Codes correcteurs derreurs sur des surfaces Hermitiennes Fr ed - - PDF document

Codes correcteurs d’erreurs sur des surfaces Hermitiennes

Fr´ ed´ eric Aka-Bil´ e EDOUKOU e.mail: edoukou@iml.univ-mrs.fr Institut de Math´ ematiques de Luminy Marseille, France JNCF 2008: Journ´ ees Nationales de Calcul Formel Lundi 20 Octobre 2008 CIRM, Luminy, Marseille, France

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Contents

I-Notations II- Construction of the code Ch(X) III- The study of the code Ch(X) on X non-degenerate Hermitian surface in P3(Fq)

• History of Ch(X) over Hermitian varieties
• Resolution of Sørensen’s conjecture (h ≤ 2

et t = pa) and some consequences

• Resolution of Sørensen’s conjecture

(h ≥ 3 and t = pa) IV-The study of the code C2(X) for X a non-degenerate Hermitian variety in P4(Fq) V- The study of the code C2(X) for X a non-degenerate Hermitian variety in P2l+1(Fq),

P2l+2(Fq)

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I-Notations

• Fq: finite field with q elements (q = pa).
• V = Am+1 the affine space of dimension

m + 1 on Fq.

Pm(Fq):the corresponding projective space

• f dimension m.
• #Pm(Fq) = πm = qm + qm−1 + ... + q + 1
• Fh(V, Fq):

vector space of forms of de- gree h on V with coefficients in Fq.

• Si f ∈ Fh(V, Fq),

Z(f): the set of zeros of f in Pm(Fq).

• Let X ⊂ Pm(Fq) a variety in Pm(Fq),

XZ(f)(Fq): the set of rational points on

Fq of the algebraic set X ∩ Z(f).

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II- Construction of Ch(X)

• Let X ⊂ Pm(Fq) and N = #X(Fq)

c : Fh(V, Fq) − → FN

q

f − → c(f) = (f(P1), . . . , f(PN)) Ch(X) = Imc

• definition Let c(f) be a codeword

cw(f) = #{P ∈ X | f(P) = 0} w(c(f)) = #X(Fq) − cw(f) distCh(X) = #X(Fq) − max

f∈Fh

cw(f)

• Proposition The parameters of Ch(X):

length Ch(X) = #X(Fq), dim Ch(X) = dim Fh − dim ker c, distCh(X) = #X(Fq) − max

f∈Fh

#XZ(f)(Fq) If c injective ⇒ dim Ch(X) =

m+h

h

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III-The study of C2(X) ( X n-degenerate Hermit. surf. in

P3(Fq))

X : xt+1 + xt+1

1

+ xt+1

2

+ xt+1

3

= 0

• 3.1 Number of points of X

#X(Fq) = (t2 + 1)(t3 + 1) , 1966

• 3.2 Injectivity of the application c

Tsfasman-Serre-Sørensen Bound ⇒ h ≤ t .

• 3.3 History of Ch(X)

h = 2, t = 2

• R. Tobias, 1985
• P. Spurr, 1986

h = 2, t = 2 A. B. Sørensen, 1991 Conjecture: #XZ(f)(Fq) ≤ h(t3 + t2 − t) + t + 1

• G. Lachaud, A.G.C.T-4, 1993

#XZ(f)(Fq) ≤ h(t3 + t2 + t + 1)

• S. H. Hansen, G. Lachaud, J. B. Little, F.

Rodier

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Weight Distribution of the code C2(X) over F4 (i.e. h = 2, t = 2) Complete Computer Search

• The code C2(X) defined over F4 is a

[45, 10, 22]4-code. And it is a even-weight code. We have the following formula: wi = (10 + i) × 2 i = 1, ..., 12

• Aw1 = 2.160 , Aw2 = 2.970, Aw3 = 4.320,

Aw4 = 40.500, Aw5 = 122.976, Aw6 = 233.415, Aw7 = 285.120, Aw8 = 233.400, Aw9 = 97.200, Aw10 = 20.574, Aw11 = 4.320, Aw12 = 1.620

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• 3.4 Resolution of Sørensen’s conjecture (h ≤

2 et t = pa) and some consequences h = 1 : Bose and Chark., 1966 Chark., 1971 h = 2 Table 1: Quadrics in PG(3,q). r(Q) Description |Q| g(Q) 1 repeated plane π2 2 Π2P0 2 pair of distinct planes 2q2 + π1 2 Π2H1 2 line π1 1 Π1E1 3 cone quadric π2 1 Π0P2 4 hyperbolic quadric π2 + q 1 H3(R, R′) 4 elliptic quadric π2 − q E3 Some values of #XZ(f)(Fq) s(t) = 2t3 + 2t2 − t + 1, s2(t) = 2t3 + t2 + 1, s3(t) = 2t3 + t2 − t + 1, s4(t) = 2t3 + 1, s5(t) = 2t3 − t + 1

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• a. Q is a pair of planes: Q = H1 ∪ H2

ˆ X1 = H1 ∩ X, ˆ X2 = H2 ∩ X et L = H1 ∩ H2 |Q ∩ X| = |H1 ∩ X| + |H2 ∩ X| − |L ∩ X|. (1) P ∩ X = L ∩ ˆ X1 = L ∩ ˆ X2. (2) a.1 Two tan planes to Q a.2 One tan and the second n-tan to Q a.3 Two n-tan planes to Q Theorem Bose-Chakravarti,1966 Let ˜ X be a degenerate Hermitian variety of rank r < n+1 in Pn(Fq) and Πr−1 a linear projective space

• f dimension r − 1 disjoint from the singu-

lar space Πn−r of ˜ X. Then Πr−1 ∩ ˜ X is a non-degenerate Hermitian variety in Πr−1.

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• b. Q is an elliptic quadric.

Table 3: Plane Hermitian curves. r(V) Description |V| g(V) 1 repeated line t2 + 1 1 Π1U0 2 cone t3 + t2 + 1 1 Π0U1 3 non-sing. Herm. curve t3 + 1 U2 Table 4: Plane Quadrics. r(V) Description |V| g(V) 1 repeated line q + 1 1 Π1P0 2 cone 2q + 1 1 Π0H1 2 point 1 Π0E1 3 conic q + 1 P2

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Rank Type #XZ(Q)(Fq) F4 wi Fq 1 1 t3 + t2 + 1 13 t5 (plane) 2 t3 + 1 9 t5 + t2 2 3 1 1 t5 + t3 + t2 (line) 4 t + 1 3 t5 + t3 + t2 − t 5 t2 + 1 5 t5 + t3 6 s4(t) 17 t5 − t3 + t2 2 s5(t) 15 t5 − t3 + t2 + t (pair of s3(t) 19 t5 − t3 + t planes) 7 s2(t) 21 t5 − t3 s(t) 23 t5 − t3 − t2 + t 8 s2(t) 21 t5 − t3 3 9 ≤ t3 + t2 + t ≤ 15 ≥ t5 − t +1 < s4(t) (cone) 10 t3 + t2 + 1 13 t5 t3 + 2t2 − t + 1 15 t5 − t2 + t 11 s2(t) 21 t5 − t3 4 12 ≤ t3 + 3t2 − t ≤ 19 ≥ t5 − t3 +1 ≤ s3(t) (hyper.) 13 ≤ t3 + 2t2 ≤ 17 ≥ t5 − t2 H(R, R′) +1 ≤ s4(t) 14 ≤ t3 + t2+ ≤ 15 ≥ t5 − t t + 1 < s4(t) 4 ≤ 2t3 + 2t ≥ t5 − t3 + t2 (ellip.) 15 +2 < s2(t) ≤ 17 −2t − 1

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Weight Distribution (wi, Awi) of C2(X) (Ft2)

• w1 = t5 − t3 − t2 + t

The codewords << w1 >>: union of 2 tan planes to X and l ∩ X = (t + 1) points. Aw1 = (t2 − 1)[1

2(t5 + t3 + t2 + 1)t5]

• w2 = t5 − t3

The codewords << w2 >> are given by: –hyperbolic containing lll of X. –union of 2 planes tan of X and l ⊂ X. –union of 2 planes one tan, the second n-tan to X and l ∩ X = 1 point. Aw2 = (t2 − 1)[1

2(t5 + t3 + t2 + 1)(3t2 − t + 1)t2]

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• w3 = t5 − t3 + t

The codewords << w3 >>: quadrics which are union of 2 planes one tan, the second non-tan to X and l ∩ X = (t + 1) points. Aw3 = (t2 − 1)(t5 + t3 + t2 + 1)(t6 − t5) Conjecture on w4 and w5

• w4 = t5 − t3 + t2

The codewords << w4 >> are given by quadrics which are union of 2 planes tan to X and l ∩ X = 1 point and particular elliptic quadrics.

• w5 = t5 − t3 + t2 + t

The codewords << w5 >>:union of 2 planes non-tan to X and l∩X = (t + 1) pts, and particular elliptic quadrics.

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• F. A. B. Edoukou, Codes defined by forms
• f degree 2 on Hermitian Surface and Sørensen’s
• conjecture. Finite Fields and Their Applica-

tions, Volume 13, Issue 3, (2007), 616-627.

• F. A. B. Edoukou,The Weight distribution
• f the functional codes defined by forms of

degree 2. To appear in J.T.N.B 2008.

Divisibility by t of the weights ???

Theorem of Ax (1964) Let r polynomials fi(x1, ..., xn) and deg (fi) = di on Fq then: if n > b r

i=1 di ⇒ qb|#Z(f1, ..., fn).

Theorem All the weights wi are divisible by t. Consequence: The conjecture on w4 and w5 is true.

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• 3.5 Resolution of Sørensen’s conjecture

(h ≥ 3 and t = pa)

• A) There is no line in Hermitian surface

∩ hypersurface of degree h. #XZ(f)(Fq) ≤ h(t3 + t2 − t) + h + 2t(h − t) (E.) #XZ(f)(Fq) ≤ h(t3 + t2 − t) + t + 1 (Sørensen)

• B) There is a line in Hermitian surface ∩

hypersurface of degree h ???

• B-1) Cubic Surface

1,519.708.182.382.116 ×1018 Tests (F9) Singular (Univ. of Kaiserlautern, July 2008) Magma (IML, Luminy, October 2008) Sage ???

• B-2) Surface of degree h > 3 ???

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IV- The study of the code C2(X) for X a non-degenerate Hermitian variety in P4(Fq)

• P4(Fq):
• F. A. B. Edoukou,Codes defined

by forms of degree 2 on non-degenerate Her- mitian varieties in P4(Fq). To appear in DCC 2008. Poids Q P ∩ X wi 1 2 n-tan H n-sin. Herm t7 − t5 − t3 − t2 2 2 n-tan

• sin. Herm (r=2)

t7 − t5 − t3 3 1t+1n-tan n-sin Herm t7 − t5 − t2 4 1t+1n-tan

• sin. Herm (r=2)

2tan line t7 − t5 5 2 tan n-sin. Herm t7 − t5 + t3 − t2

• Conjecture:

For h ≤ t #XZ(f)(Fq) ≤ h(t5+t2)+t3+1. The min weight codewords correspond to: –hypers. reaching the Tsfasman-Serre-Sørensen’s upper bound. –each hyperplane Hi is non-tangent to X –and the plane P of intersection of the h hy- perplanes intersecting X at a non-sing Herm plane curve.

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V-Generalisation: The study of the code C2(X) for X a non-degenerate Hermitian variety in P2l+1(Fq), P2l+2(Fq)

• P2l+1(Fq), P2l+2(Fq): F. A. B. Edoukou, A.

Hallez, F. Rodier and L. Storme, Codes de- fined by forms of degree 2 on non-degenerate Hermitian varieties. In preparation.

• P2l+1(Fq), P2l+2(Fq):
• F. A. B. Edoukou,
• A. Hallez, F. Rodier and L. Storme, On the

small weight codewords of the functional codes Ch(X), X a non-singular Hermitian variety. In preparation.

• F. A. B. Edoukou, Codes defined by forms
• f degree 2 on quadric Surfaces.

I.E.E.E Trans. Inf. Theo., Vo. 54, Issue 2, Pages 860-864, (2008)

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