Minjia Shi

Hongwei Zhu

Liqin Qian
email: qianliqin1108@163.com

Lin Sok
email: sok.lin@rupp.edu.kh

Patrick Solé
email: sole@enst.fr

On self-dual and LCD double circulant and double negacirculant codes over F q + uF q *

Keywords: double circulant codes, double negacirculant codes, codes over rings, self-dual codes, LCD codes, Artin conjecture 2010): 94 B15, 94 B25, 05 E30

Double circulant codes of length 2n over the semilocal ring R = F q + uF q , u 2 = u, are studied when q is an odd prime power, and -1 is a square in F q . Double negacirculant codes of length 2n are studied over R when n is even and q is an odd prime power. Exact enumeration of self-dual and LCD such codes for given length 2n is given. Employing a duality-preserving Gray map, self-dual and LCD codes of length 4n over F q are constructed. Using random coding and the Artin conjecture, the relative distance of these codes is bounded below. The parameters of examples of the modest length are computed. Several such codes are optimal.

Introduction

Double circulant and double negacirculant self-dual codes over finite fields have been studied recently in [START_REF] Alahmadi | On self-dual double negacirculant codes[END_REF] and [START_REF] Alahmadi | On self-dual double circulant codes[END_REF], respectively, from the viewpoint of enumeration and asymptotic performance. The main tool there is the CRT approach to quasi-cyclic codes as introduced in [START_REF] Ling | On the algebraic structure of quasi-cyclic codes I: finite fields[END_REF], and generalized to quasi-twisted codes in [START_REF] Jia | On quasi-twisted codes over finite fields[END_REF]. It is natural to extend these results to other classes of ring alphabets, beyond finite fields, building on the theory developed in [START_REF] Ling | On the Algebraic Structure of Quasi-cyclic Codes II: Chain Rings[END_REF]. Double circulant self-dual codes over a commutative ring can only exist if there is a square root of -1 over that ring [START_REF] Ling | On the Algebraic Structure of Quasi-cyclic Codes II: Chain Rings[END_REF]. For the ring R = F q + uF q , u 2 = u, a ring of current interest [START_REF] Shi | Two new families of two-weight codes[END_REF], this requirement leads to the condition that -1 is a square in F q . This ring is studied here because of a duality preserving Gray map that turns self-dual codes into self-dual codes, and similarly, LCD codes into LCD codes. While this Gray map was defined already in [START_REF] Zhu | A class of constacyclic codes over F p + vF p[END_REF], its duality properties were not considered there. LCD codes are defined as codes intersecting trivially with their dual. They enjoy a wealth of combinatorial and algebraic properties [START_REF] Dougherty | The combinatorics of LCD codes: linear programming bound and orthogonal matrices[END_REF][START_REF] Güneri | Quasi-cyclic complementary dual codes[END_REF]. In particular quasi-cyclic codes over fields have been proved to contain infinite families of good long LCD codes [START_REF] Güneri | Quasi-cyclic complementary dual codes[END_REF]. In this paper we study both selfdual and LCD double circulant and double negacirculant codes over R and show that all four families contain arbitrarily long codes with fixed rate and with relative distance of the Gray image bounded below by a nonzero constant. These results are based on the complete enumeration of the codes in these four families for given length 2n. While, for asymptotic purposes, only the case of n a prime with q primitive modulo n is needed, the counting results are developed for all n's. The said arithmetic condition holds for infinitely many n's by Artin conjecture [START_REF] Moree | Artin's primitive root conjecture a survey[END_REF], which is proved conditionally under GRH [START_REF] Hooley | On Artin's conjecture[END_REF]. In the case of negacirculant codes, n is supposed to be a power of 2 for asymptotics, and the truth of Artin conjecture is not needed. The factorization of x n + 1 in that case depends on properties of Dickson polynomials.

The material is organized as follows. Section 2 contains the preliminary requisite necessary to the further sections. Section 3 develops the machinery of the CRT approach to double circulant and double negacirculant codes, and derives the main enumeration results. Section 4 is dedicated to asymptotic bounds on the relative Hamming distance of the Gray image of these codes when their lengths tends to infinity. Section 5 computes some numerical examples. Section 6 concludes the article and discusses open problems.

Preliminaries

2.1 The Ring F q + uF q Let q be a prime power, and let F q be the finite field of order q. Consider the ring R = F q + uF q where u 2 = u. It is semi-local with maximal ideals (u) and (u -1). The group of units R * is isomorphic, as a multiplicative group, to the product of two cyclic groups of order (q -1). To construct double circulant codes, we need to know when the ring R = F q + uF q contains a square root of -1.

Theorem 2.1. The ring R contains a square root of -1 if one and only one of the following three conditions is true:

• q is a power of 2;

• q = p b , where p is a prime congruent to 1 mod 4;

• q = p 2b , where p is a prime congruent to 3 mod 4, and b ≥ 1 is an integer.

Proof. Note first that F q contains a square root of -1 iff one of the above three conditions is true [START_REF] Ling | On the Algebraic Structure of Quasi-cyclic Codes II: Chain Rings[END_REF]. The condition is sufficient. If one of the three cases is true, then F q , a subring of R, already contains such a root. The condition is necessary. Write such a root as x + uy with x, y ∈ F q . Then (x + uy) 2 = x 2 + u(y 2 + 2xy), showing that x 2 = -1, and that x is a square root of -1 in F q .

Remark: Note that for each x in the above proof there are two possible value of y that is y = 0, or y = -2x. There are thus zero or four distinct square roots of -1 in R.

Norm Function over Finite Field

For all x ∈ F q n , the norm of x over F q is a map Norm: F q n -→ F q defined by N orm(x) = x (q n -1)/(q-1) .

Moreover, Norm is a multiplicative homomorphism which is surjective ([11, Theorem 2.28]). N orm(0) = 0, so it maps F * q n onto F * q , where each nonzero element in F * q has a preimage of size (q n -1)/(q -1) in F * q n . x i y i , where the operation is performed in R. The Euclidean dual code of C is denoted by C ⊥ and defined as

Codes

A linear code C over R of length n is an R-submodule of R n . If x = (x
C ⊥ = {y ∈ R n | x, y = 0, ∀x ∈ C}. A linear code C of length n over R is called self-dual if C = C ⊥ . A linear code C of length n over R is called linear complementary dual (LCD) if C ∩ C ⊥ = {0}.
A matrix A over R is said to be circulant (resp. negacirculant) if its rows are obtained by successive shifts (resp. negashifts) from the first row. A code C is called double circulant (resp. double negacirculant) over R if its generator matrix G will be of the form G = (I, A), where I is the identity matrix of order n and A is a circulant (resp. negacirculant) matrix of the same order.

If C(n) is a family of codes with parameters [n, k n , d n ] over F q , the rate ρ and relative distance δ are defined as ρ = lim sup n→∞ kn n and δ = lim inf n→∞ dn n , respectively. A family of code is good if ρδ > 0. Both limits are finite as limits of bounded quantities.

Gray Map

q Odd

The Gray map φ over the ring R is defined by φ(a + ub) = (-b, 2a + b) for a, b ∈ F q from R to F 2 q . Clearly, φ is a bijection from R to F 2 q , which extends naturally to a map from R n to F 2n q . Adopting the idea of [5, Theorem 2.10.3], we can prove the following result.

Theorem 2.2. If C is a self-dual (resp. LCD) code of length n over R, then φ(C) is a self-dual (resp. LCD) code of length 2n over F q .

Proof. Consider a pair of codewords w = (w 1 , w 2 , . . . ,

w n ), v = (v 1 , v 2 , . . . , v n ) ∈ C, where w i = a i + b i u, v i = c i + d i u and a i , b i , c i , d i ∈ F q for 1 ≤ i ≤ n. If C is a self-dual code over R, then w, v = n i=1 (a i + b i u)(c i + d i u) = n i=1 (a i c i + (a i d i + b i c i + b i d i )u) = 0, which implies that n i=1 a i c i = 0 and n i=1 (a i d i + b i c i + b i d i ) = 0.
According to the definition of the Gray map, we have

φ(w), φ(v) = φ(a 1 + b 1 u, . . . , a n + b n u), φ(c 1 + d 1 u, . . . , c n + d n u) = n i=1 (b i d i + 4a i c i + 2a i d i + 2b i c i + b i d i ) = n i=1 (4a i c i + 2(a i d i + b i c i + b i d i )) = 0. It implies that φ(C ⊥ ) ⊆ φ(C) ⊥ . Since the Gray map φ is a bijection from R to F 2 q , then φ(C ⊥ ) = φ(C) ⊥ . If C is a LCD code over R, then C ∩ C ⊥ = {0}. We can easily obtain that φ(C ∩ C ⊥ ) ⊆ φ(C) ∩ φ(C ⊥ ). Because φ is a bijection from R to F 2
q , equality holds in the preceding inclusion and

φ(C) ∩ φ(C) ⊥ = φ(C) ∩ φ(C ⊥ ) = φ(C ∩ C ⊥ ) = {0}.
Thus φ(C) is a LCD code of length 2n over F q .

q Arbitrary

We can generalize the map β of [START_REF] Dougherty | Type IV codes over rings[END_REF] as follows. Define a map β : R → F 2 q by the formula

β(a + ub) = (a, a + b), for all a, b ∈ F q . Theorem 2.3. If C is a self-dual (resp. LCD) code of length n over R, then β(C) is a self-dual (resp. LCD) code of length 2n over F q .
Proof. We claim that β(C ⊥ ) = β(C) ⊥ . The rest of the proof follows in the same way as the proof of Theorem 2.2. To prove the claim note that if w, v = 0 like in the said proof, then

β(w), β(v) = n i=1 a i c i + (a i + b i )(c i + d i ) = n i=1 a i d i + b i c i + b i d i = 0. This shows that β(C ⊥ ) ⊆ β(C) ⊥ , and β being bijective, that β(C ⊥ ) = β(C) ⊥ .
3 Algebraic Structure of Double Circulant and Double Negacirculant Codes

Double Circulant Codes of Odd Length

In this subsection, we assume that n is an odd integer, q is a prime power and gcd(n, q) = 1.

We can cast the factorization of x n -1 into distinct irreducible polynomials over R in the form

x n -1 = α(x -1) s i=2 g i (x) t j=1 h j (x)h * j (x),
where α ∈ R * , g i (x) is a self-reciprocal polynomial with degree 2e i for 2 ≤ i ≤ s, and h * j (x) is the reciprocal polynomial of h j (x) with degree d j for 1 ≤ j ≤ t. By the Chinese Remainder Theorem (CRT), we have

R[x] (x n -1) ≃ R[x] (x -1) ⊕ s i=2 R[x]/(g i (x)) ⊕   t j=1 (R[x]/(h j (x)) ⊕ (R[x]/(h * j (x))))   ≃ R ⊕ s i=2 F q [u, x] (u 2 -u, g i (x)) ⊕   t j=1 F q [u, x] (u 2 -u, h j (x)) ⊕ F q [u, x] (u 2 -u, h * j (x))   ≃ R ⊕ s i=2 F q 2e i + uF q 2e i ⊕   t j=1 (F q d j + uF q d j ) ⊕ (F q d j + uF q d j )   := R ⊕ s i=2 R 2e i ⊕   t j=1 R d j ⊕ R d j   ,
where we let R ℓ = F q ℓ + uF q ℓ , for any nonegative integer ℓ. Note that all of these rings are extensions of R. This decomposition naturally extends to

R[x] (x n -1) 2 as R[x] (x n -1) 2 ≃ R 2 ⊕ s i=2 (R 2e i ) 2 ⊕   t j=1 (R d j ) 2 ⊕ (R d j ) 2   .
In particular, each linear code C of length 2 over R[x] (x n -1) can be decomposed as the "CRT sum"

C ≃ C 1 ⊕ s i=2 C i ⊕   t j=1 (C ′ j ⊕ C ′′ j )   ,
where C 1 is a linear code over R, for each 2 ≤ i ≤ s, C i is a linear code over R 2e i , and for each 1 ≤ j ≤ t, C ′ j and C ′′ j are linear codes over R d j , which are called the constituents of C. In self-reciprocal case, we give the definition of the Hermitian inner product over

R 2e i . For all z = z 1 + uz 2 ∈ R 2e i , where z 1 , z 2 ∈ F q 2e i , the conjugate of z over R 2e i is z = z q e i
1 + uz q e i 2 , and the Hermitian inner product is defined as

: (z, z ′ ) • (w, w ′ ) = zw + z ′ w ′ , where z, z ′ , w, w ′ ∈ R 2e i . Theorem 3.1.
Let n denote a positive odd integer, and let q be a prime power coprime with n. Assume that the factorization of x n -1 into irreducible polynomials over R is of the form

x n -1 = α(x -1) s i=2 g i (x) t j=1 h j (x)h * j (x), with α ∈ R * , n = 1 + s i=2 2e i + 2 t j=1 d j .
Then the total number of self-dual double circulant codes over R is

4 s i=2 (1 + q e i ) 2 t j=1 q d j -1 2 .
Proof. We can count the number of self-dual double circulant codes by counting their constituent codes. There are four self-dual codes C 1 of length 2 over R, whose generators are (1, ω), (1, -ω), (1, ω(1 -2u)) and ( 1, ω(2u -1)), respectively, where ω 2 = -1 and ω ∈ F q . More generally, a factor g i (x) of degree 2e i leads to counting self-dual Hermitian codes

C i of length 2 over R 2e i . If (1, c e i ) is the generator of C i then (1, c e i ) • (1, c e i ) = 1 + c e i c e i = 1
+ c e i c e i q e i = 0. Let c e i = x + uy, where x, y ∈ F q 2e i , we then have 1 + (x + uy)(x q e i + uy q e i ) = 0 ⇐⇒ x • x q e i = -1,

(x + y)(x + y) q e i = -1, ⇐⇒ N orm(x) = -1, N orm(x + y) = -1,
where N orm is a map from F q 2e i to F q e i . So there are q e i + 1 roots for N orm(x) = -1 and q e i + 1 roots for N orm(x + y) = -1. Therefore, the number of ordered pairs (x, y) is equal to (q e i + 1) 2 .

In the last case about reciprocal pairs, note that h j (x) and h * j (x) of both degree d j leads to counting dual pairs of codes (for the Euclidean inner product) of length 2 over R d j , that is to count the number of solutions of the equation 1 + c ′ d j c ′′ d j = 0, where (1, c ′ d j ) and (1, c ′′ d j ) are the generators of C ′ j and C ′′ j , respectively. If

c ′ d j ∈ R * d j , then c ′′ d j = -1 c ′ d j
.

There are

|R * d j | = (q d j -1) 2 choices for (c ′ d j , c ′′ d j ). If c ′ d j ∈ R d j \R * d j , then c ′ d j = ux ∈ (u) or c ′ d j = (u -1)
x ∈ (u -1). In this case, we claim that 1 + c ′ d j c ′′ d j = 0 cannot occur. Otherwise, by reducing the equation modulo u or u -1, we would get 1 = 0 in F q d j , contradiction. The proof of the theorem is now completed. Theorem 3.2. Keep the same notations as above, then the total number of LCD double circulant codes over R is

(q 2 -4) s i=2 (q 4e i -(q e i + 1) 2 ) t j=1
(q 4d j -2q 3d j + 3q 2d j -2q d j + 1).

Proof. Like in the self-dual case, we can count the number of LCD double circulant codes C by considering each constituent of C. Note that if a double circulant code of length 2 over some extension of R is not self-dual, then it is LCD because of having single-row generator matrix. Combined with the results of Theorem 3.1, the total number of LCD double circulant constituent codes C 1 over R is q 2 -4 and the total number of LCD double circulant constituent codes C i over R 2e i is q 4e i -(q e i + 1) 2 .

In the last case about reciprocal pairs, note that h j (x) and h * j (x) of degree d j leads to counting pairs of codes of length 2 over R d j with certain intersection properties [START_REF] Güneri | Quasi-cyclic complementary dual codes[END_REF]. Let (1, a) and (1, b) be the generators of C ′ j and C ′′ j , respectively.

C ′ j ∩ C ′′ j ⊥ = {0}, C ′′ j ∩ C ′ j ⊥ = {0}. ⇐⇒ 1 + ab ∈ R * d j .
Without loss of generality, we discuss on the unit character of a as follows.

• If a ∈ R * d j , then b ∈ -1 a + R * d j and | -1 a + R * d j | = |R * d j |. In this case, we have |R * d j | 2 = (q d j -1) 4 choices for (a, b). • If a ∈ R d j \{R * d j ∪ {0}}, then a = uα or a = (u -1)α. Let b = β ′ + uβ ′′ , where α ∈ F * q d j
and β ′ , β ′′ ∈ F q d j . When a = uα, we then have

1 + ab = 1 + uα(β ′ + uβ ′′ ) = 1 + uα(β ′ + β ′′ ) ∈ R * d j .
This is equivalent to α(β ′ + β ′′ ) = -1. There are q d j (q d j -1) 2 choices for (α, β ′ , β ′′ ). When a = (u -1)α, then we have

1 + ab = 1 + (u -1)α(β ′ + uβ ′′ ) = 1 + (u -1)αβ ′ ∈ R * d j .
This is equivalent to αβ ′ = 1 and β ′′ is arbitrary in F q d j . There are also q d j (q d j -1) 2 choices for (α, β ′ , β ′′ ).

• If a is zero, then b is arbitrary in R d j . There are q 2d j choices for b.

Hence, the number of the last case about reciprocal pairs is (q d j -1) 4 + q d j (q d j -1) 2 + q d j (q d j -1) 2 + q 2d j = q 4d j -2q 3d j + 3q 2d j -2q d j + 1. The proof of the theorem is now completed.

Double Negacirculant Codes of Even Length

For our purpose, we assume in this subsection that n is an even integer, q is an odd prime power, and that gcd(n, q) = 1 . We can cast the factorization of x n + 1 into distinct irreducible polynomials over R in the form

x n + 1 = α s i=1 g i (x) t j=1 h j (x)h * j (x),
where α ∈ R * , g i (x) is a self-reciprocal polynomial with degree 2e i for 1 ≤ i ≤ s, and h * j (x) is the reciprocal polynomial of h j (x) with degree d j for 1 ≤ j ≤ t. Using the same notations and arguments as above, we can easily carry out the results as follows:

R[x] (x n + 1) ≃ s i=1 R 2e i ⊕   t j=1 R d j ⊕ R d j   , and
C ≃ s i=1 C i ⊕   t j=1 (C ′ j ⊕ C ′′ j )   .
Similar to the proof of Theorem 3.1 and Theorem 3.2, we have the following two enumeration results. Their proofs are omitted. Theorem 3.3. Let n denote a positive even integer, and q a prime power coprime with n. Assume that the factorization of x n + 1 into irreducible polynomials over R = F q + uF q is of the form

x n + 1 = α s i=1 g i (x) t j=1 h j (x)h * j (x), with α ∈ R * , n = s i=1 2e i + 2 t j=1 d j .
Then the total number of self-dual double negacirculant

codes over R is s i=1 (1 + q e i ) 2 t j=1
(q d j -1) 2 .

Theorem 3.4. Keep the same notations as above, then the total number of LCD double negacirculant codes over R is s i=1 (q 4e i -(q e i + 1) 2 ) t j=1 (q 4d j -2q 3d j + 3q 2d j -2q d j + 1).

Distance Bound

Distance Bound for Double Circulant Codes

Let q be a primitive root mod n and n be an odd prime. Recall that x n -1 = (x-1)(x n-1 + . . . + x + 1) = (x -1)h(x). Note that h(x) is irreducible over R, because F q is a subring of R, and h(x) is irreducible over F q .

By the Chinese Remainder Theorem (CRT), we have

R[x] (x n -1) ≃ R[x] (x -1) ⊕ R[x] (h(x)) ≃ R ⊕ F q [u, x] (u 2 -u, h(x)) ≃ R ⊕ F q n-1 + uF q n-1 .
Let R denote the ring R[x] (h(x)) , so R is a subring of R. The nonzero codewords of the cyclic code of length n generated by h(x) is called constant vector. Lemma 4.1. If the nonzero vector z = (e, f ) with e not a constant vector, then there are at most q n+1 generators (1, a) such that z ∈ C a and C a is a double circulant code over R.

Proof. By the Chinese Remainder Theorem (CRT), (e, f ) ≃ (e 1 , f 1 )⊕(e 2 , f 2 ). Since (e, f ) ∈ C a , then f = ea, f 1 = e 1 a 1 and f 2 = e 2 a 2 , where e 1 , f 1 ,

a 1 ∈ R and e 2 , f 2 , a 2 ∈ R. Let a 1 = a ′ 1 + ua ′′ 1 , a 2 = a ′ 2 + ua ′′ 2 , where a ′ 1 , a ′′ 1 ∈ F q and a ′ 2 , a ′′ 2 ∈ F q n-1 .
In the first constituent of C a , we discuss on the unit character of e 1 as follows.

• If e 1 ∈ R * , there exists only one solution for a 1 = f 1 e 1 .

• If e 1 ∈ (u)\{0}, then e 1 = ue ′ 1 and

f 1 = uf ′ 1 , where e ′ 1 ∈ F * q , f ′ 1 ∈ F q . Since u 2 = u, then f 1 = uf ′ 1 = ue ′ 1 a 1 = ue ′ 1 (a ′ 1 + ua ′′ 1 ) = ue ′ 1 a ′ 1 + ue ′ 1 a ′′ 1 ⇐⇒ f ′ 1 e ′ 1 = a ′ 1 + a ′′ 1 .
There are q choices for a 1 .

• If e 1 ∈ (u -1)\{0}, then e 1 = (u -1)e 1 ′ and

f 1 = (u -1)f ′ 1 , where e ′ 1 ∈ F * q , f ′ 1 ∈ F q . Since u 2 = u, then f 1 = (u-1)f ′ 1 = (u-1)e ′ 1 a 1 = (u-1)e ′ 1 (a ′ 1 +ua ′′ 1 ) = (u-1)e ′ 1 a ′ 1 ⇐⇒ f ′ 1 e ′ 1 = a ′ 1 and a ′′ 1 is arbitrary in F q .
There are q choices for a 1 .

• If e 1 =0, then a 1 is arbitrary in R, there are q 2 choices for a 1 .

In the second constituent of C a , we discuss on the unit character of e 2 as follows.

• If e 2 ∈ R * , there exists only one solution of a 2 .

• If e 2 ∈ (u)\{0}, then e 2 = ue ′ 2 and

f 2 = uf ′ 2 , where e ′ 2 ∈ F * q n-1 , f ′ 2 ∈ F q n-1 . Since u 2 = u, then f 2 = uf ′ 2 = ue ′ 2 a 2 = ue ′ 2 (a ′ 2 + ua ′′ 2 ) = ue ′ 2 a ′ 2 + ue ′ 2 a ′′ 2 ⇐⇒ f ′ 2 e ′ 2 = a ′ 2 + a ′′ 2 .
There are q n-1 choices for a 2 .

• If e 2 ∈ (u -1)\{0}, then e 2 = (u -1)e 2 ′ and

f 2 = (u -1)f ′ 2 , where e ′ 2 ∈ F * q n-1 , f ′ 2 ∈ F q n-1 . Since u 2 = u, then f 2 = (u -1)f ′ 2 = (u -1)e ′ 2 a 2 = (u -1)e ′ 2 (a ′ 2 + ua ′′ 2 ) = (u -1)e ′ 2 a ′ 2 ⇐⇒ f ′ 2 e ′ 2 = a ′ 2 and a ′′ 2 is arbitrary in F q n-1 .
There are q n-1 choices for a 2 .

• If e 2 = 0, then e ≡ 0 mod h(x), then e is a constant vector, contradiction.

Thus, there are at most q n+1 generators (1, a) such that z ∈ C a .

Lemma 4.2. If z = (e, f ) ∈ R 2n with e is not a constant vector, then there are at most 4(1 + q n-1

2 ) generators (1, a) such that z ∈ C a and C a is a self-dual double circulant code over R.

Proof. Keep the same notations as Lemma 4.1. In the first constituent of C a , there are at most 4 generators (1, a 1 ) such that C 1 is self-dual double circulant code over R due to Theorem 3.1.

In the second constituent of C a , let a 2 = a ′ 2 + ua ′′ 2 , where a ′ 2 , a ′′ 2 ∈ F q n-1 , we discuss on the unit character of e 2 as follows.

• If e 2 ∈ R * , then there exists a unique solution of a 2 = f 2 e 2 .

• If e 2 ∈ (u)\{0}, then e 2 = ue ′ 2 , f 2 = uf ′ 2 and

f 2 = ua 2 e ′ 2 = u(a ′ 2 + ua ′′ 2 )e ′ 2 = u(a ′ 2 + a ′′ 2 )e ′ 2 ⇐⇒ a ′ 2 + a ′′ 2 = f ′ 2 e ′ 2
,

where e ′ 2 ∈ F * q n-1 and f ′ 2 ∈ F q n-1 . Since C a is a self-dual double circulant code, then (1, a 2 ) • (1, a 2 ) = 1 + a 2 a 2 = 1 + a 2 a 2 q n-1 2 = 0. This is equivalent to    a ′ 2 a ′ 2 q n-1 2 = -1, (a ′ 2 + a ′′ 2 )(a ′ 2 q n-1 2 + a ′′ 2 q n-1 2 ) = -1. ⇐⇒ N orm(a ′ 2 ) = -1, N orm(a ′ 2 + a ′′ 2 ) = -1,
where N orm is a map from F q n-1 to F q n-1 2

. So there are q n-1

2 +1 roots for N orm(a ′ 2 ) = -1 and q n-1 2 + 1 roots for N orm(a ′ 2 + a ′′ 2 ) = -1. But a ′′ 2 is determined by a ′ 2 , then there are at most 1 + q n-1 2
choices for a ′ 2 , hence for a 2 .

• If e 2 ∈ (u -1)\{0}, then e 2 = (u -1)e ′ 2 and f 2 = (u -1)f ′ 2 , where e ′ 2 ∈ F * q n-1 and f ′ 2 ∈ F q n-1 , which implies that

f 2 = (u -1)a 2 e ′ 2 = (u -1)(a ′ 2 + ua ′′ 2 )e ′ 2 ⇐⇒ a ′ 2 = f ′ 2 e ′ 2 .
Since C a is a self-dual double circulant code, using the same procedure as above, we have

N orm(a ′ 2 ) = -1, N orm(a ′ 2 + a ′′ 2 ) = -1.
Hence, there are 1 + q

n-1 2 choices for a 2 because a ′ 2 = f ′ 2 e ′ 2 and 1 + q n-1 2
roots for a ′′ 2 .

• If e 2 = 0, then e ≡ 0 mod h(x), then e is a constant vector, contradiction.

So there are at most 4(1+q n-1

2 ) generators (1, a) such that z ∈ C a . The proof is done.

In number theory, Artin's conjecture on primitive roots states that a given integer q which is neither a perfect square nor -1 is a primitive root modulo infinitely many primes l [START_REF] Moree | Artin's primitive root conjecture a survey[END_REF]. This was proved conditionally under the Generalized Riemann Hypothesis (GRH) by Hooley [START_REF] Hooley | On Artin's conjecture[END_REF]. Hence we can get infinite families of double circulant codes C(2n) over R where the analysis made for x n -1 with only two irreducible factors applies.

Recall the q-ary entropy function defined for 0 ≤ t ≤ q-1 q by

H q (t) =    0, if t = 0, tlog q (q -1) -tlog q (t) -(1 -t)log q (1 -t), if 0 < t ≤ q-1 q .
This quantity is instrumental in the estimation of the volume of high-dimensional Hamming balls when the base field is F q . The result we are using is that the volume of the Hamming ball of radius tn is asymptotically equivalent, up to subexponential terms, to q nHq(t) , when 0 < t < 1, and n goes to infinity [START_REF] Huffman | Fundamentals of error correcting codes[END_REF]Lemma 2.10.3]. The main result obtained in this paper is as follows.

Theorem 4.3. Let n be an odd prime, n > q, and q be a primitive root modulo n. The family of Gray images of self-dual double circulant codes over R of length 2n, of relative distance δ, and rate 1/2, satisfies H q (δ) ≥ 1 8 . The family of Gray images of LCD double circulant codes over R of length 2n, of relative distance δ, and rate 1/2, satisfies H q (δ) ≥ 1 4 . In particular, both families codes are good.

Proof. Let Ω n denote the size of the family. Thus, for n → ∞, we have, by Theorem 3.1, Ω n ∼ 4q n-1 for self-dual double circulant codes, and by Theorem 3.2, Ω n ∼ q 2n-2 for LCD double circulant codes. Assume we can prove that for n large enough Ω n > λ n B(d n ), where B(r) denotes the number of vectors in R 2n with Hamming weight of their F q image < r.

Here λ n = 4(1 + q (n-1)/2 ) for self-dual codes, and λ n = q n+1 for LCD codes. This would imply, by Lemmas 4.1 and 4.2, that there are codes of length 2n in the family with minimum Hamming distance of their F q image ≥ d n . Denote by δ the relative distance of this family of q-ary codes.

If we take d n the largest number satisfying Ω n > λ n B(d n ), and assume a growth of the form d n ∼ 4δ 0 n, and thus Ω n ∼ λ n B(d n ), for n → ∞ then, using an entropic estimate for B(d n ) ∼ q 4nHq(δ 0 ) [9, Lemma 2.10.3] yields, with the said values of Ω n and λ n the estimate H q (δ 0 ) = 1 8 for self-dual codes and H q (δ 0 ) = 1 4 for LCD codes. The result follows by observing that, by the definition of δ, we have δ ≥ δ 0 .

4.2

Distance Bound for Double Negacirculant Codes In order to describe the factorization of x n + 1 over R, where n = 2 a , we first recall the definition of Dickson polynomials. For α ∈ R, we define the Dickson polynomial of paramater α and degree n as

D n (x, α) = ⌊n/2⌋ j=0 n n -j n -j j (-α) j x n-2j .
The complete factorization of x 2 n + 1 over F q with q ≡ 3 (mod 4) is given in the following theorem [START_REF] Meyn | Factorization of the cyclotomic polynomial x 2 n + 1 over finite fields[END_REF].

Theorem 4.4.

[14] Let q ≡ 3 (mod 4), where q = 2 A m -1, A ≥ 2 and m is an odd integer.

Let n ≥ 2,

(a) if n < A, then x 2 n + 1 is the product of 2 n-1 irreducible trinomials over F q x 2 n + 1 = γ∈Γ (x 2 + γx + 1),
where Γ is the set of all roots of D 2 n-1 (x, 1).

(b) if n ≥ A, then x 2 n + 1 is the product of 2 A-1 irreducible trinomials over F q x 2 n + 1 = δ∈△ (x 2 n-A+1 + δx 2 n-A -1),
where △ is the set of all roots of D 2 A-1 (x, -1).

Example 4.5. If q = 3, i.e., q ≡ 3 (mod 4), then q = 2 2 • 1 -1. This implies A = 2, m = 1, and D 2 (x, -1) = 0, i.e., △ = {1, 2}. By Theorem 4.4

x 2 n + 1 = (x 2 n-1 + x 2 n-2 -1)(x 2 n-1 + 2x 2 n-2 -1).
Example 4.6. If q = 11, i.e., q ≡ 3 (mod 4), then q = 2 2 • 3 -1. This implies A = 2, m = 3, and D 2 (x, -1) = 0, i.e., △ = {3, 8}. By Theorem 4.4

x 2 n + 1 = (x 2 n-1 + 3x 2 n-2 -1)(x 2 n-1 + 8x 2 n-2 -1).
Example 4.7. If q = 7, i.e., q ≡ 3 (mod 4), then q = 2 3 • 1 -1. This implies A = 3, m = 1, and D 4 (x, -1) = 0, i.e., △ = {1, 3, 4, 6}. By Theorem 4.4

x 2 n + 1 = (x 2 n-2 + x 2 n-3 -1)(x 2 n-2 + 3x 2 n-3 -1)(x 2 n-2 + 4x 2 n-3 -1)(x 2 n-2 + 6x 2 n-3 -1).
Next, we need the analogous factorization theorem when q ≡ 1 (mod 4) as follows.

Theorem 4.8.

[2] Let q ≡ 1 (mod 4), where q = 2 A+1 m + 1, A ≥ 1, m is an odd integer. Denote the set of all primitive 2 k -th roots of unity in F q by U k . If n ≥ 2, then (a) if n ≤ A, then ord 2 n+1 (q) = 1, x 2 n + 1 is the product of 2 n linear factors over F q x 2 n + 1 = u∈U n+1 (x + u).

(b) if n ≥ A+1, then ord 2 n+1 (q) = 2 n-A , x 2 n +1 is the product of 2 A irreducible binomials over F q of degree 2 n-A x 2 n + 1 = u∈U A+1 (x 2 n-A + u).

Example 4.9. If q = 5, i.e., q ≡ 1 (mod 4), then q = 2 2 • 1 + 1. This implies A = 1, m = 1, and U 2 = {2, 3}. By Theorem 4.8

x 2 n + 1 = (x 2 n-1 + 2)(x 2 n-1 + 3).

Example 4.10. If q = 13, i.e., q ≡ 1 (mod 4), then q = 2 2 •3+1. This implies A = 1, m = 3, and U 2 = {5, 8}. By Theorem 4.8

x 2 n + 1 = (x 2 n-1 + 5)(x 2 n-1 + 8).

Example 4.11. If q = 41, i.e., q ≡ 1 (mod 4), then q = 2 3 •5+1. This implies A = 2, m = 5, and U 3 = {3, 14, 27, 38}. By Theorem 4.8

x 2 n + 1 = (x 2 n-2 + 3)(x 2 n-2 + 14)(x 2 n-2 + 27)(x 2 n-2 + 38).

Remark: There is an error in the case (b) of Theorem 4 in [START_REF] Alahmadi | On self-dual double negacirculant codes[END_REF], i.e., "x 2 n + 1 = γ∈△ (x n-A+1 + δx n-A -1)" should be changed to "x 2 n + 1 = γ∈△ (x 2 n-A+1 + δx 2 n-A -1)".

According to [START_REF] Meyn | Factorization of the cyclotomic polynomial x 2 n + 1 over finite fields[END_REF], there is another error of Theorem 6 in [START_REF] Alahmadi | On self-dual double negacirculant codes[END_REF], "q = 2 A m + 1, A ≥ 2" should be changed to "q = 2 A+1 m + 1, A ≥ 1".

So we can cast the factorization x n + 1 over F q into two or four irreducible polynomials by limiting the size of △ and U in Theorems 4.4 and 4.8. This factorization carries over R because F q is subring of R.

The polynomial x n + 1 factors into two irreducible polynomials that are reciprocal of each other over R, this is the case if q ≡ ±1 (mod 4), where q = 2 2 m ± 1, m odd, it happens if q = 3, 5, 11, 13, 19, 27, 29, 37, etc. The polynomial x n + 1 factors into four irreducible polynomials which are pairwise reciprocal of each other over R, this is the case if q ≡ ±1 (mod 4), where q = 2 3 m ± 1, m odd, it happens if q = 7, 23, 25, 41, etc.

4. 2 . 1

 21 Factorization of x n + 1

4. 2 . 2

 22 Distance Bounds for Decomposition I and II Decomposition I: If x n +1 = h(x)h * (x), where h(x) and h * (x) are irreducible polynomials and reciprocal of each other, where deg(h(x)) = n 2 , then R[x]/(h(x)) ≃ R[x]/(h * (x)) ≃

MSC(* This research is supported by National Natural Science Foundation of China (61672036), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133) and the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University (2015D11) and Key projects of support program for outstanding young talents in Colleges and Universities (gxyqZD2016008).

F q n 2 + uF q n 2 . For convenience, K 1 = R[x]/(h(x)) and K 2 = R[x]/(h * (x)). We have provided several examples of that situation, such as Examples 4.5, 4.6, 4.9 and 4.10. Lemma 4.12. Let q be odd, n be a power of 2, x n + 1 with Decomposition I. If z = (e, f ) ∈ R 2n is a nonzero vector, then there are at most q 3n 2 generators (1, a) such that z ∈ C a and C a is a double negacirculant code over R.

Proof. By the Chinese Remainder Theorem (CRT), (e, f ) = (e 1 , f 1 )⊕(e 2 , f 2 ). Since (e, f ) ∈ C a , then f = ea, f 1 = e 1 a 1 and f 2 = e 2 a 2 , where e 1 , f 1 , a 1 ∈ K 1 and e 2 , f 2 , a 2 ∈ K 2 . Let

In the first constituent of C a , we discuss on the unit character of e 1 as follows.

• If e 1 ∈ K * 1 , there exists only one solution for a 1 = f 1 e 1 .

• If e 1 ∈ (u)\{0}, then e 1 = ue ′ 1 and

There are q n 2 choices for a 1 .

a ′ 1 and a ′′ 1 is arbitrary in F q . There are q n 2 choices for a 1 .

• If e 1 = 0, then a 1 is arbitrary in K 1 , there are q n choices for a 1 .

Using the same argument as above in the second constituent of C a , there are also at most q n choices for a 2 . But if z is not zero, then e 1 and e 2 cannot both be zero. Hence there are only at most q 3n 2 generators (1, a) such that z ∈ C a . We have thus proved the lemma.

Lemma 4.13. Let q be odd, n be a power of 2. Assume x n + 1 satisfies decomposition I. If z = (e, f ) ∈ R 2n is a nonzero vector, then there are at most q n 2 generators (1, a) such that z ∈ C a and that C a is self-dual double negacirculant code of length 2n over R.

Proof. Keep the same notations as Lemma 4.12. Since C a is a self-dual code, then (1, a 1 ), (1, a 2 ) = 1 + a 1 a 2 = 0. Clearly, a 1 , a 2 are units and a 2 is determined by a 1 . Since e 1 and e 2 can not both be zero, without loss of generality, we may assume e 1 = 0, then we just need to consider the first constituent of C a by symmetry. Next, we discuss on the unit character of e 1 as follows.

• If e 1 ∈ K * 1 , there exists only one solution of a 1 = f 1 e 1 .

• If e 1 ∈ (u)\{0}, then e 1 = ue ′ 1 and

There are q n 2 choices for a 1 .

• If e 1 ∈ (u -1)\{0}, then e 1 = (u -1)e 1 ′ and

There are q n 2 choices for a 1 .

Hence there are at most q n 2 generators (1, a) such that z ∈ C a . We have thus proved the lemma.

). We have provided several examples of that situation, such as Examples 4.7 and 4.11.

Lemma 4.14. Let q be odd, n be a power of 2, and x n + 1 satisfying Decomposition II . If z = (e, f ) ∈ R 2n is a nonzero vector, then there are at most q In the first constituent of C a , Let a 1 = a ′ 1 + ua ′′ 1 , where a ′ 1 , a ′′ 1 ∈ F q n 4 , we discuss on the unit character of e 1 as follows.

• If e 1 ∈ I * 1 , then there exists only one solution for a 1 that is

• If e 1 ∈ (u)\{0}, then e 1 = ue ′ 1 and

There are q n 4 choices for a 1 .

• If e 1 ∈ (u -1)\{0}, then e 1 = (u -1)e 1 ′ and

1 , while a ′′ 1 is arbitrary in F q . Thus there are q n 4 choices for a 1 .

• If e 1 = 0, then a 1 is arbitrary in I 1 , there are q n 2 choices for a 1 .

Using the same argument as above in the three remaining constituents of C a . There are also at most q n 2 choices for a i , where i = 2, 3, 4. But z is not zero, then e 1 , e 2 , e 3 and e 4 can not be zero at the same time. Hence there are at most q 3n 2 × q n 4 generators (1, a) such that z ∈ C a . We have thus proved the lemma. Lemma 4.15. Let q be odd, n be a power of 2, and x n + 1 satisfying Decomposition II. If z = (e, f ) ∈ R 2n is a nonzero vector, then there are at most q 3n 4 generators (1, a) such that z ∈ C a , and C a is a self-dual double negacirculant code over R.

Proof. Keep the same notations as Lemma 4.14. Since C a is a self-dual code, we then have

So we just need to consider a 1 and a 3 , because a 2 and a 4 are determined by a 1 and a 3 , respectively. Since z is not zero, then we may assume e 1 = 0. Using a similar argument as in the proof of Lemma 4.13, we see that there are only at most q n 4 choices for a 1 and q n 2 choices for a 3 . Hence, there are at most q 3n 4 generators such that z ∈ C a .

The following results can be proved by the same method as employed in the last subsection.

Theorem 4.16. Under the condition of the Decomposition I (resp. Decomposition II), if q is an odd prime power, and n is a power of 2, then the family of Gray images of self-dual double negacirculant codes over R of length 2n, of relative distance δ, and rate 1/2, satisfies H q (δ) ≥ 1 8 (resp. H q (δ) ≥ 1 16 ). In particular, these families of codes are good.

Proof. The proof follows the method of the proof of Theorem 4.3 with Ω n ∼ q n for both Decomposition I and Decomposition II, by Theorem 3.4 and λ n = q n 2 and λ n = q 3n 4 by Lemmas 4.13 and 4.15, respectively. The details are omitted. Theorem 4.17. Under the condition of the Decomposition I (resp. Decomposition II), if q is an odd prime power, and n is a power of 2, then the family of Gray images of LCD double negacirculant codes over R of length 2n, of relative distance δ, and rate 1/2, satisfies H q (δ) ≥ 1 8 (resp. H q (δ) ≥ 1 16 ). In particular, these families of codes are good.

Proof. The proof follows the method of the proof of Theorem 4.3 with Ω n ∼ q 2n for both Decomposition I and Decomposition II, by Theorem 3.4 and λ n = q 3n 2 and λ n = q 7n 4 by Lemmas 4.12 and 4.14, respectively. The details are omitted.

Numerical Examples

We need the following lemma, before doing any computation.

Lemma 5.1. If C has generator matrix G = (I, A), with A = A 1 + uA 2 , where A 1 , A 2 are q-ary matrices of order n, and I denotes the identity matrix of order n, then φ(C) has generator matrix

Proof. The first row of the first matrix is φ(uG), and its second row is φ(G). The matrix spans subsets of φ(C). Since the rank is 2n, the result follows.

In Table 1 and Table 2, we have provided some examples of LCD and self-dual double circulant codes respectively with the best parameters obtained by Magma search and Lemma 5.1. The coefficients of degree n polynomial a 1 (x) and a 2 (x) in F 5 [x] are written in decreasing powers of x. For example for n = 3, the entry 311 means 3x 2 + x + 1. The parameters over F 5 are given in the form [4n, 2n, d] where d is the minimum distance. The entry in the rightmost column is the best known distance of a [4n, 2n] (self-dual) code over F 5 , obtained by looking up the tables in [START_REF] Grassl | Bounds on the minimum distance of linear codes and quantum codes[END_REF] and [19]. When the distance of the code constructed reaches that value the parameters are starred.

Conclusion and Open Problems

In this paper we have studied double circulant and double negacirculant codes over the ring R = F q + uF q . It might be worth looking at quasi-cyclic codes of higher index, or more generally, at quasi-twisted codes of higher index. Four-circulant or four-negacirculant codes are natural candidates for this exploration [START_REF] Shi | On the self-dual four-circulant codes[END_REF].

While passing from fields to rings increases significantly the complexity of the proofs and calculations, it might still be worth looking at other polynomial rings with u having a minimal polynomial of higher degree, or defined by several variables.


Rallou Thomopoulos
email: rallou.thomopoulos@inra.fr

A Practical Application Approach to Argumentation for Multicriteria Analysis and Decision Support

Keywords: Argumentation, Application, Decision support, Multicriteria analysis

The paper explores, through an application example, connections between argumentation and another type of methods often involved in decision problems, namely multicriteria analysis. A novel argumentative analysis framework is proposed and applied to assess a short supply chain device for food aid, regarding different categories of criteria. The analysis leads to some feedback concerning argumentation systems.

Introduction

Making a decision involving several stakeholders with different objectives requires to take into account various types of information: the consequences of each possible decision, the stakeholders' viewpoints and preferences on the decisions, and the indicators they consider as decision criteria. Argumentation [START_REF] Besnard | Elements of Argumentation[END_REF][START_REF] Rahwan | Argumentation in Artificial Intelligence[END_REF] deals with situations where information is incoherent because it comes from several sources or corresponds to several points of view that possibly have different priority levels. It is a reasoning model based on the construction and evaluation of interacting arguments. It has been formalized both in philosophy and in computer science [START_REF] Mackenzie | Begging the question in non-cumulative systems[END_REF][START_REF] Rescher | The role of rhetoric in rational argumentation[END_REF][START_REF] Besnard | Elements of Argumentation[END_REF] and applied to nonmonotonic reasoning (e.g. [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF]), decision making (e.g. [START_REF] Bonet | Arguing for decisions: A qualitative model of decision making[END_REF][START_REF] Fox | Safe and sound -artificial intelligence in hazardous applications[END_REF]), for modeling different types of dialogues including negotiation (e.g. [START_REF] Kraus | Reaching agreements through argumentation: A logical model and implementation[END_REF][START_REF] Sycara | Persuasive argumentation in negotiation[END_REF]), in deductive argumentation [START_REF] Amgoud | Final review and report on formal argumentation system[END_REF][START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF] and defeasible logic programming [START_REF] García | Defeasible logic programming: An argumentative approach[END_REF]. The system introduced in [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] consists of a set of arguments and a binary relation on that set, expressing conflicts among arguments. An argument gives a reason for believing a claim, or for doing an action. Historically, the prototypical application field of argumentation in computer science was the legal domain [START_REF] Prakken | Law and logic: A review from an argumentation perspective 227[END_REF]. More recently, several works proved its relevance in a larger context, in social-related concerns, medicine, food systems, chains, policies and controversies, especially for decision-making purposes [START_REF] Fox | Argumentation-based inference and decision making-a medical perspective[END_REF][START_REF] Croitoru | A Practical Application of Argumentation in French Agrifood Chains[END_REF][START_REF] Tamani | An argumentation system for eco-efficient packaging material selection[END_REF][START_REF] Thomopoulos | Information for decision-making is ubiquitous: Revisiting the reverse engineering mode in breadmaking technology[END_REF][START_REF] Tremblay | Value-based argumentation for policy decision analysis: methodology and an exploratory case study of a hydroelectric project in Québec[END_REF].

Although international research communities are active both in the argumentation and in the decision fields, most often these domains have been studied separately. Several works have proposed attempts to combine both [START_REF] Jarke | Mediator: Towards a negotiation support system[END_REF][START_REF] Brewka | How to Buy a Porsche: An Approach to Defeasible Decision Making[END_REF][START_REF] Parsons | Argumentation and decision making: A position paper[END_REF][START_REF] Gordon | The Zeno Argumentation Framework[END_REF][START_REF] Karacapilidis | Computer Supported Argumentation and Collaborative Decision Making: The HERMES System[END_REF][START_REF] Amgoud | Using arguments for making and explaining decisions[END_REF]. Applications in agronomy have emerged a few years ago and are growing. Recent works have dealt with the interest of argumentation in decisions about agri-food chain steering [START_REF] Bourguet | An artificial intelligence-based approach to deal with argumentation applied to food quality in a public health policy[END_REF][START_REF] Thomopoulos | Artificial intelligence-based decision support system to manage quality of durum wheat products[END_REF][START_REF] Thomopoulos | Decision support for agri-food chains: A reverse engineering argumentation-based approach[END_REF]. Among public policy decision problems, agri-food chain arbitrations involve various actors, from production to consumption through processing, distribution and recycling. Consequently, the interests of all the stakeholders of the chain interfere. Given the diversity of their viewpoints, they pursue possibly divergent goals.

In recent food-related concerns, short supply chains [START_REF] Chiffoleau | Le développement des circuits courts à l'épreuve de l'analyse des réseaux sociaux[END_REF][START_REF] Paturel | Alimentation et lien social : les circuits courts comme alternative ?[END_REF][START_REF] Paturel | Approvisionnement local pour les restaurants du coeur de l'Hérault. Etude de faisabilité réalisée dans le cadre du PROA-LR[END_REF] are defined by the close proximity of production to consumption, both geographically and relationally. In 2011-2012, a food aid association in the South of France experimented with a short supply chain. The aim was to test the feasability of local fruit and vegetable supply for the association, denoted 'AD34'. The analysis of this experimentation examined four criteria: technical, economic, social and participative. A need for explanation, analysis and rationalization of the collected results motivated the approach presented in this paper. Understanding the system involves a high number of interconnected factors that are difficult for a human to grasp. In [START_REF] Thomopoulos | Decision support for agri-food chains: A reverse engineering argumentation-based approach[END_REF], the relevance of the argumentative approach was highlighted regarding cognitive considerations. In this paper, we revisit argumentation systems to discuss the meaning of notions such as argument acceptability in a concrete case.

Section 2 recalls some elements of argumentation. Section 3 formalizes the short supply case in terms of argumentation systems. Section 4 proposes a novel analysis framework. Section 5 applies this model to the case study and presents the results. Section 6 provides some conclusive feedback and discussion. A very first stage of this study is consultable in [START_REF] Thomopoulos | Un système d'argumentation avec contextes et indicateurs pour analyser un dispositif d'approvisionnement en circuit court[END_REF].

Elements of argumentation systems

Classically, an argumentation process follows three main steps: (1) constructing arguments and counter-arguments, (2) evaluating the acceptability of the different arguments, and (3) defining the justified conclusions. In Dung's framework [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF], to complete the first step, a binary relation on the set of arguments called "attack" is defined, reflecting conflicts among arguments. This section briefly recalls Dung's abstract argumentation framework and mentions some further works based on it.

Definition 1 (Dung's argumentation framework). An argumentation framework is a pair AF = (A, R) where A is a set of arguments and R ⊆ A × A is an attack relation. An argument α attacks an argument β if and only if (α, β) ∈ R.

In the above definition, arguments are abstract entities. Their origin and structure are left unknown. Note that we can associate each argumentation system with a directed graph whose nodes are the different arguments, and the edges represent the attack relation between them. Among all the conflicting arguments, one has to define which arguments to keep for inferring conclusions or for making decisions. In [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF], different semantics for the notion of acceptability have been proposed. For the purpose of this paper, we only recall admissible and preferred semantics.

Definition 2 (Conflict-free, Defense, Admissibility). Let B ⊆ A.
-B is conflict-free if and only if α i , α j ∈ B such that (α i , α j ) ∈ R;

3 Problem formalization

The experimental device

The partners involved in the short food chain experimentation (also referred to as the 'device') were the AD34 association, the managers of the National Wholesale Market (MIN), and the "Innovation" French joint research team. In a first stage, the criteria to be analyzed in the study were defined: technical (logistic), economic (added value for producers and wholesalers), social (relations and the sharing of information between stakeholders) and participative (involvement of the actors). In a second stage, information was collected: semi-structured interviews were conducted with producers participating in the device (10 interviews with 8 producers), groupings of producers, coordinators of the agri-food networks in MIN (4 interviews), the two wholesalers (4 interviews), the volunteers responsible for receiving the fruit and vegetables from AD34's warehouse (4 interviews), and other AD34 volunteers (10 interviews with 10 volunteers responsible for the distribution centers). By the end of the campaign, surveys were conducted with volunteers at the distribution centers (response rate of 77% covering 56 volunteers) and recipients of the food aid (122 people answered, from a sample of 10 representative centers), essentially concerned with the technical criterion.

Formalizing arguments and attacks

From the surveys conducted in the way presented in the previous section, the various arguments put forward by the different stakeholders were collected and organized. Then, the argument analysis was performed in five steps:

the main arguments in favor ('pro') and against ('con') the experimental device were identified; arguments were refined and structured in argument/decision tables which establish a link with the decisions (support or oppose the device, corresponding to 'pro' and 'con' arguments respectively) that the involved actors may make. In this step, the hypothesis and conclusion parts of each argument were identified, as depicted in Table 1 in the "Description" column (the hypothesis is in italic, the conclusion is underlined). We may also need to express refinements of a given argument. For example in Table 1 after Argument B we see two arguments, C and D, refining it with more specialized hypotheses: 'nonstandard' (in Argument C) and 'overproduction' (in Argument D) provide sub-cases of 'hard to sell' (in Argument B). This will play a role in the attack definition; attacks are then defined based on logical contradiction. It is not the aim of this paper to introduce a logical framework. Interested readers may refer for instance to [START_REF] Thomopoulos | Decision support for agri-food chains: A reverse engineering argumentation-based approach[END_REF][START_REF] Tamani | An argumentation system for eco-efficient packaging material selection[END_REF] for logical approaches in practical applications. As a brief propositional-logic illustration, in Table 1 we have priceLoss → ¬wellAccepted (Argument A) and hardT oSell ∧ priceLoss → wellAccepted (Argument B), therefore (B, A) ∈ R. This step implies a terminological homogenization, for instance 'no price loss' and 'lower prices' are considered as synonyms; the contents of the argument/option tables were displayed in the form of argumentation graphs that can be evaluated according to the method of [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF]; finally, we analysed the selected argumentation graph in the light of the analytical model introduced in this paper (Sections 4 and 5).

Several groups of arguments were identified. The first group (see Table 1) examines which food products were likely to lead to the success or failure of the device and why.

Arg.

Description: Hypothesis, Conclusion Decision Criterion A the device is well accepted only when it induces no price loss pro economic for producers compared to the classical system B the device requires little effort for producers and succeeds even pro economic with lower prices for products that are hard to sell in the classical system C the device succeeds with lower prices for non-standard products, pro economic which are hard to sell in the classical system D the device succeeds with lower prices for overproduction due to pro economic climatic reasons, which is hard to sell in the classical system since it leads to an imbalance between supply and demand E the device failed when it turned out to be too adverse compared con economic to the classical system Table 1 Arguments about the interest of the device (first series)

We can notice that arguments C and D are particular cases of argument B, and that the three of them (B, C, D) disagree with argument A since they express that price is not the only element that may lead to the success of the device. Thus we can note that the attack relation R contains the following attacks: (B,A),(C,A),(D,A).

Argument E, which is less detailed than the previous ones since it only mentions "adverse" conditions without more precise elements (price, etc.), expresses no contradiction with the previous arguments, although it is expressed as a negative argument. However it is refined by the following new arguments, given in Table 2.

Argument F is counterbalanced by the arguments listed in Table 3.

Arg. Description Decision Criterion F the device failed when it was too adverse in terms of price con economic G the device failed when it was too adverse in terms of storage con technical capacity Arguments I, J, K, L are different variations of argument H. In this set {H, I, J, K, L}, all arguments contradict with the prior argument F. Thus we can add the following attacks to the attack relation R: (H, F), (I, F), (J, F), (K, F), (L, F).

Finally, the technical and social arguments provided in Table 4 complete the above description. We can notice several oppositions related to these last five arguments. Indeed, argument N is a counter-example of argument M based on a social benefit of the device. Thus we can add to the attack relation R: (N, M).

Arg.

Description Decision Criterion M the device failed when it did not take into account the quantities con technical available on the local market N for some products, the device was a success despite the pro social inadequacy of the planned dates to the reality of the local market, thanks to the wholesalers ' Argument O mentions a technical difficulty associated with large-size products. This contrasts with argument J which considers the handling of non-standard products as a positive feature of the device. It also contrasts with argument A, which tends to reduce to price issues the difficulties encountered by the device. Thus the following attacks can be added to R: (O, A), (O, J).

Moreover, arguments O and P both focus on logistical aspects of the device. O is a negative one, since it addresses the necessity of cutting large-size products, which is not only time-consuming but also implies space and equipment to do it. On the opposite, P is a positive aspect, since the sorting of damaged products can be avoided in the device. Thus a mutual attack between O and P is declared in R: (O, P), (P, O).

Finally, argument P highlights waste reduction, which counterbalances the economic argument F. Thus we can add to the attack relation R: (P, F).

The full argumentation graph is given in Figure 1. ) of this argumentation system. The arguments displayed in green (light grey in case of black and white display) belong to the extension, those in red (dark grey in case of black and white display) do not. An immediate remark that can be noted is that there is more than one extension, here two extensions which correspond to coherent sets of arguments. Thus there are divergences in the system that led to these two viewpoints.

Fig. 2 First preferred extension

There are three rejected arguments. These are the arguments that belong to no extension. Technically, they appear in red (dark grey) both in Figures 2 and3. They are, namely, A, F, Three other arguments belong to one extension but not to both. These are the strictly credulously accepted arguments, namely O -in green (light grey) in Figure 2 but not 3as well as J and P -in green (light grey) in Figure 3 but not 2. This means that, to adopt a coherent point of view, one has to choose between either O, or the pair {J, P }. Thus we have SCred = {J, O, P }.

The remaining arguments belong to both extensions: they appear in green (light grey) both in Figures 2 and3. They are skeptically accepted and thus form Skept.

Figure 4 synthesizes the acceptability of arguments. Arguments belonging to Rej are depicted with a black dash, arguments of SCred with a grey dash, arguments of Skept are white (with no dash). Cred is the union of SCred and Skept. At this stage, we lack of features to analyze these divergences. Indeed, we would expect from the argumentation system to provide explanations about the origin of the discordances that led to different viewpoints, highlighted by the existence of several extensions. These explanations could be provided by tracing not only the conflicting arguments, but also the criteria and rationale they rely on. To go further we need to use the analytical argumentation system AAS and the indicators presented in next section.

The proposed analysis framework

This section introduces a model denoted analytical argumentation system together with several indicators. It includes several ways of organizing the set of arguments in contexts. Each way provides a partition of A.

Definition 4 (Analytical Argumentation System

). An Analytical Argumentation System is a tuple ASS = (A; R; C 1 . . . Cn) where:

-A is a set of arguments; -R ⊆ A × A is an attack relation; -each C i is a partition of A.
It is thus a breakdown of the set of arguments A into subsets called contexts. By definition of a partition: ( c∈Ci c = A) and ( c∈Ci c = ∅).

We may note that the case n = 0 corresponds to the classic Dung system. The case n = 1 reminds of the CP AF model [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF], without preference relations, though. Indeed, among the various works that were proposed in order to enhance the expressivity of argumentation systems, some envisaged contexts as a way to equip argumentation systems with multiple non-conflicting preference relations [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF][START_REF] Bourguet | An artificial intelligence-based approach to deal with argumentation applied to food quality in a public health policy[END_REF]. However, in the present work, we are interested in structuring the set of arguments A without expressing any value judgement regarding the relative importance of arguments. Consequently the framework we propose does not include preferences. Indeed, the introduction of contexts has another purpose here, providing several ways to categorize the arguments.

Several indicators are then associated with the AAS in order to make an analysis. • the number of internal attacks is computed as

|{(a i , a j ) ∈ R, a i ∈ c, a j ∈ Rej c}|;
• the number of external attacks is computed as

|{(a i , a j ) ∈ R, a i ∈ A \ c, a j ∈ Rej c}|.
-Indicators concerning the origin of divergent viewpoints:

-Ratio of skeptically versus credulously accepted arguments: |Skept|/|Cred| ∈ [0; 1]; -Number and proportion of strictly credulously accepted arguments per context. For each context c:

• the number of strictly credulously accepted arguments is computed as |{a ∈ SCred c}|;

• the proportion of strictly credulously accepted arguments is computed as |{a ∈ SCred c}|/|c|.

-Number of internal and external attacks towards strictly credulously accepted arguments, per context. For each context c:

• the number of internal attacks towards strictly credulously accepted arguments is computed as |{(a i , a j ) ∈ R, a i ∈ c, a j ∈ SCred c}|; • the proportion of external attacks towards strictly credulously accepted arguments is computed as |{(a i , a j ) ∈ R, a i ∈ A \ c, a j ∈ SCred c}|.

In the following we explain the rationale of these indicators.

Indicators concerning the polemical status of the system.

-Ratio of rejected arguments.

Rejected arguments are those that are attacked and not defended, thus belonging to none of the extensions and defined as not acceptable. Beyond this concept, there is the idea that these arguments are not reliable, since they receive non-countered attacks. However, as illustrated by the case study in the following, these arguments can express first rough ideas or biases that serve as initial pillars which are then criticized, refined and thus rejected afterwards. They can be of great importance for the construction of the argumentation system. The ratio of rejected arguments informs about the extent to which the situation is polemical. Note that rejected arguments are, in a sense, consensual, since all the viewpoints that emerged agree to reject them.

-Number and proportion of rejected arguments per context.

Since a partition is a breakdown of the set of arguments in different contexts (for example, the criteria studied, the decisions supported, the authors of the arguments, etc.), it is important to note that contexts are angles of analysis. They are different from extensions, which bring together coherent arguments, possibly from different contexts. Basic statistics about the repartition of the rejected arguments in the different contexts of a partition (number and proportion of rejected arguments per context) provides information about the contexts that are the more subject to polemic. The higher these indicators, the more polemical the context is, in a given partition. -Number of internal and external attacks towards rejected arguments.

Finally, in a given partition, rejected arguments can be further analyzed by determining the number and the ratio of "internal" attacks, i.e. attacks from arguments of the same context, and "external" attacks, i.e. from arguments of other contexts (and which of them in particular). These indicators inform on the nature of the polemic, whether it stems from similar or from other types of considerations than those of the rejected arguments.

Indicators concerning the origin of divergent viewpoints.

-Ratio of skeptically versus credulously accepted arguments (|Skept|/|Cred| ∈ [0; 1]). Skeptically accepted arguments are those that are accepted in all the extensions, that is to say, in all coherent sets of arguments, which can be interpreted as viewpoints. They represent arguments that, whatever the viewpoint, are not attacked or, if they are attacked, are defended in all viewpoints. They are thus consensual arguments.

Credulously accepted arguments are accepted in some extensions, at least one, but not necessarily in all of them. The ratio of skeptically versus credulously accepted arguments measures the balance between consensual and dividing arguments. The closer to 1, the more consensual the situation. The closer to 0, the more divided the situation. Arguments that are not accepted in all extensions, i.e. credulously but not skeptically accepted, are incompatible with some arguments in the other viewpoints. As stated in Definition 3, we denote these arguments "strictly credulously accepted arguments". They are thus defined as the elements of Cred \ Skept. They correspond to arguments about which a choice had to be made when adopting a viewpoint. These arguments correspond to divergences in the opinions. -Number and proportion of strictly credulously accepted arguments per context.

To understand the origin of divergent viewpoints, strictly credulously accepted arguments are key features, since they constitute the differences between viewpoints. Strictly credulously accepted arguments are in conflict with some arguments of the system, but supported by others. The number and proportion of strictly credulously accepted arguments in each context of a partition provides information about the contexts that are involved in divergences. The higher these indicators, the more divergences about the context, in a given partition. -Number of internal and external attacks towards strictly credulously accepted arguments.

In a given partition, strictly credulously accepted arguments can be further analyzed by determining the number and the ratio of "internal" attacks, i.e. attacks from arguments of the same context, and "external" attacks, i.e. from arguments of other contexts (and which of them in particular). These indicators inform on the nature of discordances, whether they stem from similar or from other types of considerations than those of the strictly credulously accepted arguments.

The use of these indicators is illustrated on the case study in the following section.

Analysis of the device through the analytical argumentation framework

This section proposes an analysis and feedback of the experimental device based on the AAS framework. An analytical argumentation system AAS = (A; R; C1; C2) is instantiated with the following elements:

-A is composed of the arguments from A to Q described in Section 3.2; -R is composed of the attacks indicated in Section 3.2; two partitions are defined:

according to the criteria studied in the device, thus C1 = {economic context, technical context, participative context, social context}; according to the aspects, favorable ('pro') or unfavorable ('con') to the device, of the arguments, thus C2 = {'pro' context, 'con' context}.

In the following, we review the two families of indicators proposed in Definition 5.

Indicators concerning the polemical status of the system.

-Ratio of rejected arguments.

3 arguments out of 17 are rejected. The ratio of rejected arguments is thus 0.18, which means that a minority of arguments (18%) are attacked without being defended. One could think that these are weak arguments that are not receivable. However, looking into the content of these arguments shows that they appeal to common sense. The first two arguments (A and F) express that the device has to be economically viable to be of interest, the third one (M) claims it has to be aware of the market quantities. The three of them recommend a practical view of the market reality, and are far from being nonsense. However, these arguments need to be refined, examined with particular situations showing they could be overcome. That was the role of the experimental device, and the reason why these arguments were rejected. In summary, the system shows a moderate polemic linked to the refinement of initial common-sense arguments about consideration of the market reality.

-Number and proportion of rejected arguments per context.

In order to further understand the sources of polemic and divergences, let us now display the system according to the partition C1 proposed in the analytical argumentation system AAS. Figures 5 and6 show the two preferred extensions partitioned according to C1. Figures 7 and8 show the two preferred extensions partitioned according to C2. The contexts of C1 contain, respectively, 9, 5, 1 and 2 arguments, showing the prevalence of economic and technical motivations. The contexts of C2 contain, respectively, 12 and 5 arguments, highlighting the amount of opinions expressed in support to the device. In C1 = {economic context, technical context, participative context, social context}, the numbers of rejected arguments per context are respectively 2, 1, 0 and 0, and their proportions 22%, 20%, 0% and 0%. We can conclude that the polemic mainly regards economic concerns, and secondarily technical concerns. Indeed, among the rejected arguments, A and F deal with prices, whereas M deals with market quantities.

In C2 = {'pro' context, 'con' context}, the numbers of rejected arguments per context are respectively 1 and 2 and their proportions 8% and 40%. We can see that the "unfavorable context" in C2 concentrates rejected arguments. They represent 40% of this context. This is not surprising, since the initial arguments, which were refined and rejected by the experimental device, were not in favor of the device. The device had to prove it had a chance to work, despite the initial unfavorable arguments.

-Number of internal and external attacks towards rejected arguments. In C1 = {economic context, technical context, participative context, social context}, the numbers of internal attacks (i.e. from the same context) towards rejected arguments (i.e. A, F, M) are respectively 6, 0, 0 and 0, and the numbers of external attacks (i.e. from other contexts) are 4, 1, 0 and 0.

It may be surprising to note that the economic arguments are mainly internally attacked by other economic arguments, which tends to demonstrate that initial reluctances to consider a possible economic viability of the device had to be revised in the light of the practical implementation of the device and specific conditions (e.g. re-duced transportation cost, context of low demand, etc.). External attacks to the economic arguments are quite balanced between the different other contexts (2 from the technical context, 1 from the participative context, 1 from the social context). Their meaning is that economic drawbacks can be counterbalanced by benefits in other concerns (e.g. keeping good business contacts with wholesalers, as expressed by the social argument L). It is worth noticing that the social context, although composed only of 2 arguments, is highly involved in the polemic against rejected arguments, since both social arguments attack a rejected argument. For instance, the technical rejected argument is only attacked by a social one.

In C2 = {'pro' context, 'con' context}, the numbers of internal attacks are respectively 3 and 0, and the numbers of external attacks are 1 and 7.

These figures show that the attacks to the rejected arguments essentially came from arguments favorable to the device, whether or not the rejected arguments were themselves favorable. This is understandable, since the attacks can have different meanings: either a refinement of a 'pro' argument such as A, or a contradiction to a 'con' argument such as F or M.

Indicators concerning the origin of divergent viewpoints.

-Ratio of skeptically versus credulously accepted arguments.

14 arguments are credulously accepted, among which 11 are skeptically accepted. The ratio of skeptically versus credulously accepted arguments is 0.79, which ex-presses a rather consensual debate (79% consensual), although a 21% divergence remains. As mentioned above, this divergence is due to the three arguments that are strictly credulously accepted (J, O and P). The divergence regards two points: (i) whether or not the sale of non-standard size products is a possible advantage, with a divergence between J and O, and (ii) whether the device brings logistic advantage, with a divergence between O and P.

-Number and proportion of strictly credulously accepted arguments per context.

In C1 = {economic context, technical context, participative context, social context}, the numbers of strictly credulously accepted arguments per context are respectively 1, 2, 0 and 0, and their proportions 11%, 40%, 0% and 0%. We can conclude that the technical context plays an important part in divergences. Indeed, among the strictly credulously accepted arguments, O and P are technical and deal with logistical issues, whereas J considers the device as an economic opportunity for non-standard products.

In C2 = {'pro' context, 'con' context}, the numbers of strictly credulously accepted arguments per context are respectively 2 and 1, and their proportions 17% and 20%. These similar proportions express that divergences in viewpoints are balanced between favorable or unfavorable opinions regarding the device.

-Number of internal and external attacks towards strictly credulously accepted arguments.

In C1 = {economic context, technical context, participative context, social context}, the numbers of internal attacks (i.e. from the same context) are respectively 0, 2, 0 and 0, and the numbers of external attacks (i.e. from other contexts) are 1, 0, 0 and 0. These figures provide important information: divergences in viewpoints are all related to technical considerations. Indeed, there are no internal discordances except for the technical context, which has an internal dilemma about the logistical benefit of the device (arguments O and P). Moreover, there is only one external attack directed against the economic argument, and this attack is again coming from the technical context (argument O). Thus the technical argument O appears to be a backbone of the divergences expressed in the system. In C2 = {'pro' context, 'con' context}, the numbers of internal attacks are respectively 0 and 0, and the numbers of external attacks are 2 and 1. Unsurprisingly we can conclude on a dichotomy between favorable and unfavorable arguments, since in this partition there are no internal divergences. All of them stem from the opposite context.

Conclusion

This paper introduced a novel analysis framework based on argumentation, called Analytical Argumentation System, presented its application to evaluate an innovative experimentation of a short food supply chain and analysed the results obtained regarding the controversial aspects of the experimental device. This model uses contexts and defines several indicators to evaluate the system studied.

The analysis led to the following main feedback:

-Rejected arguments are not to be interpreted as non-receivable, but rather as rough ideas that needed to be refined. The idea that rejected arguments are not reliable, is not validated by the study, since they show to be initial pillars of great importance for the construction of the argumentation system. This corroborates results obtained in previous studies, such as [START_REF] Bourguet | An artificial intelligence-based approach to deal with argumentation applied to food quality in a public health policy[END_REF] where the argument supporting a nutritional recommendation by the Ministry of Health was attacked on all sides as a defensive response. On the contrary, in different types of applications such as in big data analysis, social networks, etc., the number and structure of attacks to an argument may be relevant indicators of its strength [START_REF] Amgoud | Measuring the intensity of attacks in argumentation graphs with shapley value[END_REF], which contrasts with the human-interpretable, sparse information context which occurs in the context of the present study. -Unsurprisingly, extensions, that is to say, coherent sets of arguments, can be interpreted as standpoints. -Skeptically accepted arguments are consensual arguments, whereas credulously accepted arguments are dividing arguments. -We introduced the concept of "strictly credulously accepted arguments" to denote arguments which are credulously but not skeptically accepted. These arguments correspond to divergences in the opinions. They are key features to understand the origin of divergent standpoints.

Concerning the case study, the following conclusions can be highlighted:

-Two standpoints can be distinguished in the system.

-Initial prejudices about short supply chains concerned mainly economic aspects, and secondarily technical aspects. -Economic drawbacks are counterbalanced by benefits in other concerns.

-The social dimension is highly involved in the reevaluation of initial ideas.

-The divergences between the two standpoints are all related to technical considerations, and more specifically to one technical argument about the logistical benefit of the device.

Several extensions to this work are considered, in a decision support perspective. One of them consists in taking into account other kinds of relations between arguments, such as specializations, expressing that some of them are particular cases or variations of others, as evoked in Section 3.2 of the paper. The consideration of such relations and their impact on the system are open questions. This paper illustrated how an argument may refine a previous one, in particular by considering more specialized hypotheses. In a complementary way, [START_REF] Bourguet | An artificial intelligence-based approach to deal with argumentation applied to food quality in a public health policy[END_REF] considered the case where decisions may be more specialized than others. As an example, an argument giving a reason to support saltless bread with increased flour yield, decision denoted "F.Y & Saltless", was considered as a specialization of the argument supporting the decision "F.Y.". This had consequences on the attack relation definition and on its symmetricity property. In the presence of specialized decisions, the attack relation is not symmetric: in [START_REF] Bourguet | An artificial intelligence-based approach to deal with argumentation applied to food quality in a public health policy[END_REF] if a decision d 1 is more specialized than a decision d 2 , then arguments supporting d 1 attack arguments supporting d 2 but the reverse attacks do not hold. On these bases, specialization may be exploited in decision support, such as policy making, in several directions: (i) for context-specific purposes, by refining recommended decisions according to the particular conditions considered, e.g. a given segment of the population, and (ii) for innovation, which is another promising use of specialization, through the invention of new options, namely specialized decisions, for mutual gain in a multi-actor context.

Fig. 1

 1 Fig. 1 Graph of arguments and attacks

Fig. 3

 3 Fig. 3 Second preferred extension

Fig. 4

 4 Fig. 4 Argument acceptability

Definition 5 (-

 5 Indicators). Given an Analytical Argumentation System ASS = (A; R; C 1 . . . Cn), let Rej, Skept, Cred and SCred denote respectively the set of rejected, skeptically accepted, credulously accepted and strictly credulously accepted arguments in (A, R) (see Definition 3). The following indicators are computed: Indicators concerning the polemical status of the system: -Ratio of rejected arguments: |Rej|/|A| ∈ [0; 1]; -Number and proportion of rejected arguments per context. For each context c: • the number of rejected arguments is computed as |{a ∈ Rej c}|; • the proportion of rejected arguments is computed as |{a ∈ Rej c}|/|c|. -Number of internal and external attacks towards rejected arguments, per context. For each context c:

Fig. 5

 5 Fig. 5 First preferred extension partitioned according to C1 (the contexts are the criteria of the study)

Fig. 6 Fig. 7 Fig. 8

 678 Fig. 6 Second preferred extension partitioned according to C1 (the contexts are the criteria of the study)

Table 2

 2 Arguments about the interest of the device (second series)

	Arg.	Description	Decision	Criterion
	H	the device was a success even with low prices when it took	pro	economic
		place in a context of low demand, combined with an advantage
		for producers
	I	reduced transportation cost is a possible advantage	pro	economic
	J	the sale of non-standard (e.g. large-size) products is a possible	pro	economic
		advantage
	K	motivation to participate in a solidarity project is a possible	pro	participative
		advantage
	L	keeping good business contacts with the wholesalers is a	pro	social
		possible advantage

Table 3

 3 Arguments about the interest of the device (third series)

Table 4

 4 Arguments about the interest of the device (fourth series)

		good knowledge of the local
		market, leading to a new planning proposition
	O	large-size products brought logistical difficulties to the	con	technical
		volunteers of the distribution centers
	P	local fresh products facilitated the volunteers' work from a	pro	technical
		logistical viewpoint, by avoiding them to sort damaged
		products
	Q	the disposal of products was achieved in reduced time	pro	technical