will be called the distinguished tangent at p.

F 4 U i ∨ F 4 U i ′ with i ∈ I m,0 ( 13
)
is a G S (m) (2) -invariant subset. The 3 • 2 m-1 real points on these lines comprise an orbit of G S (m) (2) .
3. If m is even then span B + m and span B - m are real subspaces; each of the lines from [START_REF] Glynn | The modular counterparts of Cayley's hyperdeterminants[END_REF] is contained in precisely of them.

If m is odd then span B +

m and span B - m are complex-conjugate subspaces. All lines from (13) meet span B + m and span B - m at precisely one point, respectively.

All distinguished tangents of the Segre S (m) (2) meet span B +

m and span B - m at precisely one point, respectively. Proof. Ad 1 and 2: The assertions on the invariance of span B + m ∪ span B - m and on the invariance of the union of all lines from (13) are a direct consequence of Theorem 3.

We denote the set of all real points on the lines from (13) by R. Let j ∈ I m,0 and let p be an arbitrary real point on the line F 4 U j ∨ F 4 U j ′ . Any collineation from G S (m) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] takes p to some real point on a line from [START_REF] Glynn | The modular counterparts of Cayley's hyperdeterminants[END_REF], whence the orbit of p is contained in R.

Conversely, let q ∈ R. So there is a k ∈ I m,0 with q ∈ F 4 U k ∨ F 4 U k ′ . By Theorem 3, there exists a collineation in G S (m) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] which maps F 4 U j to F 4 U k and, consequently, also F 4 U j ′ to F 4 U k ′ . Furthermore, p is mapped to some real point p on the line F 4 U k ∨ F 4 U k ′ . There exists f 1 ∈ GL(V 1 ) with sgn 2 f 1 = 0. Then u (1) 0 and u (1) 1 are eigenvectors of f 1 with eigenvalues λ and λ 2 , where λ ∈ {ω, ω 2 }. (See the proof of Proposition 2.) The linear bijection

f := f 1 ⊗ id V 2 ⊗ • • • ⊗ id V m has
U k and U k ′ as eigenvectors with eigenvalues λ and λ 2 , respectively, due to k 1 = 0. Thus the collineation arising from f induces a non-identical even permutation on the three real points of the line F 4 U k ∨ F 4 U k ′ . Such a permutation has only one cycle. So one of f , f 2 or f 3 yields a collineation from G S (m) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] which maps p to q.

Ad 3 and 4: As the opposite points of B m are complex-conjugate, they share the same parity for m even, but have different parity for m odd.

Ad 5: First, we exhibit the distinguished tangent T of the Segre S (m) (2) at the point F 2 E 1,1,...,1 . On each of the m generators of the Segre through this point we select one more real point, namely F 2 E 0,1,1,...,1 , F 2 E 1,0,1,...,1 , . . . , F 2 E 1,1,...,1,0 for facilitating our subsequent reasoning. So, the distinguished tangent T contains the real point

F 2 (E 0,1,1,...,1 + E 1,0,1,...,1 + • • • + E 1,1,...,1,0 ). ( 14
)
By [START_REF] Doković | On polynomial invariants of several qubits[END_REF], we have e

(k) 0 = ω 2 u (k) 0 + ωu (k) 1 and e (k) 1 = u (k) 0 + u (k) 1 for all k ∈ {1, 2, . . . , m}. So E 1,1,...,1 = i ∈ I m U i and E 0,1,...,1 = i ∈ I m
x (1) i U i with x (1) i =

ω 2 for i 1 = 0, ω for i 1 = 1.
Mutatis mutandis, we obtain linear combinations for E 1,0,1,...,1 , . . . , E 1,1,...,1,0 with coefficients x (2) i , . . . , x (m) i ∈ {ω 2 , ω}. Summing up gives

E 0,1,1,...,1 + E 1,0,1,...,1 + • • • + E 1,1,...,1,0 = i ∈ I m y i U i , (15)
where

y i = ω 2 + ω 2 + • • • + ω 2 # of zeros in i + ω + ω + • • • + ω # of ones in i .
There are two cases: m even: Due to Char F 4 = 2 we have y i = 0 for all i with even parity and y i = ω 2 + ω = 1 for all i with odd parity. So T meets span B - m at the point ( 14). The sum of the tensor from ( 15) and E 1,1,...,1 determines the point of intersection of T with span B + m . m odd: Due to Char F 4 = 2 we have y i = ω 2 for all i with even parity and y i = ω for all i with odd parity. So T meets span B + m at the point

F 4 j ω 2 U j = F 4 j U j , (16)
where j ranges over all elements of I m with even parity, and the subspace span B -

m at the point F 4 ( k ωU k ) = F 4 ( k U k ),
where k ranges over all elements of I m with odd parity.

In either case the two points of intersection are unique, because span B + m and span B - m are skew. Next, we consider an arbitrary distinguished tangent of the Segre. As all points of the Segre comprise a point orbit of G S (m) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] , also all distinguished tangents are in one line orbit of G S (m) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] . So, all distinguished tangents share the properties of the tangent T .

Recall that any pair of skew and complex-conjugate subspaces of P m k=1 W k determines a geometric line spread of P m k=1 V k . This spread comprises all real lines which meet one of the subspaces (and hence both of them in complexconjugate points). Any of these subspaces is called an indicator set of the spread. See [2, p. 74] and [23, p. 29 2) and all lines given by [START_REF] Glynn | The modular counterparts of Cayley's hyperdeterminants[END_REF] belong to this spread.

m k=1 V k = PG(2 m -1, 2). All distinguished tangents of the Segre S (m) (
It is now a straightforward task to establish connections between the fundamental polarity of the Segre S (m) (4) and the quadric Q(4) which arises according to Theorem 1. The reader will easily verify the following: The fundamental polarity maps each of the lines from [START_REF] Glynn | The modular counterparts of Cayley's hyperdeterminants[END_REF] to the span of the remaining ones. For any even m the subspaces span B + m and span B - m are interchanged under the fundamental polarity, whereas for m odd each of them is invariant (totally isotropic). The subspaces span B + m and span B - m are contained in Q(4) precisely when m ≥ 2 is odd. [START_REF] Benz | Cross ratios and a unifying treatment of von Staudt's notion of reeller Zug[END_REF][START_REF] Baer | Partitionen abelscher Gruppen[END_REF]. The 15 points of this projective space fall into two orbits under the action of G 1,1 (2): The first orbit is S 1,1 (2) (nine points), the remaining six points form the second orbit. It comprises the real points of the lines span B + m and span B - m . There are nine distinguished tangents of the quadric. Together they form the hyperbolic linear congruence of lines which arises by joining every real point of span B + m with every real point of span B - m .

Example 1. Let m = 2. The Segre S 1,1 (2) is a hyperbolic quadric of P(V 1 ⊗ V 2 ) = PG
The previous example is clearly just an easy exercise, and it could be mastered without our results about the general case.

The invariant Hermitian variety

The symplectic form [•, •] on m k=1 V k can be extended in exactly two ways to a non-degenerate sesquilinear form2 on m k=1 W k . The bilinear extension is symplectic. In accordance with the notation used elsewhere, it is also denoted by

[•, •].
The only other extension is sesquilinear with respect to the Frobenius automorphism z → z 2 of F 4 . Such Hermitian extension will be written as

[•, •] H . We have [X, Y] H = [X, Y] (17)
for all tensors X, Y ∈ m k=1 W k , where X denote the complex-conjugate tensor of X. While [•, •] describes the fundamental polarity of the Segre S (m) (4), the Hermitian sesquilinear form [•, •] H yields a unitary polarity of P( m k=1 W k ) and, moreover, the Hermitian variety H comprising all its absolute points. By its definition, H is a G S (m) (2) -invariant notion, whence we call it the invariant Hermitian variety of the Segre S (m) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF]. Note that H, like the invariant basis and the invariant line spread, is an invariant notion only for S (m) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF], but not for S (m) [START_REF] Bichara | Primes and order structure in the product spaces[END_REF].

We remark that the invariant basis B m is self-polar with respect to the unitary polarity given by [•, •] H . Indeed, given i, j ∈ F m 2 we have

[U i , U i ] H = m k=1 [u (k) i k , u (k) i k ] = (ω + ω 2 ) m k=1 [e (k) 0 , e (k) 1 ] = 1, (18)
[U i , U j ] H = m k=1 [u (k) i k , u (k) j k ] = 0 for all i j, (19)
since, for example, i 1 j 1 implies u (1) i 1 = u (1) j 1 , whence [u (1) i 1 , u (1) j 1 ] = 0. The following Proposition establishes a link among the invariant line spread from Corollary 1, the invariant quadric Q(2), and the invariant Hermitian variety H.

Proposition 3. Let m ≥ 2 be odd. A line L of the invariant geometric line spread

L is a generator of the invariant quadric Q(4) if, and only if, the intersection point L ∩ span B + m belongs to the invariant Hermitian variety H. Otherwise that line L is a bisecant of Q(4), whence it has no points in common with Q(2).

Proof. Suppose that L ∩ span B + m = F 4 X. By a remark at the end of Section 4, we have

F 4 X ∈ span B + m ⊂ Q(4) and F 4 X ∈ span B - m ⊂ Q(4). So the line L = F 4 X ∨ F 4 X
is either a generator or a bisecant of Q(4). The first possibility occurs precisely when F 4 X lies in the tangent hyperplane of Q(4) at F 4 X. Employing [START_REF] Hirschfeld | General Galois Geometries[END_REF], this in turn is equivalent to 0

= [X, X] = [X, X] H which characterises F 4 X as a point of H.
The above Proposition is a special case of [START_REF] Dye | Maximal subgroups of finite orthogonal groups stabilizing spreads of lines[END_REF]Theorem 1] on quadrics which admit a spread of lines. In our context the invariant spread from Corollary 1 yields a spread of lines on Q(2), since over F 2 each line of the invariant spread is either external to or contained in that quadric.

6 The Segre variety S 1,1,1 [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] In this section we exhibit the ambient space [START_REF] Baer | Partitionen abelscher Gruppen[END_REF]. This space has 2 8 -1 = 255 points. Furthermore, we have the cardinalities #S 1,1,1 (2 4 -1) = 135 (see [START_REF] Hirschfeld | Projective Geometries over Finite Fields[END_REF]Theorem 5.21]), and #L = 255/3 = 85. Proposition 4. Under the action of the stabiliser group G S 1,1,1 (2) of the Segre S 1,1,1 (2) the lines of the invariant spread

P(V 1 ⊗ V 2 ⊗ V 3 ) = PG(7, 2) of the Segre S 1,1,1
) = 3 3 = 27, #Q(2) = (2 3 + 1)(2
L of P(V 1 ⊗ V 2 ⊗ V 3 ) = PG(7, 2) fall into four orbits L 1 , L 2 , L 3 , L 4 .
In terms of the invariant basis B 3 the following characterisation holds: A line from L is in orbit L r if, and only if, its (imaginary) point of intersection with the subspace span B + 3 lies in 4r planes of the tetrahedron B + 3 .

Proof. (a) Throughout this proof the pointwise stabiliser and the stabiliser of B + 3 in the group G S 1,1,1 [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] are abbreviated by G + pw and G + , respectively. We observe that G + pw acts transitively on the points of the Segre S 1,1,1 (2): We fix the point F 2 E 111 = F 2 (e (1) 1 ⊗ e (2) 1 ⊗ e (3) 1 ). Given any point of the Segre, say F 2 A, where A = a 1 ⊗ a 2 ⊗ a 3 , there are linear bijections f k ∈ GL(V k ) satisfying sgn 2 f k = 0 and e (k) 1 → a k for k = {1, 2, 3}. So f 1 ⊗ f 2 ⊗ f 3 induces a collineation which sends F 2 E 111 to F 2 A and belongs to G + pw by [START_REF] Gel'fand | Discriminants, resultants, and multidimensional determinants[END_REF]. We write M r , r ∈ {1, 2, 3, 4}, for the subset of span B (d) We know from Proposition 2 that G + acts transitively on the set of vertices of B + 3 via translations i → i+s on multi-indices. From Theorem 3 the G + -stabiliser of F 4 U 000 acts transitively on the remaining vertices of B + 3 via permutations i → σ(i) on multi-indices. Together with our previous results this means that each of the four subsets M r is a G + -orbit. Consequently, each of the sets L r is contained in an orbit under the action of G S 1,1,1 [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] on the line spread L.

Any collineation from G S 1,1,1 (2) \ G + also preserves each of the sets L r , as it commutes with the Baer involution of PG [START_REF] Burau | Mehrdimensionale projektive und höhere Geometrie[END_REF][START_REF] Bichara | Primes and order structure in the product spaces[END_REF] [START_REF] Burau | Mehrdimensionale projektive und höhere Geometrie[END_REF][START_REF] Baer | Partitionen abelscher Gruppen[END_REF] pointwise. This completes the proof.

fixing P(V 1 ⊗ V 2 ⊗ V 3 ) = PG
From ( 18) and ( 19) the equation of the Hermitian variety H ∩ span B + 3 with respect to the basis (U 000 , U 011 , U 101 , U 110 ) reads

x 3 000 + x 3 011 + x 3 101 + x 3 110 = 0.
Because of3 = 1 for all z ∈ F 4 \ {0}, we get H ∩ span B + 3 = M 2 ∪ M 4 . By Proposition 3, the lines from L 2 ∪ L 4 are on the invariant quadric Q(4). More precisely, the lines from L 2 are those generators of Q(4) which do not contain any point of the Segre S 1,1,1 (2), whereas the lines from L 4 are the distinguished tangents of S 1,1,1 [START_REF] Baer | Partitionen abelscher Gruppen[END_REF]. Figure 1, left 3 , displays the polar space (point-line incidence structure) on H. This Hermitian variety consists of 45 points and carries 27 lines (represented by segments and curves), with five points on any line and three lines on any point. The 27 points represented by small circles are those from M 4 , the remaining 18 points are represented by bullets and belong to M 2 . The 27 points of M 4 can be viewed as a skew projection of the Segre S 1,1,1 (2) (Figure 1, right) into H along the invariant line spread. Under this projection collinearity of points is being preserved.

The lines from L 1 are the four lines from [START_REF] Glynn | The modular counterparts of Cayley's hyperdeterminants[END_REF]. Like the remaining 36 lines from L 3 they are exterior lines (over F 2 ) of the invariant quadric Q(2). Proof. It is clear that O 5 is an orbit under G S 1,1,1 [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] . The points of O 1 form an orbit according to Theorem 4. In order to show that O 2 and O 3 are orbits, we shall select one line of L 2 and L 3 , respectively. By Proposition 4, it suffices then to show that all real points of this line are in one orbit. This task will be accomplished with mappings f k ∈ GL(V k ), k ∈ {1, 2, 3}, given by e (k) 0 → e (k) 1 , e (k) 1 → e (k) 0 + e (k) 1 . From (10), we have

f k (u (k) 0 ) = ωu (k) 0 and f k (u (k) 1 ) = ω 2 u (k) 1 .
Let L 2 ∈ L 2 be the line joining F 4 (U 000 + U 011 ) ∈ M 2 with its complexconjugate point F 4 (U 111 + U 100 ). The mapping f 1 ⊗ id V 2 ⊗ id V 3 has U 000 + U 011 as eigentensor with eigenvalue ω. Its complex-conjugate tensor is therefore an eigentensor with eigenvalue ω 2 . From the proof of Proposition 2, this implies that f 1 ⊗ id V 2 ⊗ id V 3 induces a non-trivial even permutation on the set of real points of L 2 . So, under the powers of this permutation the three real points of L 2 are permuted in one cycle.

Let L 3 ∈ L 3 be the line joining F 4 (U 011 + U 101 + U 110 ) ∈ M 3 with its complexconjugate point. Here f 1 ⊗ f 2 ⊗ f 3 possesses U 011 + U 101 + U 110 as eigentensor with eigenvalue ω 5 = ω 2 . Its complex-conjugate tensor is therefore an eigentensor with eigenvalue ω. Now the assertion follows as above.

The distinguished tangent of the Segre S 1,1,1 (2) at the point F 2 E 111 contains two precisely two points of O 4 . From [START_REF] Glynn | The geometry of additive quantum codes[END_REF], these points are F 2 (E 011 + E 101 + E 110 ) and F 2 (E 111 + E 011 + E 101 + E 110 ). Let g 1 ∈ GL(V 1 ) be defined by e (1) 0 → e (1) 0 + e (1) 1 , e (1) 1 → e (1) 1 . Then g 1 ⊗ id V 2 ⊗ id V 3 will interchange these two points, whence we may argue as before.

Let us close this section with a few remarks: The orbits of the stabiliser group G S 1,1,1 (2) are described (without proof) in a completely different way in [14, p.

) 20
The square of the left hand side of ( 20) is Cayley's hyperdeterminant of the 3×3×3 array (x i ) i ∈ I 3 ; see [START_REF] Glynn | The geometry of additive quantum codes[END_REF]Theorem 5.45] and compare with [START_REF] Gel'fand | Discriminants, resultants, and multidimensional determinants[END_REF] and [START_REF] Glynn | The modular counterparts of Cayley's hyperdeterminants[END_REF].

By virtue of the fundamental polarity of S 1,1,1 [START_REF] Baer | Partitionen abelscher Gruppen[END_REF], Proposition 5 provides a classification of the hyperplanes of P(V 1 ⊗ V 2 ⊗ V 3 ) under the action of the group G S 1,1,1 [START_REF] Baer | Partitionen abelscher Gruppen[END_REF] . Moreover, it gives a classification of the geometric hyperplanes (or primes) of S 1,1,1 [START_REF] Baer | Partitionen abelscher Gruppen[END_REF], since any geometric hyperplane of this Segre arises as intersection with a unique hyperplane of the ambient space [START_REF] Ronan | Embeddings and hyperplanes of discrete geometries[END_REF]. This is a rather particular property of Segre varieties S (m) (2) which is not shared by Segre varieties S (m) (F) in general [START_REF] Bichara | Primes and order structure in the product spaces[END_REF].

The Segre S 1,1,1 (2) (as a point-line geometry) appears in the literature in various guises, namely as the (27 3 , 27 3 ) Gray configuration [START_REF] Marušič | The genus of the GRAY graph is 7. European[END_REF] or as the smallest slim dense near hexagon [START_REF] Brouwer | Near polygons and Fischer spaces[END_REF]. It is also a point model of the chain geometry based on the F 2 -algebra F 2 × F 2 × F 2 , the chains being the twisted cubics on S 1,1,1 (2) (i. e. triads of points with mutual Hamming distance 3); see [3, (5.4)] or [5, p. 272]. We add in passing that the tangent lines of these twisted cubics are just our distinguished tangents of S 1,1,1 (2).

Conclusion

We established several invariant notions for Segre varieties S (m) (2) over the field F 2 . For m ≤ 3 these invariants provide sufficient information for the classification of the points and hyperplanes of the ambient space of S (m) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF]. For larger values of m the situation seems to be much more intricate. For example, when m is odd then the lines of the invariant spread will fall into at least 2 m-1 orbits, as follows from a straightforward generalisation of Proposition 4. However, this gives only a lower bound for the number of orbits. Indeed, for m ≥ 3 there are 3 m distinguished tangents of S (m) [START_REF] Baer | Partitionen abelscher Gruppen[END_REF], but 3 2 m-1 -1 points of span B + m which belong to no face of the simplex B + m . These two cardinalities coincide only when m = 3, whence for all odd m > 3 we no longer have a one-one correspondence between the set of distinguished tangents and the set of all points of span B + m which belong to no face of B + m .

Theorem 3 .

 3 The stabiliser group G S (m)[START_REF] Baer | Partitionen abelscher Gruppen[END_REF] of the Segre S (m) (2) acts on the invariant basis B m (via the multi-indices of its points) as the group of all affine transformations of F m 2 having the form i → σ(i) + s, where σ ∈ S m and s ∈ F m 2 . Hence, Hamming distances on B m are preserved under G S (m)

Theorem 4 . 1 .

 41 The stabiliser group G S (m)[START_REF] Baer | Partitionen abelscher Gruppen[END_REF] of the Segre S (m)[START_REF] Baer | Partitionen abelscher Gruppen[END_REF] has the following properties: The union of the skew subspaces span B + m and spanB - m is a G S (m) (2) -invariant subset of P m k=1 W k .2. The union of the 2 m-1 real lines

+ 3 =

 3 PG(3, 4) comprising all points which lie in precisely 4r planes of the tetrahedron B + 3 . So we have #M 1 = 4 vertices, #M 2 = 3 • 6 = 18 edge points, #M 3 = 4 • 9 = 36 face points, and #M 4 = 27 general points. Clearly, the G + -orbit of any point from span B + 3 is contained in one of the sets M r . (b) We show that M 4 is an orbit under G + pw : By (16), the distinguished tangent of the Segre at F 2 E 111 meets span B + 3 at the point p := F 4 (U 000 + U 011 + U 101 + U 110 ) ∈ M 4 . We infer from the transitive action of G + pw on the Segre S 1,1,1 (2) that all distinguished tangents meet span B + 3 in points of M 4 . Since #S 1,1,1 (2) = 27 = #M 4 , the group G + pw acts transitively on M 4 . (c) Any edge of B + 3 contains precisely three points of M 2 .We obtain all of them by projecting M 4 from the opposite edge, whence G + pw acts transitively on the set of these three points. Likewise, G + pw acts transitively on the nine points of M 3 in any face of B + 3 .

Figure 1 :

 1 Figure 1: The Hermitian variety M 2 ∪ M 4 (left) and the Segre S 1,1,1 (2) (right).

Proposition 5 .

 5 Under the action of the stabiliser group GS 1,1,1 (2) of the Segre S 1,1,1 (2) the points of P(V 1 ⊗V 2 ⊗V 3 ) = PG(7, 2) fall into five orbits O 1 , O 2 , . . . , O 5 .For r ∈ {1, 2, 3} the points of O r are precisely the real points on the lines of L r . The orbit O 4 comprises those real points on the lines from L 4 which are off the Segre S 1,1,1 (2), whereas O 5 equals the Segre S 1,1,1[START_REF] Baer | Partitionen abelscher Gruppen[END_REF].

  82]. The a, b, c, d, e-orbits from there are in our terminology the sets O 5 (27 points), O 2 (54 points), O 3 (108 points), O 4 (54 points), and O 1 (12 points), respectively. The union O 2 ∪ O 4 ∪ O 5 is the invariant quadric Q(2). With respect to the tensor basis (1) the equation of Q(2) reads x 000 x 111 + x 001 x 110 + x 010 x 101 + x 011 x 100 = 0. (

  ]. So, part of Theorem 4 can be reformulated as follows:

	Corollary 1. If m is odd then the complex-conjugate subspaces span B + m and span B -

m are indicator sets of a G S (m) (2) -invariant geometric line spread L of P

We use the same symbol for all these forms.

We assume such forms to be linear in the right and semilinear in the left argument.

The style of our figure is taken from[21, p. 61].

Acknowledgements. This work emerged out of vivid discussions amongst the members of the Cooperation Group "Finite Projective Ring Geometries: An Intriguing Emerging Link Between Quantum Information Theory, Black-Hole Physics, and Chemistry of Coupling" at the Center for Interdisciplinary Research (ZiF), University of Bielefeld, Germany. It was carried out within the "Slovak-Austrian Science and Technology Cooperation Agreement" under grants SK 07-2009 (Austrian side) and SK-AT-0001-08 (Slovak side), being also partially supported by the VEGA grant agency projects 2/0092/09 and 2/0098/10.


Serge Bernard

Florence Azaïs

Mariane Comte

Olivier Potin

Vincent Kerzérho

Michel Renovell

Adaptive LUT-Based System for In Situ ADC Auto-correction

I. INTRODUCTION

Nowadays, most part of the signal processing in areas like instrumentation, telecommunications, control and consumable electronics is carried out at the digital level. The role of Analog-to-Digital Converters (ADCs) placed at the borders of the digital domain acquires a particular relevance, since the signal degradation introduced by these components cannot normally be recovered by subsequent processing. As the new generations of ADCs provide increasing speed and resolution, design requirements become more and more stringent, resulting in ever more difficulty to keep a high production yield. At the same time, the strong demand for integrated systems suitable for different standards or application field forces designer to develop wideband ADC. Finally, due to sensitivity of new technology to the environment (temperature, power supply value…), the design have to take into account all the possible user cases. This increases again the design constraints and leads designer to find new strategy for design. An attractive solution to make both the design and the manufacturing of high performance devices easier consists in relaxing the design constraints and calibrating the devices after the devices manufacturing. The main parameter suitable for post correction is the Integral Non-Linearity (INL). Several techniques exist for off-line INL post correction [START_REF] Taherzadeh-Sani | Digital background calibration of interstage-gain and capacitor-mismatch errors in pipelined ADCs[END_REF][START_REF] Tsimbinos | Results of A/D converter compensation with a VLSI chip[END_REF][START_REF] Lundin | Post-correction of analog-to-digital converters[END_REF][START_REF] Irons | Improved compensation for analog-to-digital converters[END_REF][START_REF] Handel | A calibration scheme for imperfect quantizers[END_REF][START_REF] Lundin | Analog-todigital converter error correction using frequency selective tables[END_REF][START_REF] Lundin | Optimal index-bit allocation for dynamic postcorrection of analog-to-digital converters[END_REF][START_REF] Lundin | A criterion for optimizing bitreduced postcorrection of AD converters[END_REF]. Some of them are architecture-dependent and some others consider the ADC as a black box. These techniques can significantly increase the manufacturing yield but a strong limitation is the fact that the correction definition can be made only once for a given value of the ADC parameters. Moreover, no new calibration is possible even in case of a later change in the ADC parameters due to the environment or aging for example. In this context, we propose a solution for "on-line" autocalibration of ADC. In the literature some techniques can be found for digital on-line correction of ADC [START_REF] Davis | Digital correction of circuit imperfections in cascaded Σ-Δ modulators composed of 1st-order sections[END_REF][START_REF] Sarhang-Nejad | A high-resolution multibit ΣΔ ADC with digital correction and relaxed amplifier requirements[END_REF][START_REF] Kiss | Adaptive digital correction of analog errors in MASH ADCs-partII. Correction using test signal injection[END_REF], but, they are dedicated to specific ADC architecture and they imply a large additional circuitry. For auto-correction purpose, it is necessary to evaluate on-chip the converter INL. And, one of the main issues here is to perform the INL estimation with a minimal silicon area overhead.

In the remainder of the paper, section 2 gives the fundamental principle of the widely used digital correction: the Look-Up-Table based correction. The state of the art of LUT-based techniques with its associated limitations is described. Section 3 gives details on the proposed solution allowing the on-chip auto-correction. In section 4, this adaptive LUT-based correction technique is validated through simulations with two different case studies. Finally, section 5 gives some concluding remarks

II. LUT-BASED CORRECTION

A. Principle

As explained in the introduction several techniques have been developed for ADC correction. The Look-Up-Table (LUT) based technique [START_REF] Lundin | Post-correction of analog-to-digital converters[END_REF][START_REF] Irons | Improved compensation for analog-to-digital converters[END_REF][START_REF] Handel | A calibration scheme for imperfect quantizers[END_REF][START_REF] Lundin | Analog-todigital converter error correction using frequency selective tables[END_REF][START_REF] Lundin | Optimal index-bit allocation for dynamic postcorrection of analog-to-digital converters[END_REF] is clearly one of the most efficient solutions because it works only in the digital domain after conversion of the signal. As presented in Figure 1, the fundamental principle consists in using the output code of the ADC to address a table (LUT) and substitutes this output code with the one in the table which includes a pre-computed correction of the output codes.

( ) ( ) ( i HCF i LCF i INL (1)
Where LCF is the Low Code Frequency component and HLC is the High Code Frequency component of INL. The LCF shows the smooth variation of the INL versus codes and HCF emphasizes the sharp variations of INL. Generally, the LCF component can be modeled with a quite simple 3 rd or 4 th order polynomial; on the contrary HCF is strongly architecture dependent and cannot be described with a general analytical equation. In fact, the previous model assumes that the INL is only static but in reality the INL can be affected by the input signal frequency [START_REF] Björsell | Achievable ADC Performance by Postcorrection Utilizing Dynamic Modeling of the Integral Nonlinearity[END_REF] [START_REF] Medawar | ADC Characterization By Dynamic Integral Nonlinearity[END_REF]. To be more precise, INL doesn"t depend on the frequency of the input signal but on the speed of signal variation at any point of the dynamic range of the ADC. To take into account this effect, the INL of the code i can be now expressed by equation (2):

) , ( ) ( ) ( ) , ( d i DLCF i HCF i SLCF d i INL (2)
Where d is the slope:

t Vin d
of the input signal at the input signal value corresponding to the code i. The LCF component is divided into a static part (SLCF) independent of the input signal variation and a dynamic part (DLCF) which corresponds to the variation of the INL according to the input signal slope. This last component is related to the frequencydependent elements of the converters such as some parts of the amplifiers or comparators… Especially when no samplingand-hold circuit is used for the converter. The HCF component is related to the imperfections of the ADC quantizer for which the output is fairly independent of the frequencies within the bandwidth. Although the INL model presented in equation ( 2) is more precise it doesn"t take into account the influence of the environment or the aging effect. We consequently consider now a complete INL model as described in equation (3):

) , , ( ) , ( ) , ( ) , , ( k d i DLCF k i HCF k i SLCF k d i INL (3)
Where k emphasizes the external parameters such as temperature, drift due to aging, new application setup with a different sampling frequency, variation of the power supply… Generally, it is very difficult to estimate how all these "external" parameters might affect the INL nevertheless for high performance and large bandwidth ADC these parameters have to be considered. The objective is not to perform a perfect measurement but to estimate the INL with the required accuracy for the correction process. In this context, an accurate estimation of the SLCF component is mandatory, but a precise estimation of the HCF component is not required even if this component remains essential for efficient correction in most cases. Finally, the DLCF component has to be taken into account because ADCs are generally defined for large bandwidth although each application uses a small part of this bandwidth.

C. Static versus dynamic correction

In the classical LUT-based correction techniques there are two different approaches: static or dynamic correction. The static correction of an n-bit ADC uses an n-bit LUT of 2 n words. In the nominal operation of the ADC, the LUT is simply addressed by the current output code [START_REF] Lundin | Post-correction of analog-to-digital converters[END_REF][START_REF] Irons | Improved compensation for analog-to-digital converters[END_REF][START_REF] Handel | A calibration scheme for imperfect quantizers[END_REF]; this implies a one-to-one correspondence between the ADC and the LUT giving a 2 n words LUT for an n-bit ADC. Some solutions exist to reduce the size of the LUT by correcting only a selected subset of the ADC codes. It is to note that static correction is efficient only if the input frequencies are close to the frequency used to fill the LUT [START_REF] Lundin | Analog-todigital converter error correction using frequency selective tables[END_REF] or if ADC INL are relatively insensitive to input frequency. Indeed, this static correction technique cannot correct the DLCF component of the INL. The dynamic correction has been developed to overcome the limitations of the static one. In order to take into account the dynamic non-linearity (DLCF), it is necessary to consider not only the current output codes but also the previous ones. Consequently, additional computed data must be used to fill and address the LUT. Two different dynamic correction approaches have been proposed in the literature. Firstly, the phase-plan technique [START_REF] Lundin | Analog-todigital converter error correction using frequency selective tables[END_REF] for which the input signal slope is used as an additional addressing data. The resulting table has two dimensions. The main issue is to be able to obtain a good estimation of the slope of the input signal. The second dynamic correction method is called state-space correction [START_REF] Lundin | Optimal index-bit allocation for dynamic postcorrection of analog-to-digital converters[END_REF][START_REF] Lundin | A criterion for optimizing bitreduced postcorrection of AD converters[END_REF]. The basic principle is to consider not just the current output code but instead the K last output codes that may be, in some way, an image of the input slope. The size of the correction table is related to the number K of considered samples: K previous samples plus the current sample gives a space of dimension K+1, and, consequently, a K+1 dimension table of 2 n words. The drawback of this method is the huge size of the table and the long time to address it using (K+1)*n bits for an n-bit ADC. Again, solutions have been proposed to reduce this size: the samples may be under-sampled and truncated [START_REF] Lundin | A criterion for optimizing bitreduced postcorrection of AD converters[END_REF] or certain bits can be masked after a learning phase. These different addressing techniques have been compared in [START_REF] Lundin | A criterion for optimizing bitreduced postcorrection of AD converters[END_REF] and it appears that the effectiveness of the correction decreases as the size of the LUT decreases to the point where the "dynamic" correction is not better than the static one!

D. Summary

The static LUT-based correction is very efficient because it is possible to achieve a good correction with a quite small LUT. But, the static technique can not take into account the dynamic part of the INL. In the other hand, although dynamic correction is a promising technique, the global benefits can be discussed because of the complexity and required silicon area. None of the static or dynamic techniques is able to estimate and correct the influence of the environment or aging expressed by equation (3).

III. IN SITU LUT-BASED AUTO-CORRECTION

A. Principle

In this context, we propose a new solution for auto-correction of ADCs using a small LUT but with the capability to take into account the dynamic effects as well as the environment and aging effects. The fundamental idea is to propose an in situ LUT-based autocorrection scheme with the on-chip capability of computing and filling the LUT. By completing the LUT "in situ", i.e. directly in the application, the corrected codes are computed according to the input signal dynamic, aging and environment conditions. The proposed solution needs the on-chip generation of an input signal and INL estimation as illustrated figure 2. During the calibration phase a control signal disconnects the primary input of the ADC and connects the integrated signal generator. The output INL is estimated and the computed corrected codes are stored into the LUT. After this calibration phase the ADC is disconnected from the embedded generator and the ADC works with its normal primary inputs, the output code being corrected on-line by the values stored into the LUT. Classical calibration processes are usually made only once after manufacturing. This means that classical calibrations are made with no information on the targeted application and environment conditions. They are consequently performed for a wide band of frequencies, standard environment conditions, and independently from the aging effects. In our solution, because the calibration circuitry is integrated into the chip, the above calibration phase can be repeated at any time and at any location into a given application and for local and aging conditions. Concerning the dynamic correction, it has been previously commented that the corresponding multi-dimensional LUTs are very large. In addition, the integration of the BIST circuitry for generation of the input signal, analysis of the output codes and generation of the correcting codes would represent a prohibitively large additional circuitry. For this reason, we propose to use a static correction but using for the calibration an input signal exactly in the frequency range of the targeted application. To be able to cover all possible applications, the generator must be flexible and able to generate an input signal in the corresponding frequency bandwidth. In other words, we assume that in the narrow frequency band of the targeted application, the INL is not very sensitive to small dynamic effects.

B. Integrated module requirements

As explained in the previous paragraph, the corrected codes are computed using an estimated INL curve. The computation simply consists in rounding down the measured INL curve after subtraction of the ideal INL. However, the INL estimation cannot be limited to the LCF part of the INL. The integrated module must be able to estimate sharp variations of the INL which correspond to an approximation of the Differential Non-Linearity (DNL). The histogram-based method is a conventional test method mainly used to measure the Integral Non-Linearity (INL) of an ADC under test [START_REF]IEEE standard for terminology and test methods for analog-to-digital converters[END_REF] [START_REF] Dynad | Methods and draft standards for the DYNamic characterization and testing of analog to digital converters[END_REF]. The process is made of three steps. The first step consists in building the experimental histogram that represents the number of times each ADC output code appears for a given input signal. Then, this experimental histogram is compared with an ideal histogram obtained in the case of both an ideal ADC and an ideal analog input. Note that for a linear input signal this ideal histogram is perfectly flat along ADC codes. Finally, the main ADC parameters (such as INL) are extracted from the comparison between the two histograms. Other techniques have been developed [START_REF] Adamo | FFT Test of A/D Converters to Determine the Integral Nonlinearity[END_REF][START_REF] Csizmadia | Estimating the Integral Non-Linearity of AD-Converters via the Frequency Domain[END_REF][START_REF] Peralıas | Simple Evaluation of the Nonlinearity Signature of an ADC Using a Spectral Approach[END_REF][START_REF] Janik | A Spectral approach to estimate the INL of A/D converter[END_REF][START_REF] Kerzérho | Comparison between spectral-based methods for INL estimation and feasibility of their implantation[END_REF][START_REF] Kerzérho | A First Step for an INL Spectral-based BIST: the Memory Optimization[END_REF] for INL estimation; they are based on an INL modeling using polynomial or Fourier series expansion. In [START_REF] Kerzérho | Comparison between spectral-based methods for INL estimation and feasibility of their implantation[END_REF][23], the authors showed that only the technique based on Fourier series expansion can give local information on the INL. Unfortunately, the silicon integration of such Fourier based techniques is impossible because it needs the integration of both FFT computation and complex computations for INL extraction. As a consequence, the only possible solution for silicon integration is the histogram-based method. Concerning the analog signal generator, it must be able to generate signal with different dynamic properties. As explained in paragraph III.A the generated signal has to be inside the required bandwidth of the targeted application. To be more precise, the slope of the generated signal must be in the range of the slope of the possible input signal in the considered application. In addition, we want all the codes to be corrected using this range of input signal slope. It is important to note that we don"t need a large variety of input signal slopes, but we want all the codes to be corrected using the application range of slope. From this point of view, the classical sine wave signal is not satisfactory because it exhibits a large variety of slopes but not for all the codes. We consequently prefer to use a linear input signal with a slope selected in the range of the application slopes.

C. Histogram-based Analyzer

As explained in the previous section, the proposed correction scheme requires an on-chip estimation of the INL and the histogram-based method is the most efficient in terms of additional silicon. Many ADC BIST solutions have been proposed in the literature [START_REF] Roberts | Analog Signal Generation for Built-In Self-Test of Mixed-Signal Integrated Circuits[END_REF][START_REF] Lu | An Analog Multi-Tone Signal Generator for Built-In Self-Test Applications[END_REF][START_REF] Provost | Auto-Calibrating Analog Timer for On-Chip Testing[END_REF][START_REF] Azaïs | A lowcost adaptive ramp generator for analog BIST applications[END_REF][START_REF] Bernard | A High Accuracy Triangle-Wave Signal Generator for On-Chip ADC Testing[END_REF][START_REF] Vries | Built-In Self-Test Methodology for A/D Converters[END_REF][START_REF] Azaïs | A Low-Cost BIST Architecture for Linear Histogram Testing of ADCs[END_REF][START_REF] Renovell | Method and device for Integrated Testing for an Analog-to-Digital Converter[END_REF][START_REF] Sunter | A simplified polynomial-fitting algorithm for DAC and ADC BIST[END_REF][START_REF] Xu | A New Approach for the Nonlinearity Test of ADCs/DACs and its application for BIST[END_REF] mainly for test and characterization purpose including a solution patented by the authors [START_REF] Renovell | Method and device for Integrated Testing for an Analog-to-Digital Converter[END_REF]. We consequently propose to re-use these fundamental existing BIST concepts but of course they need to be modified and adapted to the purpose of auto-correction. In the remaining part of this section this BIST concepts are reminded and the required modifications are presented. A straightforward implementation of the histogram test technique requires two memories of 2 n words to store the experimental and the ideal histogram, plus a DSP to compute the ADC parameters from the stored histograms. Of course, this would not be a viable solution in a BIST or BISR (Builtin-Self Repair) context. In order to reduce the required hardware resources, the digital analyzer [START_REF] Azaïs | A Low-Cost BIST Architecture for Linear Histogram Testing of ADCs[END_REF] is implemented using the concept of on time decomposition. First, the classical parallel computation of the different ADC parameters is replaced by a sequential computation of each parameter. Second, the computation of every parameter is itself decomposed in a sequential process. This time-spread computation allows us to drastically reduce not only the required memory hardware but also the digital analyzer able to perform the different computations. Indeed, due to this time decomposition concept, the digital analyzer allowing the computations on the stored histograms is only composed of two small modules. The first module is the Detector Module (DM) is an n-bit register (for n-bit ADC) configurable in either a counter or a comparator and allows us to initialize the register. This module positions the code to be processed and compares it with the current ADC code. The second bloc is the Exploitation Module (EM) is composed of a P-bit register together with a configuration logic controlled by two signals. Basically, the EM module can either operate as an up/down counter to realize addition and subtraction or provide the 2's complement of the value stored in the register to manage the sign of results. Note that obviously, this module can compute division because the divisor is a power of 2. Indeed, the binary division by 2 P simply corresponds to a P-bit shift. The number of bits of the Exploitation Module is given by the following equation:

n INL P ) ( log 2
(4) Where INL is the required accuracy for the Integral Non-Linearity measurement and n the ADC resolution. For INL correction with LUT accuracy of 0.2LSB is sufficient. The number of blocs is thus equal to 3+n. As illustration, the Figure 3 gives the circuitry needed for implementing a 3 bit-EM register.

[ i ] 1-bit block

EM_Out[i] [ i-1] 1-bit block EM_Out[i-1] [ i+1] 1-bit block EM_Out[i+1] S1[i-2] S2[i-2] EM_Clear Ext_Clock c2 c1 S1[i+1] S2[i+1]

D. Analog signal generator

As for the histogram BIST, many papers have been published proposing efficient implementation for on-chip input signal generation. Consequently, an existing published solution is reused and modified to be compatible with the specific requirements of our on-chip auto-correction technique. In [START_REF] Azaïs | A lowcost adaptive ramp generator for analog BIST applications[END_REF] a solution is proposed for ramp and triangle wave analog signal generation. For our auto-correction scheme, only the ramp generator is interesting and will be adapted. This analog generator is based on the very simple principle of a constant current charge of a capacitor. Figure 4 shows the conceptual scheme of the ramp signal generator [START_REF] Azaïs | A lowcost adaptive ramp generator for analog BIST applications[END_REF] where Ic, C, Init and Step are respectively the charging current, the charge capacitor, the initialization signal and the signal for ramp duration. In order to achieve a satisfactory quality in terms of linearity, the generator is based on Wide Swing current mirrors. These mirrors copy the current generated by a single transistor with a controlled gate-source voltage. This architecture allows one to achieve 15 bits of linearity. In addition, an adaptive scheme is implemented to calibrate the slope of the ramp in situ, in the application. It is made of a comparator and a ramp rate control circuit which control the value of the charging current. Figure 5 shows the transistor level representation of the adaptive ramp generator.

Using such an analog adaptive scheme, the calibration process starts with a number of iterations in which the control voltage is progressively incremented until the ramp voltage reaches the reference voltage (Vref) within the given period. This control voltage then oscillates around the proper value in the following iterations, indicating that the calibration is completed has illustrated in Figure 6 for low frequency stimulus.

The basic idea is obviously to use this adaptive scheme to adapt the slope of the ramp for the in situ calibration into the application. The ramp period is fixed by a digital signal and the adaptive system manages the charging current to obtain the period for a dynamic range equal to the ADC under calibration.

E. Global implementation and operation

The algorithm for the LUT filling is depicted in Figure 7. For each ADC output code O(i), we compare the absolute value of its INL to 0.5LSB. If the |INL| is smaller than 0.5LSB, we fill the LUT with the expected code O(i) at the address O(i). If the INL is higher than 0.5, we calculate a corrected output code which will be stored in the LUT at the address O(i). To do so, we round the INL value to the nearest integer. Then we subtract the expected code O(i) to the computed compensation C p (i) to obtain the corrected output code C r (i). Obviously, C r (i) is bounded by 0 and 2 N -1. The controller used to manage the INL estimation and the correction algorithm is very simple with a negligible impact in terms of area overhead in comparison to the LUT and the ADC.

IV. VALIDATION OF THE IN SITU AUTO-CORRECTION

A. Platform

Under Labview, we have developed a platform where an ADC is defined by its resolution and its INL, and we can extract its transfer function and its response to different input signals such as sine wave, triangle, ramp… The platform is divided into 4 parts: the first defines the test stimuli, the second sets the ADC characteristics, the third computes the dynamics parameters (SINAD, THD, ENOB and SFDR) and the last displays the output responses with or without correction.

B. Validation

The validation is performed with a ADC. We studied two conditions (#1 and #2 in Table 1) corresponding to two different applications with two-targeted bandwidths. For the first condition, the maximum INL is equal to 2.7LSB. For the second condition, the maximum INL increases to 0.3LSB due to the dynamic part of the INL. Tableau 1, Dynamic parameters for ADC with and without postcorrection

1) Correction of INL in condition #1

As depicted in Figure 8, the INL is closed to 0 for lower code and reach 2.7 LSB for upper code. In such case, the uncorrected ADC has an ENOB of 8.71 bits, a THD equal to -57.38dB and a SFDR equal to -59.49dB. After applying postcorrection, the ENOB is 9.45bits, the THD is equal to -77.07 and the SFDR is equal to -72.66dB. As soon as the INL is outside the bandwidth of -0.5LSB and +0.5LSB, the output is corrected as shown in Figure 8.

2) Correction of INL in condition #2

As exposed in [START_REF] Lundin | Analog-todigital converter error correction using frequency selective tables[END_REF], the post-correction technique is better for input signal frequencies near from the frequency used for the histogram test. Consequently, the LUT filled in previous condition is less effective in condition #2 (see dynamic parameters in Tableau 1 for LUT #1 in condition #2). However, due to this adaptive LUT-based correction approach, we can apply a new calibration for a better correction in this condition. So, for the INL depicted in Figure 9, the ENOB decreases to 8.27bits rather than after postcorrection, the ENOB is still 9.34bits. The THD is -54.62 and -71.24 without and with post-correction respectively. The SFDR is -57.63dB rather than -70.38dB after post-correction. In the context of high speed and high resolution ADCs with calibration capabilities, we have proposed in this paper an original auto-correction scheme which is able to perform the auto-calibration in situ, i.e. directly in the application. This in situ auto-calibration approach has the main advantage of taking into account implicitly the environment conditions and the aging effects. In addition, the calibration is performed with an integrated adaptive signal generator providing an input signal tuned according to the application, i.e. taking into account the dynamic effects coming from the application signal speed. Both adaptive input signal generator and output INL estimator based on the histogram technique are described. The whole correction scheme is proved to be effective through extensive simulations.

Figure 1 .

 1 Figure 1. LUT-based post-correction of ADC For this purpose, the LUT is usually filled during the production phase of the ADC after estimation of its nonlinearity. The first step of the process is thus the estimation of the ADC INL.

)

Figure 2 .

 2 Figure 2. Adaptive LUT-based post-correction

Figure 3 .

 3 Figure 3. The Exploitation Module register

Figure 5 .

 5 Figure 4. Ramp generator

Figure 6 .

 6 Figure 6. Adaptive ramp generator simulation

Figure 7 ,

 7 Figure 7, LUT training algorithm

Figure 8 ,

 8 Figure 8, measured INL with and without post-correction

Figure 9 ,

 9 Figure 9, measured INL with and without post-correction V. CONCLUSION


André Paquier

Flood studies and effects of climate change: two examples

INTRODUCTION

Various floods

Usually, various kinds of floods are distinguished based on the origin or on the main processes. Basically, the cause of the flood can be overflow over the banks of the main channel, direct runoff or failure of a levee or other structure. Flash floods are typical of steep beds and / or high intensity rainfalls. In mountain areas, they can be associated to debris flows or highly concentrated bed load. In urban areas, they often follow the reduction of infiltration and of drainage network. All these kinds of floods could have their parameters modified because of climate change but effects will be obviously varied in terms of consequences.

Laws and floods

In France, floods constitute most part of the damages linked to natural risks. Thus, in the 1970s, a system of public funding was set up to make easier the reimbursement of damages by insurance companies. At the same time, measures to reduce the impact of floods develop including protection works and risk mapping. Two kinds of maps were set. The atlas of historical floods aims at informing people of past events that are in some way similar to the ones that will occur. With the same objective, some municipalities increase the number of signs or information places in which inhabitants can be warned of past floods.

The other documents are the maps showing the existing risks. Law changes several times but nowadays Central Government carried out flood studies in the main river basins in order to set Plans for Mitigating Flood Risks (abbreviated in PPRI in French). These PPRI include a flood hazard map and a flood risk map. In case of river overflow, the flood hazard map is set for the discharge hydrograph of the more catastrophic historical flood or, if peak discharge is higher, the theoretical flood of return period equal to 100 years (MATE and METL, 1999). For other causes of flood, this latter return period remains also an objective although calculation methods can be different. Because levees protecting from river floods are more than 8000 kilometres long in France (http://bardigues.aix.cemagref.fr), special treatment is planned for levee breaching. The additional hazard caused by the levee failure is also mapped to prevent building too close to the levee. Taking into account also stakes, the hazard map is transformed in a flood risk map with a few types of flooded areas: for instance, the main channel with high velocities during flood will be distinguished from retention areas with high flow depth in which buildings can be found. In each kind of area, specific rules are set up, for instance, to stop urban development or permit it with special construction rules.

The use of the reference to the 100 year return period was always at the origin of some questions about the method to calculate such flood. The number of measured years at gauging stations is often less than 100 years and sometimes only 20 or 30 years. Thus determining the 100 year flood discharge is highly uncertain.

Climate change and floods

Considering the calculation or estimate of reference discharges, climate change added some questions about the hypothesis of stationarity that was present in most of the hydrological classical methods. This question of stationarity was not a new one because the change of land use and particularly urbanization already raised this question. For instance, in Yzeron basin (a research experimental basin about 130 km 2 near Lyon, France), the increase of flood discharges from the 1970s to the 1990s was attributed [START_REF] Radojevic | Méthode d'évaluation de l'influence urbaine sur le régime des crues d'un bassin versant de 130 km²[END_REF] to the increase of urban areas (6 % to 19 %) rather than to the increase of rainfall.

In France, (Renard et al., 2009) studied a lot of hydrometric stations that show long series of observations. After selection of the stations that do not show any abnormal behaviour, trends were studied. Although no general trend can be pointed out, a slight increase of maximal annual flow can be noticed for North Eastern France which means that apart this latter result, the hydrological variations of floods from one year to the other one were higher than the variations due to the climatic trend. Conclusions on the consequences of global change (both climatic and land use changes) should take into account this variability. More over, global change has effects on the whole hydrological cycle and by this way indirect effects on floods. Change in land use of the whole river basin (for instance because summers are warmer) has direct effect on runoff but also on the sedimentary inputs and the morphological features through the sediment balance.

This integration of the effects of global change (more relevant than only climate change for future policies) is to be planned when defining the rules that will apply when complementing PPRI to reach European standards. In fact, European Flood Directive (http://ec.europa.eu/environment/water/flood_risk/) will be the frame for future approach of flood risk in France and in the other countries of the European Union. Three steps are planned:

In 2011, identify the river basins at risk of flooding In 2013, draw up flood risk maps for three levels of floods (usual, medium (return period about 100 years), extreme).

In 2015, establish flood risk management plans focused on prevention, protection and preparedness. In parallel to the European Flood Directive, the European Water Framework Directive insists on the Good State of the Water Bodies, which implies an equilibrium that depends of climatic parameters. For both directives, it is thus important to clarify the objectives in relation with future climatic trends and the key parameters such as land use of the basin that are connected with these latter trends. If the question is a general one, some areas are more sensitive to a slight change in hydraulic conditions.

To illustrate these latter aspects, we select the cases of a flat flood plain protected by levees (lower Agly. River) and an urban area (Vieux Port in the city of Marseille) that is often flooded from high intensity rainfalls.

AGLY RIVER CASE

Presentation of the studied area

The high flood that occurred in the southern part of France near the Spanish Border in November 1999 was one of the highest of the 20 th century in the lower reach of Agly River. The coastal area received rainfalls between 200 and 400 mm during less than 24 hours.

In Rivesaltes, the main city of this plain, the peak discharge reached about 2000 m3/s, which is as high as the historical 1940 flood. This discharge was higher than the conveyance of the main channel between the levees in the lower reach. The levees and the channel widening carried out in the 1970's were designed for a discharge of 1250 m3/s and may accommodate a discharge of about 1500 m3/s without freeboard.

In November 1999, several overflows occurred along the levees. These overflows were the cause of the flooding of large areas but generally only with a few centimetres of water. However, one of these overflows was the origin of a breach that suddenly released a 2 metres high wave that destroyed a sewage plant in Saint Laurent de la Salanque. This breach, which lowered the water levels in the downstream reach of the main channel, avoided other breaches by reducing or stopping the overflows.

These events demonstrated the vulnerability of the inhabited areas, although the levees protect them. From a more general point of view, the high rate of increase of the population in the areas protected by the levees is a sign of the failure of a strategy which aims at building structures which guarantees safety till a project discharge and does not take into account higher discharges and complementary risks. In the case of the lower reach of Agly River, more than 20,000 inhabitants are concerned.

Hydrodynamic modelling

In order to simulate flooding of this area, 2-D hydrodynamic modelling using the code Rubar 20 was performed (Paquier et al., 2002). Two meshes were successively built. The first one (8848 cells) rely on a calibration of the main channel conveyance using the 1992 flood event that passed bankful in most part of the area. The part played by the structures enabling the flow through the embankments that cross the flood plain (highway A9, railway line, etc) was quite well understood. However, the mesh was not refined enough to represent the variability of the main channel (cross sections) due to three factors: Deposits in the main channel occurred along the years and were vegetated. Sediment transport during the 1999 flood completely modified some cross sections because of either erosion or deposition. Blockage of flow evacuation to the sea occurred following deposition of debris on structures and rise of sea level reinforced by wind. Thus a more detailed grid (18568 cells) was set up permitting to simulate overflow in a more realistic way. The elevation of the crest of the levees was a main factor but sediment transport or, in other words, geometry and conveyance of the main channel during the flood was also key factor. In Saint Laurent, the modelling of the breach in the levee supposed a very rapid opening (corresponding to the related 2 metres high wave) with the full cross section of the breach as measured (about 40 m wide and 3 m high) and a relatively low Manning coefficient (0.02). The peak discharge through the breach was estimated to 170 m3/s. Other modelling by a hydraulic structure, either a weir (from the time of the failure) or a specific structure modelling a progressive breaching, provided very similar results. After calibration, the water levels remain low, which may be related to the presence (before the opening of the breach) of a few centimetres of water from rain and from infiltration through (or below) the levees; the wind (from South East, with an average velocity of about 20 m/s) also may have raised the level by a few centimetres in this relatively large inundated area.

Questions linked to climate change

Flooding processes and particularly overflow and breaching are very sensitive to all the parameters that influence water elevation along the main channel. The choice of a design flood is thus essential. In Agly flood plain, the 100 year flood with peak discharge similar to the one of 1999 flood was selected. Taking into account climate change would mean a change in this design flood but this change should also consider the sediment transport impact and its consequences on the morphology of the main channel, the sea level rise, the wind parameters, etc.

MARSEILLE CASE

Presentation of the studied area

In the Mediterranean coast of France, at fall, severe storms can generate rainfall intensities of more than 100 mm during one hour. Because such events are rather rare, the drainage or sewage networks are not designed to convey the corresponding discharges. Thus, in dense urban areas, the streets become rivers during a few hours. For instance, in Marseille, such an event occurs in September 2000 causing severe damages near the Vieux Port (city centre). To study flood hazard, the basin (about 2 km from West to East and 3 km from North to South) including this area is modelled to obtain the hydrodynamic parameters (water depth and velocity). The slopes are very steep with the peak elevation close to 120 m (Fig. 2), which means specific care to model the flows [START_REF] Mignot | Modeling floods in a dense urban area using 2D shallow water equations[END_REF].

Rainfall simulator and hydrodynamic modelling

Hydrodynamic model is directly fed with inputs representing rainfall. A rainfall simulator produced these inputs. The simulator is based of the turning band simulation technique. The simulation of precipitation fields is build as a composite, with one field of nonzero precipitation multiplied by one field of binary rainfall indicator function. The nonzero precipitation is simulated according to the observed distribution of non-zero precipitation (usually one with a strong dissymmetry, like a GPA, Inverse Gaussian, gamma or log-normal). The indicator field is obtained thresholding an independently simulated Gaussian field.

The simulation approach makes an explicit distinction between the perceived time-series relevant to fixed locations at ground level like rain gauges or sub basins, and the simulated rainfall field subject to atmospheric advection. As we assume that rainfall-generating processes are simpler to describe at the cloud level we technically come to simulate along backtracks. Advection is described as a horizontal incompressible field.

The method to make the inference of parameters is based on the integrative properties of random field. Given a basic point structure for both the indicator and the non-zero precipitation fields, the combination of them, aggregated over a given duration, has some known properties deduced from field observations.

For Marseille case, the information concerning the rainfalls consists in a series of 24 rain gauges (over about 20 years) that are distributed over the whole area of the municipality of Marseille. The parameters of the rainfall simulator are calibrated to take into account the variability of the rainfall and the main advection direction. Using the calibrated rainfall simulator, several possible rainfall events are constituted; each one provides -on every rain gauge -the same amount of rainfall as registered every six minutes. For the September 2000 event, one of these possible events is illustrated on Figure 3 for a six-minute period near the first peak of the rainfall. Figure 4 compares the rainfall spatially averaged on the Vieux Port basin for various runs of the simulator; the 31 runs show variability around 50% in rainy time steps (>1 mm over 6 minutes) although, at the rain gauge located in the basin, they provide exactly the same amount (close to 200 mm for the whole event). One of these runs (in bold on Fig. 4) was randomly selected to be used as input to hydrodynamic models. Rainfall reaches the streets, the roofs of the buildings, the yards, etc. In the hydrodynamic modelling, we choose to represent only the street network that covers about one third of the whole area. A simple way is to consider the parcel blocks totally impervious and to distribute the rainfall input homogeneously over the street around each block. But alternatively, the analysis of urban database permits [START_REF] Rodriguez | Surface runoff in urban catchments: morphological identification of unit hydrographs from urban databanks[END_REF] to attribute the corresponding roof rainfall input to different street locations considering also the infiltration into green areas or to sewage network. Note that considering the high intensity of the rainfall on September 2000, at Marseille, sewage network is considered to have been full rapidly and thus can be neglected at the peak of the flood.

The flow propagation along the streets is modelled using 2-D shallow water equations (code Rubar 20 as presented in (El Kadi Abderrezzak et al., 2006)) on a regular mesh formed of quadrilaterals and triangles with a mean cell size of 5 meters (leading to about 72,000 cells on the whole Vieux Port basin).

Resulting peak water depths are low on the slopes and in the places in which no water can accumulate. On the eastern and southern parts of the Vieux Port, peak water depths are generally between 0.5 and 1 m but can go beyond in local depressions. These results agree with observations collected from citizen interviews or videos. Velocities can be also high (>2 m/s) in a few steep main streets.

Questions linked to climate change

Comparison between calculations using uniform rainfall (from the rain gauge) and varied rainfall (from the simulator) shows [START_REF] Paquier | Influence of rainfall distribution on urban flood: a case study[END_REF]) that the differences can be about 0.1 m in a wide area. The differences can be even higher at specific times because the two peaks are moved.

Thus, using the rainfall simulator can provide more realistic flooding but permits also to deal with climate change by modifying the parameters to represent rainfall trends for the future. Then, the effect of the variability of rainfall can be easily compared to the effect of climate change in terms of floods parameters. Of course, the change in downstream conditions due to an eventual rise of sea level (in the harbour) can also be integrated in the hydrodynamic modelling.

CONCLUSIONS

The example of Agly lower flood plain emphasizes that the eventual change in flood discharges could be one among other factors of the changes in flood risk. The main factor remains the type of land use: in the last 50 years, vulnerability increases a lot because of the development of urbanization mainly in connection with recreation activities or because of the proximity of a large city (Perpignan). This trend together with climate change effects on hydrological and morphological parameters could lead to a new design of flood protection because people cannot bear an increasing risk. The solutions can be higher levees but also defining new structures such as diversion channels or measures in the upstream basin. Of course, reducing vulnerability could be a very efficient alternative.

In Marseille, the rainfalls analysis permits to calibrate a rainfall simulator that directly feeds the hydrodynamic model. Thus, using the same approach and passing to a rainfall analysis in a non-stationary approach could permit to study the effects of climate change. Beyond, the reorganization of urban areas also necessary to adapt to other consequences of climate change could be used to decide measures to mitigate floods.

All the aspects met in the latter two cases are more general. They emphasize that any measure (for instance, in order to meet the requirements of the European Flood Directive) should take into account the eventual trend induced by climate change. This climate change should be expressed in terms of rainfall distribution (space and time) and of river discharge chronicle but land use evolution will be a key factor to estimate evolution of flood risk.

Figure 1 .

 1 Figure 1. Computational grid (8848 cells): the numbers indicate the hydraulic structures, the map is north oriented.

Figure 2 .

 2 Figure 2. Ground elevation of the basin of Vieux Port (with the street network in grey).

Figure 3 .

 3 Figure 3. Rainfall during six minutes at 6:12 p.m. on September 19, 2000 provided by one run of the rainfall simulator (Centre of Marseille municipality).

Figure 4 .

 4 Figure 4. Reconstructed rainfall of September 19, 2000 (spatial average over the Vieux Port Basin). 31 runs set, the selected one shown in bold.