Some constructions of maximal witness codes Bertrand M EYER Tlcom - PowerPoint PPT Presentation

Some constructions of maximal witness codes Bertrand M EYER Télécom ParisTech Aachen, August 25th 2011 B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 1 / 24

Introduction Progression Introduction 1 Some constructions 2 Semidefinite programming bounds. 3 B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 2 / 24

Introduction Witness codes Definition A code C is a subset of F n 2 . A witness of a codeword c is a set of coordinates W ⊆ [ n ] such that on the restriction ∀ c ′ � = c ∈ C , c | W � = c ′ | W . A w-witness code C is a code with witnesses of length � w. Example : A 2-witness code of length 6 with 4 codewords   1 0 0 1 0 1 ,       0 0 0 1 0 0 ,   C = . 1 1 0 1 1 0 ,       0 0 1 0 1 1   B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 3 / 24

Introduction State of the art Our question What is f ( n , w ) = max |C| ? where C is a w -witness code of length n . Example : The Hamming sphere S n ( w ) of radius w is a w -witness code. So � n � � f ( n , w ) . w Theorem (Cohen, Randriam, Zémor, 08) � n � For any fixed w, the sequence n �→ f ( n , w ) / is decreasing, w B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 4 / 24

Some constructions Progression Introduction 1 Some constructions 2 Semidefinite programming bounds. 3 B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 5 / 24

Some constructions The double sphere For n = 2 w − 1, take C 0 = S n ( w − 1 ) ∪ S n ( w ) . Witnesses : support or co-support. Proposition � 2 w � � f ( 2 w − 1 , w ) w By extension : take C = { cx ; c ∈ C 0 , x ∈ F ℓ 2 } Proposition � 2 ( n − w + 1 ) � ∀ w > n / 2 , � f ( n , w ) n − w + 1 B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 6 / 24

Some constructions A construction for n = 2 w Codewords of C are Type Codeword Witness A 111 x , with x ∈ S n − 3 ( w − 2 ) { 1 , 2 } ∪ Supp ( x ) B 011 x , with x ∈ S n − 3 ( w − 2 ) { 1 , 3 } ∪ Supp ( x ) C 010 x , with x ∈ S n − 3 ( w − 1 ) { 2 } ∪ Supp ( x ) D 101 x , with x ∈ S n − 3 ( w − 1 ) { 3 } ∪ Supp ( x ) E 100 x , with x ∈ S n − 3 ( w ) Supp ( x ) Proposition � 2 w � � 2 w − 2 � ∀ w � 2 , + � f ( 2 w , w ) w w − 1 B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 7 / 24

Some constructions Translates of classical codes Let C 0 ⊆ F n 2 a code with Hamming minimal distance d � 2 ( n − w ) + 1 and C = { c 0 + x ; c 0 ∈ C 0 , x ∈ S n ( n − w ) } . Witness of c 0 + x ∈ C : complement of Supp x . Proposition � n � ∀ w > ( n + 1 ) / 2 , A 2 ( n , 2 ( n − w ) + 1 ) � f ( n , w ) , w where A 2 ( n , d ) is the largest size of a length n code with minimal distance d. Example : Let r � 3. With C 0 the [ 2 r − 1 , 2 r − r − 1 , 3 ] 2 Hamming code, we get ( 2 r − 1 ) 2 2 r − r − 1 � f ( 2 r − 1 , 2 r − 2 ) . With C 0 = extended Golay code, we get 8290304 � f ( 24 , 21 ) . B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 8 / 24

Semidefinite programming bounds. Progression Introduction 1 Some constructions 2 Semidefinite programming bounds. 3 B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 9 / 24

Semidefinite programming bounds. f ( n , w ) as an independance number � [ n ] Let Γ = ( V , E ) be the proximity graph with edges V = F n � 2 × and w vertices ( x , W ) et ( x ′ , W ′ ) if x | W = x ′ | W or x | W ′ = x ′ | W ′ . f ( n , w ) = α (Γ) (independance number) B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 10 / 24

Semidefinite programming bounds. A semidefinite program through Lovász ϑ number ϑ ′ (Γ) = X ∈ R V × V � J , X � max  � I , X � = 1  X ( v , v ′ ) if ( v , v ′ ) ∈ E  = 0  such that X 0 �   X 0  � where R is the real field, I the identity matrix, J the all-one matrix, X � 0 means X is positive semidefinite. Proposition f ( n , w ) = α (Γ) � ϑ ′ (Γ) B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 11 / 24

Semidefinite programming bounds. Symmetry of the program The graph Γ is invariant under the hyperoctaedral group H = F n 2 ⋊ S n (translations and permutations). So we can consider only H -invariant matrices X in the semidefinite program. Entries in the matrix X depend only on � | W ∩ W ′ | , d W ∩ W ′ ( x , x ′ ) , d = ( ι, d i , d l , d r , d t ) = � d W � W ∩ W ′ ( x , x ′ ) , d W ′ � W ∩ W ′ ( x , x ′ ) , d [ n ] � W ∪ W ′ ( x , x ′ ) where d A the Hamming distance on A ⊆ [ n ] . B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 12 / 24

Semidefinite programming bounds. Characterisation of positive semidefinite matrix Theorem (Bochner) There exists a basis of matrix valued functions d �→ Z k ( d ) called zonal functions such that X � 0 is equivalent to � ∀ d ∈ D , X ( d ) = � Φ k , Z k ( d ) � , k where the (Φ k ) k are � 0 matrix. “A semidefinite positive kernel has � 0 Fourier coefficients and vice versa.” B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 13 / 24

Semidefinite programming bounds. Theoretic expression of the zonal functions Let H k be a familly of representatives of the irreducible representations of H and d k their dimensions. Let C V = � � m k j = 1 H k , j be a decomposition of C V into irreducible k subrepresentations with multiplicity denoted m k . Let e k , j ,ℓ be the copy of an orthonormal basis of H k dans H k , j . Then the zonal function Z k is the m k × m k -matrix-valued function defined over V × V by d k � Z k ( v , v ′ ) � � e k , j 1 ,ℓ ( v ) e k , j 2 ,ℓ ( v ′ ) . j 1 , j 2 = ℓ = 1 B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 14 / 24

Semidefinite programming bounds. Decomposition into irreducible’s of C V under H = F n 2 ⋊ S n For I ⊆ [ n ] , we denote 2 �→ ( − 1 ) wgt ( x I ) ∈ C . χ I := x ∈ F n We denote S s [ i ] the irreducible representation (Specht module) of S [ i ] associated with the partition i = ( i − s ) + s . We recall that C ( [ i ] j ) = � i j = 0 S s [ i ] , j . Then we get the irreducible representation H � �� T j χ [ i ] ⊗ S s [ i ] , j ⊗ S t i , s , t = 2 ⋊ ( S [ i ] × S [ i + 1 , n ] ) .  [ i + 1 , n ] , w − j  F n and the decomposition can be written as min ( w , i ) n C V = T j � � � i , s , t . i = 0 j = max ( 0 , w + i − n ) 0 � s � min ( j , i − j ) , 0 � t � min ( j , n − i − j ) B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 15 / 24

Semidefinite programming bounds. Computation of the zonal functions We deduce Z k ( x , W ) , ( x ′ , W ′ ) � � j 1 , j 2 ( − 1 ) wgt ( x | I − x ′ i , j 1 , j 2 ( | W ∩ W ′ ∩ I | ) Hahn t j 2 ( | W ∩ W ′ ∩ I c | ) | I ) Hahn s � = i , ˘ ˘ j 1 , ˘ I ∈ ( [ n ] w ) | W ∩ I | = j 1 , | W ′ ∩ I | = j 2 where ˘ i = n − i , ˘ j 1 = w − j 1 , ˘ j 2 = w − j 2 . B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 16 / 24

Semidefinite programming bounds. Computation of the zonal functions Zonal functions are naturally orthogonal as the subrepresentations are. We just need to focus on their dominant term. We expand the previous expression and compare with the expansion of the suggested monomials in the basis of the functions χ i � 1 if x k = 1 χ k ( x ) = 0 if x k = 0 and their product. B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 17 / 24

Semidefinite programming bounds. Zonal functions In our case : Proposition The zonal functions Z ( i , s , t ) , j 1 , j 2 can be computed by a Gram-Schmidt process on the monomials � ι t d t α d l β d r γ d i δ � δ � . s α ! β ! γ ! δ ! α + β + γ + δ = i , β + δ = j 1 ,γ + δ = j 2 ordered by lex order on ( i , t , s , j 1 , j 2 ) . B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 18 / 24

Semidefinite programming bounds. A new equivalent program � ϑ ′ (Γ) = max y d y ∈ R |D| d ∈D  y ( w , 0 , 0 , 0 , 0 ) = 1   y d = 0 if d ∈ D o  such that y d 0 for any d ∈ D �   � d y d Z i , s , t ( d ) 0 for any ( i , s , t )  � Linear number of variables in n . B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 19 / 24

Semidefinite programming bounds. Numerical results (1) n w = 2 w = 3 w = 4 w = 5 w = 6 w = 7 3 6.00 8.00 6 b 8 b 4 8.00 12.56 16.00 8 c 12 b 16 b 5 10.56 20.00 26.15 32.00 10 a 20 b 25 e 32 b 6 15.00 28.03 42.66 54.35 64.00 15 a 26 c 40 b 52 e 64 b 7 21.00 39.31 70.00 90.30 112.00 128.00 21 a 35 a 70 b 80 b 112 d 128 b 8 28.00 56.65 102.15 150.61 189.27 226.86 28 a 56 a 90 c 140 b 160 b 224 b 9 36.00 84.00 149.05 252.00 321.31 393.34 36 a 84 a 126 a 252 b 280 b 320 b B. Meyer (Télécom Paris) Witness codes Aachen, 25.IX.2011 20 / 24