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 1
Contents I-Notations II- Construction of the code C h ( X ) III- The study of the code C h ( X ) on X non-degenerate Hermitian surface in P 3 ( F q ) • History of C h ( X ) over Hermitian varieties • Resolution of Sørensen’s conjecture ( h ≤ 2 et t = p a ) and some consequences • Resolution of Sørensen’s conjecture ( h ≥ 3 and t = p a ) IV-The study of the code C 2 ( X ) for X a non-degenerate Hermitian variety in P 4 ( F q ) V- The study of the code C 2 ( X ) for X a non-degenerate Hermitian variety in P 2 l +1 ( F q ) , P 2 l +2 ( F q ) 2
I-Notations • F q : finite field with q elements ( q = p a ). • V = A m +1 the affine space of dimension m + 1 on F q . P m ( F q ):the corresponding projective space of dimension m . • # P m ( F q ) = π m = q m + q m − 1 + ... + q + 1 • F h ( V, F q ): vector space of forms of de- gree h on V with coefficients in F q . • Si f ∈ F h ( V, F q ), Z ( f ): the set of zeros of f in P m ( F q ). • Let X ⊂ P m ( F q ) a variety in P m ( F q ), X Z ( f ) ( F q ): the set of rational points on F q of the algebraic set X ∩ Z ( f ). 3
II- Construction of C h ( X ) • Let X ⊂ P m ( F q ) and N = # X ( F q ) → F N c : F h ( V, F q ) − q → c ( f ) = ( f ( P 1 ) , . . . , f ( P N )) f �− C h ( X ) = Im c • definition Let c ( f ) be a codeword cw ( f ) = # { P ∈ X | f ( P ) = 0 } w ( c ( f )) = # X ( F q ) − cw ( f ) dist C h ( X ) = # X ( F q ) − max cw ( f ) f ∈F h • Proposition The parameters of C h ( X ): length C h ( X ) = # X ( F q ), dim C h ( X ) = dim F h − dim ker c, dist C h ( X ) = # X ( F q ) − max # X Z ( f ) ( F q ) f ∈F h � m + h � If c injective ⇒ dim C h ( X ) = h 4
III-The study of C 2 ( X ) ( X n-degenerate Hermit. surf. in P 3 ( F q ) ) X : x t +1 + x t +1 + x t +1 + x t +1 = 0 0 1 2 3 • 3.1 Number of points of X # X ( F q ) = ( t 2 + 1)( t 3 + 1) , 1966 • 3.2 Injectivity of the application c Tsfasman-Serre-Sørensen Bound ⇒ h ≤ t . • 3.3 History of C h ( X ) h = 2 , t = 2 R. Tobias, 1985 P. Spurr, 1986 h = 2 , t = 2 A. B. Sørensen, 1991 Conjecture: # X Z ( f ) ( F q ) ≤ h ( t 3 + t 2 − t ) + t + 1 G. Lachaud, A.G.C.T-4, 1993 # X Z ( f ) ( F q ) ≤ h ( t 3 + t 2 + t + 1) S. H. Hansen, G. Lachaud, J. B. Little, F. Rodier 5
Weight Distribution of the code C 2 ( X ) over F 4 (i.e. h = 2 , t = 2 ) Complete Computer Search • The code C 2 ( X ) defined over F 4 is a [45 , 10 , 22] 4 -code. And it is a even-weight code. We have the following formula: w i = (10 + i ) × 2 i = 1 , ..., 12 • A w 1 = 2 . 160 , A w 2 = 2 . 970, A w 3 = 4 . 320, A w 4 = 40 . 500, A w 5 = 122 . 976, A w 6 = 233 . 415, A w 7 = 285 . 120, A w 8 = 233 . 400, A w 9 = 97 . 200, A w 10 = 20 . 574, A w 11 = 4 . 320, A w 12 = 1 . 620 6
• 3.4 Resolution of Sørensen’s conjecture ( h ≤ 2 et t = p a ) 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 2 π 2 Π 2 P 0 2 q 2 + π 1 2 pair of distinct planes 2 Π 2 H 1 2 line 1 π 1 Π 1 E 1 3 cone quadric 1 π 2 Π 0 P 2 4 hyperbolic quadric π 2 + q 1 H 3 ( R , R ′ ) 4 elliptic quadric π 2 − q 0 E 3 Some values of # X Z ( f ) ( F q ) s ( t ) = 2 t 3 + 2 t 2 − t + 1, s 2 ( t ) = 2 t 3 + t 2 + 1, s 3 ( t ) = 2 t 3 + t 2 − t + 1, s 4 ( t ) = 2 t 3 + 1, s 5 ( t ) = 2 t 3 − t + 1 7
a. Q is a pair of planes: Q = H 1 ∪ H 2 X 1 = H 1 ∩ X , ˆ ˆ X 2 = H 2 ∩ X et L = H 1 ∩ H 2 |Q ∩ X| = | H 1 ∩ X| + | H 2 ∩ X| − |L ∩ X| . (1) P ∩ X = L ∩ ˆ X 1 = L ∩ ˆ X 2 . (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 P n ( F q ) and Π r − 1 a linear projective space of dimension r − 1 disjoint from the singu- ˜ Then Π r − 1 ∩ ˜ lar space Π n − r of X . X is a non-degenerate Hermitian variety in Π r − 1 . 8
b. Q is an elliptic quadric. Table 3: Plane Hermitian curves. r( V ) Description |V| g( V ) t 2 + 1 1 repeated line 1 Π 1 U 0 t 3 + t 2 + 1 2 cone 1 Π 0 U 1 t 3 + 1 3 non-sing. Herm. curve 0 U 2 Table 4: Plane Quadrics. r( V ) Description |V| g( V ) 1 repeated line q + 1 1 Π 1 P 0 2 cone 2 q + 1 1 Π 0 H 1 2 point 1 0 Π 0 E 1 3 conic q + 1 0 P 2 9
Rank Type # X Z ( Q ) ( F q ) F 4 w i F q t 3 + t 2 + 1 t 5 1 1 13 t 3 + 1 t 5 + t 2 (plane) 2 9 t 5 + t 3 + t 2 2 3 1 1 t 5 + t 3 + t 2 − t (line) 4 t + 1 3 t 2 + 1 t 5 + t 3 5 5 t 5 − t 3 + t 2 6 s 4 ( t ) 17 t 5 − t 3 + t 2 + t 2 s 5 ( t ) 15 t 5 − t 3 + t (pair of s 3 ( t ) 19 t 5 − t 3 planes) 7 s 2 ( t ) 21 t 5 − t 3 − t 2 + t s ( t ) 23 t 5 − t 3 8 s 2 ( t ) 21 ≤ t 3 + t 2 + t ≥ t 5 − t 3 9 ≤ 15 +1 < s 4 ( t ) t 3 + t 2 + 1 t 5 (cone) 10 13 t 3 + 2 t 2 − t + 1 t 5 − t 2 + t 15 t 5 − t 3 11 s 2 ( t ) 21 ≤ t 3 + 3 t 2 − t ≥ t 5 − t 3 4 12 ≤ 19 +1 ≤ s 3 ( t ) ≤ t 3 + 2 t 2 ≥ t 5 − t 2 (hyper.) 13 ≤ 17 H ( R , R ′ ) +1 ≤ s 4 ( t ) ≤ t 3 + t 2 + ≥ t 5 − t 14 ≤ 15 t + 1 < s 4 ( t ) ≤ 2 t 3 + 2 t ≥ t 5 − t 3 + t 2 4 (ellip.) 15 +2 < s 2 ( t ) ≤ 17 − 2 t − 1 10
Weight Distribution ( w i , A w i ) of C 2 ( X ) ( F t 2 ) • w 1 = t 5 − t 3 − t 2 + t The codewords << w 1 >> : union of 2 tan planes to X and l ∩ X = ( t + 1) points. 2 ( t 5 + t 3 + t 2 + 1) t 5 ] A w 1 = ( t 2 − 1)[ 1 w 2 = t 5 − t 3 • The codewords << w 2 >> 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. A w 2 = ( t 2 − 1)[ 1 2 ( t 5 + t 3 + t 2 + 1)(3 t 2 − t + 1) t 2 ] 11
• w 3 = t 5 − t 3 + t The codewords << w 3 >> : quadrics which are union of 2 planes one tan, the second non-tan to X and l ∩ X = ( t + 1) points. A w 3 = ( t 2 − 1)( t 5 + t 3 + t 2 + 1)( t 6 − t 5 ) Conjecture on w 4 and w 5 • w 4 = t 5 − t 3 + t 2 The codewords << w 4 >> are given by quadrics which are union of 2 planes tan to X and l ∩ X = 1 point and particular elliptic quadrics. • w 5 = t 5 − t 3 + t 2 + t The codewords << w 5 >> :union of 2 planes non-tan to X and l ∩ X = ( t + 1) pts, and particular elliptic quadrics. 12
F. A. B. Edoukou , Codes defined by forms of 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 of 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 f i ( x 1 , ..., x n ) and deg ( f i ) = d i on F q then: if n > b � r i =1 d i ⇒ q b | # Z ( f 1 , ..., f n ). Theorem All the weights w i are divisible by t . Consequence : The conjecture on w 4 and w 5 is true. 13
• 3.5 Resolution of Sørensen’s conjecture ( h ≥ 3 and t = p a ) • A) There is no line in Hermitian surface ∩ hypersurface of degree h . # X Z ( f ) ( F q ) ≤ h ( t 3 + t 2 − t ) + h + 2 t ( h − t ) (E.) # X Z ( f ) ( F q ) ≤ h ( t 3 + t 2 − 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 × 10 18 Tests ( F 9 ) Singular (Univ. of Kaiserlautern, July 2008) Magma (IML, Luminy, October 2008) Sage ??? • B-2) Surface of degree h > 3 ??? 14
IV- The study of the code C 2 ( X ) for X a non-degenerate Hermitian variety in P 4 ( F q ) • P 4 ( F q ): F. A. B. Edoukou,Codes defined by forms of degree 2 on non-degenerate Her- mitian varieties in P 4 ( F q ). To appear in DCC 2008. Poids Q P ∩ X w i t 7 − t 5 − t 3 − t 2 1 2 n-tan H n-sin. Herm t 7 − t 5 − t 3 2 2 n-tan sin. Herm (r=2) t 7 − t 5 − t 2 3 1t+1n-tan n-sin Herm 4 1t+1n-tan sin. Herm (r=2) t 7 − t 5 2tan line t 7 − t 5 + t 3 − t 2 5 2 tan n-sin. Herm • Conjecture: # X Z ( f ) ( F q ) ≤ h ( t 5 + t 2 )+ t 3 +1 . For h ≤ t The min weight codewords correspond to: –hypers. reaching the Tsfasman-Serre-Sørensen’s upper bound. –each hyperplane H i 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. 15
V-Generalisation: The study of the code C 2 ( X ) for X a non-degenerate Hermitian variety in P 2 l +1 ( F q ) , P 2 l +2 ( F q ) • P 2 l +1 ( F q ), P 2 l +2 ( F q ): 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. • P 2 l +1 ( F q ), P 2 l +2 ( F q ): F. A. B. Edoukou, A. Hallez, F. Rodier and L. Storme, On the small weight codewords of the functional codes C h ( X ), X a non-singular Hermitian variety. In preparation. F. A. B. Edoukou , Codes defined by forms of degree 2 on quadric Surfaces. I.E.E.E Trans. Inf. Theo., Vo. 54, Issue 2, Pages 860-864, (2008) 16
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