. Complex spherical codes and linear programming bounds Sho Suda - PowerPoint PPT Presentation

. . Complex spherical codes and linear programming bounds Sho Suda (Aichi University of Education) April 23, 2015 2015 Workshop on Combinatorics and Applications at SJTU Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 1 / 23

Introduction ▶ Spherical designs and codes on complex unit sphere, by Roy and S. (2014) Journal of Combin. Des. with a connection to association schemes. ▶ Ω( d ) : the unit sphere of C d . ▶ For X ⊂ Ω( d ) , define A ( X ) := {⟨ x, y ⟩ | x, y ∈ X, x ̸ = y } . ▶ A complex s -code X is a finite subset of Ω( d ) with | A ( X ) | = s . . What is the upper bounds for complex s -codes? Classify complex spherical codes which attain the upper bound. . Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 2 / 23

Motivation ▶ Spherical designs and codes on real unit sphere, by Delsarte, Goethals and Seidel (1977). ▶ Delsarte, Goethals and Seidel also showed that tight (or close to tight) spherical designs form symmetric association schemes. . Let X be a spherical t -design with degree s satisfying t ≥ 2 s − 2 . Then X with the inner products of points in X determines a Q -polynomial . association scheme To obtain nice commutative association schemes, we want to establish the theory of codes and designs on complex unit sphere. Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 3 / 23

Complex spherical s -codes ▶ Ω( d ) = { z = ( z 1 , . . . , z d ) ∈ C d | z 1 ¯ z 1 + · · · + z d ¯ z d = 1 } : the complex unit sphere. ▶ a complex spherical code X : a finite subset of Ω( d ) ▶ A ( X ) = {⟨ x, y ⟩ | x, y ∈ X, x ̸ = y } : the inner product set of X , where ⟨ x, y ⟩ = ∑ i x i ¯ y i . . Definition . A complex spherical code X is an s -code if | A ( X ) | = s holds. . A fundamental problem is to find upper bounds for complex spherical s -codes. Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 4 / 23

Complex spherical S -codes ▶ Let X be a finite subset in Ω( d ) . ▶ A ( X ) = {⟨ x, y ⟩ | x, y ∈ X, x ̸ = y } . ▶ F ( x ) ∈ C [ x, ¯ x ] is an annihilator polynomial if F ( α ) = 0 for all α ∈ A ( X ) . ▶ S is a finite subset of N × N . . Definition . A complex spherical code X is an S -code if an annihilator polynomial is in x l : ( k, l ) ∈ S} . Span { x k ¯ . . Example . Let X be a 2 -code with A ( X ) = { α, ¯ α } with α ̸∈ R . Then ▶ F ( x ) := ( x − α )( x − ¯ α ) is an annihilator, thus X is a { (2 , 0) , (1 , 0) , (0 , 0) } -code. ▶ F ( x ) := x + ¯ x − α − ¯ α is an annihilator, thus X is a { (1 , 0) , (0 , 1) , (0 , 0) } -code. . Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 5 / 23

Example for complex spherical ▶ A complex projective code L is a set of lines through the origin in C d . ▶ For a complex projective code L , let X be a finite set in Ω( d ) such that the points of X span the lines of L . In the case, we write L = L ( X ) . ▶ An equiangular line set L = L ( X ) is a complex projective code such that there exists 0 ≤ α ∈ R , | ⟨ x, y ⟩ | = α for any distinct x, y ∈ X . . Remark . A complex projective code L = L ( X ) is equiangular iff X is a complex spherical S -code with S = { (0 , 0) , (1 , 1) } . . Recall that a complex spherical code X is an S -code if there exists x l such that F ( α ) = 0 for any α ∈ A ( X ) . ( k,l ) ∈S a k,l x k ¯ F ( x ) = ∑ For S = { (0 , 0) , (1 , 1) } , F ( x ) = a 0 , 0 + a 1 , 1 x ¯ x . Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 6 / 23

Linear programming bounds for complex spherical codes Define Jacobi polynomials g k,l recursively as follows: g 0 , 0 ( x ) = 1 xg k,l ( x ) = a k,l g k +1 ,l ( x ) + b k,l g k − 1 ,l ( x ) , xg k,l ( x ) = a k,l g k,l +1 ( x ) + b k,l g k,l − 1 ( x ) ¯ k +1 d + l − 2 d + k + l − 2 and set g k,l ( x ) = 0 unless ( k, l ) ∈ N 2 . where a k,l = d + k + l , b k,l = . Proposition . Let X be a complex spherical code. Suppose that F ( x ) = ∑ k,l f k,l g k,l ( x ) is a polynomial such that f 0 , 0 > 0 , f k,l ≥ 0 for all k and l , and F ( α ) ≤ 0 for every α ∈ A ( X ) , then | X | ≤ F (1) . f 0 , 0 . Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 7 / 23

Complex spherical designs in Ω( d ) ▶ Hom ( k, l ) : the set of homogeneous polynomials on Ω( d ) of degree k in { z 1 , . . . , z d } and of degree l in { ¯ z 1 , . . . , ¯ z d } . ▶ T ⊆ N 2 is a lower set if ( k, l ) ∈ T then so is ( m, n ) for all 0 ≤ m ≤ k , 0 ≤ n ≤ l . . Definition (Complex spherical designs) . For a lower set T , define a complex spherical T -design in Ω( d ) to be a finite subset in Ω( d ) such that for all ( k, l ) ∈ T and all polynomials f ∈ Hom ( k, l ) , 1 ∫ 1 ∑ f ( x ) dσ ( x ) = f ( x ) | Ω( d ) | | X | Ω( d ) . x ∈ X Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 8 / 23

Link with real spherical designs ▶ Y is a spherical t -design in S d − 1 = { x ∈ R d | | x | = 1 } if for any polynomial f ( x ) with degree at most t , 1 1 ∫ ∑ S d − 1 f ( x ) dσ ( x ) = f ( x ) | S d − 1 | | Y | x ∈ Y ▶ We define a map φ : C d → R 2 d as ( x 1 , . . . , x d ) �→ ( Re ( x 1 ) , Im ( x 1 ) , . . . , Re ( x d ) , Im ( x d )) . . Theorem . Let X be a finite set of Ω( d ) , t be a positive integer, T = { ( k, l ) ∈ N 2 | k + l ≤ t } . The following are equivalent: 1. X is a complex spherical T -design. 2. φ ( X ) is a spherical t -design in S 2 d − 1 . . Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 9 / 23

Bounds on designs and codes ▶ U ∗ U := { ( k + l ′ , k ′ + l ) | ( k, l ) , ( k ′ , l ′ ) ∈ U} . ▶ Harm ( k, l ) = Hom ( k, l ) ∩ ker (∆) , where ∆ = ∑ d ∂ 2 z i . i =1 ∂z i ∂ ¯ ( k + d − 1 )( l + d − 1 ( k + d − 2 )( l + d − 2 ▶ dim ( Harm ( k, l )) = ) ) − . d − 1 d − 1 d − 1 d − 1 . Proposition(Absolute bound) . 1. If X is a U ∗ U -design, then | X | ≥ ∑ ( k,l ) ∈U dim(Harm( k, l )) . 2. If X is an S -code, then | X | ≤ ∑ ( k,l ) ∈S dim(Harm( k, l )) . . ▶ Tight U ∗ U -design: a U ∗ U -design X such that ∑ | X | = dim(Harm( k, l )) . ( k,l ) ∈U ▶ Tight S -code: an S -code X such that ∑ | X | = dim(Harm( k, l )) . ( k,l ) ∈S Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 10 / 23

Equivalence of tightness . Theorem . The following are equivalent: 1. X is an S -code and S ∗ S -design. 2. X is a tight S ∗ S -design. 3. X is a tight S -code. . Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 11 / 23

Commutative association schemes Let X be a finite set and { R 0 , R 1 , . . . , R s } be a set of non-empty subsets of X × X . For 0 ≤ i ≤ s , A i is the (0 , 1) -adjacency matrix of the graph ( X, R i ) ; { 1 if ( x, y ) ∈ R i , A i ( x, y ) = 0 otherwise . . Definition . ( X, { R i } s i =0 ) is a commutative association scheme if the following conditions hold; 1. A 0 = I , 2. ∑ s i =0 A i = J , where J is the all ones matrix, 3. A T i ∈ { A 1 , . . . , A s } for 1 ≤ i ≤ s , 4. for 1 ≤ i, j ≤ s , A i A j is a linear combination of A 0 , A 1 , . . . , A s , 5. A i A j = A j A i for any i, j . . Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 12 / 23

Krein number ▶ A = Span { A i | 0 ≤ i ≤ s } is said to be the Bose-Mesner algebra of an association scheme. ▶ Since A is commutative, there is another basis 1 { E 0 = | X | J, E 1 , . . . , E s } consisting of primitive idempotents. ▶ Since A is closed under the entrywise product ◦ , we define Krein numbers as follows: s 1 ∑ q k E i ◦ E j = i,j E k . | X | k =0 ▶ It is known that Krein numbers q k i,j are nonnegative real numbers. Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 13 / 23

T -design with degree s ▶ Let X be a T -design in Ω( d ) with degree s . ▶ The inner product set A ( X ) = { α 1 , . . . , α s } and α 0 = 1 . ▶ Define R i = { ( x, y ) ∈ X 2 : ⟨ x, y ⟩ = α i } for 0 ≤ i ≤ s . ▶ Let U be a subset T such that U ∗ U ⊂ T . . Theorem . 1. |U| ≤ s + 1 . 2. If s ≤ |U| holds, then ( X, { R i } s i =0 ) forms a commutative association scheme. 3. If |U| = s + 1 holds, then X is a tight U ∗ U design. . Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 14 / 23

Examples satisfying s ≤ |U| ▶ Doubly regular tournaments = nonsymmetric association schemes of 2 -classes. ▶ Skew-Symmetric Hadamard matrices = specific nonsymmetric association schemes of 3 -classes. ▶ Specific SIC-POVMs in C 2 , C 3 . ▶ A SIC-POVM in C d is a finite set X in Ω( d ) such that | X | = d 2 and √ |⟨ x, y ⟩| = 1 / d + 1 for any distinct x, y ∈ X . ▶ A specific complex MUB in C 2 . ▶ Complex MUBs in C d is a set of orthonormal bases in Ω( d ) such that √ for any x, y belonging to different bases, |⟨ x, y ⟩| = 1 / d . Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 15 / 23