. 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 Cd. ▶ 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 ≥ 2s − 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 = (z1, . . . , zd) ∈ Cd | z1 ¯

z1 + · · · + zd ¯ zd = 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 xi ¯

yi. .

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 Span{xk¯ xl : (k, l) ∈ S}. .

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 Cd. ▶ 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 F(x) = ∑

(k,l)∈S ak,lxk¯

xl such that F(α) = 0 for any α ∈ A(X). For S = {(0, 0), (1, 1)}, F(x) = a0,0 + a1,1x¯ 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 gk,l recursively as follows: g0,0(x) = 1 xgk,l(x) = ak,lgk+1,l(x) + bk,lgk−1,l(x), ¯ xgk,l(x) = ak,lgk,l+1(x) + bk,lgk,l−1(x) where ak,l =

k+1 d+k+l, bk,l = d+l−2 d+k+l−2 and set gk,l(x) = 0 unless (k, l) ∈ N2.

.

Proposition

. . Let X be a complex spherical code. Suppose that F(x) = ∑

k,l fk,lgk,l(x)

is a polynomial such that f0,0 > 0, fk,l ≥ 0 for all k and l, and F(α) ≤ 0 for every α ∈ A(X), then |X| ≤ F(1) f0,0 .

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 {z1, . . . , zd} and of degree l in { ¯ z1, . . . , ¯ zd}.

▶ T ⊆ N2 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 |Ω(d)| ∫

Ω(d)

f(x)dσ(x) = 1 |X| ∑

x∈X

f(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 Sd−1 = {x ∈ Rd | |x| = 1} if for any

polynomial f(x) with degree at most t, 1 |Sd−1| ∫

Sd−1 f(x)dσ(x) =

1 |Y | ∑

x∈Y

f(x)

▶ We define a map φ : Cd → R2d as

(x1, . . . , xd) → (Re(x1), Im(x1), . . . , Re(xd), Im(xd)). .

Theorem

. . Let X be a finite set of Ω(d), t be a positive integer, T = {(k, l) ∈ N2 | k + l ≤ t}. The following are equivalent:

• 1. X is a complex spherical T -design.
• 2. φ(X) is a spherical t-design in S2d−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 i=1 ∂2 ∂zi∂ ¯ zi . ▶ dim(Harm(k, l)) =

(k+d−1

d−1

)(l+d−1

d−1

) − (k+d−2

d−1

)(l+d−2

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| = ∑

(k,l)∈U

dim(Harm(k, l)).

▶ Tight S-code: an S-code X such that

|X| = ∑

(k,l)∈S

dim(Harm(k, l)).

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 {R0, R1, . . . , Rs} be a set of non-empty subsets

• f X × X.

For 0 ≤ i ≤ s, Ai is the (0, 1)-adjacency matrix of the graph (X, Ri); Ai(x, y) = { 1 if (x, y) ∈ Ri,

• therwise .

.

Definition

. . (X, {Ri}s

i=0) is a commutative association scheme if the following

conditions hold;

• 1. A0 = I,
• 2. ∑s

i=0 Ai = J, where J is the all ones matrix,

• 3. AT

i ∈ {A1, . . . , As} for 1 ≤ i ≤ s,

• 4. for 1 ≤ i, j ≤ s, AiAj is a linear combination of A0, A1, . . . , As,
• 5. AiAj = AjAi for any i, j.

Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 12 / 23

Krein number

▶ A = Span{Ai | 0 ≤ i ≤ s} is said to be the Bose-Mesner algebra of

an association scheme.

▶ Since A is commutative, there is another basis

{E0 =

1 |X|J, E1, . . . , Es} consisting of primitive idempotents. ▶ Since A is closed under the entrywise product ◦, we define Krein

numbers as follows: Ei ◦ Ej = 1 |X|

s

∑

k=0

qk

i,jEk. ▶ It is known that Krein numbers qk 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) = {α1, . . . , αs} and α0 = 1. ▶ Define Ri = {(x, y) ∈ X2 : ⟨x, y⟩ = αi} 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, {Ri}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 C2, C3.

▶ A SIC-POVM in Cd is a finite set X in Ω(d) such that |X| = d2 and

|⟨x, y⟩| = 1/ √ d + 1 for any distinct x, y ∈ X.

▶ A specific complex MUB in C2.

▶ Complex MUBs in Cd 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

A sufficient condition for a finite set to be association schemes

Define the intersection numbers for x, y ∈ X, 1 ≤ i, j ≤ s as pi,j(x, y) = |{z ∈ X | ⟨x, z⟩ = αi, ⟨z, y⟩ = αj}|. .

Theorem

. . Let X be a T -design with U ∗ U ⊆ T and the inner product set A(X) = {α1, . . . , αs}. Assume there exists I ⊆ {1, 2, . . . , s} such that |I| = |U| and pi,j(x, y) depends only on i, j, x∗y and pi,j(x, y) = pj,i(x, y) for (i, j) ̸∈ I2. If the matrix G = (gk,l(αi))

i∈I (k,l)∈U

is nonsingular, X carries a commutative association scheme.

Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 16 / 23

Examples satisfying the previous sufficient condition

▶ Equiangular tight frames with third root of unity. ▶ A specific SIC-POVM in C8. ▶ Complex MUBs in Cd from Z4-Kerdock codes, d = 2k, odd k. ▶ 240 vectors in C4 (The minimal vectors of the complex E8 lattice). ▶ 756 vectors in C6 (The minimal vectors of the complex Coxeter-Todd

lattice).

▶ 16240 vectors in C28 (The minimal vectors of the lattice in C28 on

which the Rudvalis group acts. The Rudvalis group is one of the sporadic finite simple groups.)

Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 17 / 23

Embedding schemes into complex unit sphere

▶ Fix a primitive idempotent E1 such that ET 1 ̸= E1. ▶ E1 is a positive semidefinite, we regard E1 as the Gram matrix of a

finite set ˜ X in Ω(d), where d = rank(E1).

▶ Eˆ i = ET i .

.

Theorem

. . Let (X, {Ri}s

i=0) be a commutative association scheme, T a lower set.

The following are equivalent;

• 1. ˜

X is a T -design in Ω(d).

• 2. for each (i, j) ∈ T , the following holds;

s

∑

l0,l1,...,li=0 s

∑

h0,h1,...,hj=0

ql0

0,0ql1 1,l0 · · · qli 1,li−1qh0 0,0qh1 1,h0 · · · qhj 1,hj−1q0 li, hj

=   

d2i

(m+i−1

i

) if i = j, 0 if i ̸= j.

Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 18 / 23

Embedding schemes into complex unit sphere

.

Corollary

. . Let (X, {Ri}s

i=0) be a commutative association scheme. Then the

following hold;

• 1. ˜

X is a {(1, 1)}-design.

• 2. ˜

X is a {(2, 0)}-design if and only if ˆ 1 ̸= 1.

• 3. ˜

X is a {(2, 1)}-design if and only if ˆ 1 ̸= 1 and q1

1,1 = 0.

• 4. ˜

X is a {(3, 0)}-design if and only if ˆ 1 ̸= 1 and qˆ

1 1,1 = 0.

In the case symmetric association schemes, Cameron, Goethals and Seidel showed

▶ ˜

X is a 2-design.

▶ ˜

X is a 3-design if and only if q1

1,1 = 0.

Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 19 / 23

Tight complex spherical 2-codes

A complex spherical 2-code is a {(0, 0), (1, 0), (0, 1)}-code. .

Theorem

. . Let X be a complex spherical 2-code in Ω(d). Then the following hold. 1. |X| ≤ { 2d + 1 if d is odd, 2d if d is even.

• 2. For d odd, equality holds if and only if (X, {Ri}2

i=0) is a

nonsymmetric association scheme.

• 3. For d even, equality holds if and only if I + A − AT is a

skew-symmetric Hadamard matrix. For a 2-code X with A(X) = {α, ¯ α}, A is defined to be A(x, y) = { 1 if ⟨x, y⟩ = α,

• therwise .

Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 20 / 23

Tight complex spherical 3-codes

A complex spherical 3-code is a {(0, 0), (1, 0), (0, 1), (1, 1)}-code. .

Theorem

. . Let X be a complex spherical 3-code in Ω(d). Then the following hold.

• 1. Then

|X| ≤ { 4 if d = 1, d2 + 2d if d ≥ 2.

• 2. Equality holds if and only if d = 1, 2.

Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 21 / 23

Future works

▶ Construct complex spherical U ∗ U designs with degree s such that

s ≤ |U|.

▶ Improve the upper bounds for complex spherical 3-codes. ▶ Establish the theory of SDP.

Reference:

• A. Roy and S, Complex spherical designs and codes, J. Combin. Des. 22

(2014), 105-148.

• H. Nozaki, S, Complex spherical codes with two inner products, preprint,

arXiv:1503.01575.

Thank you for your attention!

Sho Suda (Aichi Univ. of Edu.) Complex spherical codes and LP April 23, 2015 22 / 23