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Current progress in the Delsarte theory
Hajime Tanaka
Research Center for Pure and Applied MathematicsGraduate School of Information Sciences
Tohoku University
November 24, 2018International Workshop on Algebraic Combinatorics
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The Delsarte Theory (1973)
Philippe Delsarte,An algebraic approach to the association schemes of coding theory,Philips Res. Rep. Suppl. No. 10 (1973).
Warninga very quick survey!!not at all comprehensive!!
Story in a nutshellLP −→ SDPadjacency algebra −→ Terwilliger algebrasingle code/design −→ two (or more) codes/designs
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Notation
X = (X, {Ri}di=0) : a symmetric association schemeA0, . . . , Ad : the adjacency matricesA = ⟨A0, . . . , Ad⟩ : the adjacency algebra (over C)E0, . . . , Ed : the primitive idempotents of AP,Q : the first and second eigenmatrices, i.e.,
Ai =
d∑j=0
Pj,iEj , Ei =1
|X|
d∑j=0
Qj,iAj
ki : the valency (or degree) of (X,Ri) (0 ⩽ i ⩽ d)
mi = rankEi = traceEi (0 ⩽ i ⩽ d)
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D-codes
D ⊂ {1, 2, . . . , d}C ⊂ X : a D-codedef⇐⇒ C : an independent set of the graph
(X,
∪i∈D Ri
)⇐⇒ (C × C) ∩Ri = ∅ (i ∈ D)
ExampleX = H(d, q) : the Hamming schemeX = Fd
q
C : e-error correcting ⇐⇒ C : a {1, . . . , 2e}-code
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D-codes
χ = χC ∈ RX : the (column) characteristic vector of C, i.e.,
χx =
{1 if x ∈ C
0 if x ∈ C(x ∈ X)
χTAiχ =∣∣(C × C) ∩Ri
∣∣C : a D-code ⇐⇒ χTAiχ = 0 (i ∈ D)
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D-designs
D ⊂ {1, 2, . . . , d}
C ⊂ X : a D-design def⇐⇒ χTEiχ = 0 (i ∈ D)
Theorem (Delsarte, 1973)X = J(v, d) : the Johnson schemeC : a t-(v, d, λ) design (for some λ) ⇐⇒ C : a {1, . . . , t}-design
Theorem (Delsarte, 1973)X = H(d, q) : the Hamming schemeC : an orthogonal array of strength t ⇐⇒ C : a {1, . . . , t}-design
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The LP bound for D-codes
C ⊂ X : a D-code
ei :=χTAiχ
|C|(0 ⩽ i ⩽ d)
e = (e0, . . . , ed) : the inner distribution of C
ei = 0 (i ∈ D)
e0 = 1
∵ A0 = I
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The LP bound for D-codes
View the ei as real variables !!
Theorem (Delsarte, 1973)Consider the following LP problem:
|C|
maximize ϑ′ =
d∑i=0
ei
subject to e0 = 1
ei = 0 for i ∈ D
ei ⩾ 0 for i ∈ {1, 2, . . . , d}\D(eQ)i ⩾ 0 for i ∈ {1, 2, . . . , d}
χTEiχ ⩾ 0
If C is a D-code, then |C| ⩽ ϑ′.
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Codes in P -polynomial schemes
X : P -polynomialdef⇐⇒ ∃v0(ξ), . . . , vd(ξ) ∈ R[ξ], ∃θ0, . . . , θd ∈ R s.t.
deg vi(ξ) = i (0 ⩽ i ⩽ d)
Pj,i = vi(θj) (0 ⩽ i, j ⩽ d)
Suppose X is P -polynomial.δ := min{i = 0 : χTAiχ = 0} : the minimum distance of Cs∗ :=
∣∣{i = 0 : χTEiχ = 0}∣∣ : the dual degree of C
Theorem (Delsarte, 1973)δ ⩽ 2s∗ + 1
If δ ⩾ 2s∗ − 1, then C is completely regular.
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Codes in P -polynomial schemes
Recall δ ⩽ 2s∗ + 1.
|C| ⩽ |X|k0 + · · ·+ k⌊ δ−1
2⌋
: the sphere-packing boundthe volume of a ball of radius ⌊ δ−1
2⌋
C : perfect def⇐⇒ “=” holds above
Lj(ξ) := v0(ξ) + · · ·+ vj(ξ) : the Lloyd polynomial of degree j
Theorem (Delsarte, 1973)δ = 2s∗+1 ⇐⇒ C : perfect =⇒ Ls∗(ξ) has s∗ simple roots
in {θ0, . . . , θd}.
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Designs in Q-polynomial schemes
X : Q-polynomialdef⇐⇒ ∃v∗0(ξ), . . . , v∗d(ξ) ∈ R[ξ], ∃θ∗0, . . . , θ∗d ∈ R s.t.
deg v∗i (ξ) = i (0 ⩽ i ⩽ d)
Qj,i = v∗i (θ∗j ) (0 ⩽ i, j ⩽ d)
Suppose X is Q-polynomial.δ∗ := min{i = 0 : χTEiχ = 0} : the dual distance of Cs :=
∣∣{i = 0 : χTAiχ = 0}∣∣ : the degree of C
Theorem (Delsarte, 1973)δ∗ ⩽ 2s+ 1
If δ∗ ⩾ 2s− 1, then C induces a Q-polynomial scheme.
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Designs in Q-polynomial schemes
Recall δ∗ ⩽ 2s+ 1.
|C| ⩾ m0 + · · ·+m⌊ δ∗−12
⌋ : the Fisher type inequality
C : tight def⇐⇒ “=” holds above
Wj(ξ) := v∗0(ξ) + · · ·+ v∗j (ξ) : the Wilson polynomial of degree j
Theorem (Delsarte, 1973)δ∗ = 2s+ 1 ⇐⇒ C : tight =⇒ Ws(ξ) has s simple roots
in {θ∗0, . . . , θ∗d}.
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Width and dual width (2003)
Suppose X is P -polynomial.
Theorem (Delsarte, 1973)δ ⩽ 2s∗ + 1
If δ ⩾ 2s∗ − 1, then C is completely regular.
w := max{i : χTAiχ = 0} : the width of C
Theorem (Brouwer–Godsil–Koolen–Martin, 2003)w ⩾ d− s∗
If w = d− s∗, then C is completely regular.
# = s
· · ·e = (1, 0, . . . , 0, ∗, 0, . . . , 0, ∗, . . . , ∗, 0, . . . , 0)
δ w
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Width and dual width (2003)
Suppose X is Q-polynomial.
Theorem (Delsarte, 1973)δ∗ ⩽ 2s+ 1
If δ∗ ⩾ 2s− 1, then C induces a Q-polynomial scheme.
w∗ := max{i : χTEiχ = 0} : the dual width of C
Theorem (Brouwer–Godsil–Koolen–Martin, 2003)w∗ ⩾ d− s
If w∗ = d− s, then C induces a Q-polynomial scheme.
# = s∗
· · ·eQ = (|C|, 0, . . . , 0, ∗, 0, . . . , 0, ∗, . . . , ∗, 0, . . . , 0)
δ∗ w∗
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Width and dual width (2003)
Suppose X is both P -polynomial & Q-polynomial.
w ⩾ d− s∗, w∗ ⩾ s∗ =⇒ w + w∗ ⩾ d
w∗ ⩾ d− s, w ⩾ s =⇒ w + w∗ ⩾ d
Corollary (Brouwer–Godsil–Koolen–Martin, 2003)w + w∗ ⩾ d
If w + w∗ = d, then C is completely regular and induces aQ-polynomial scheme.
C : a descendent def⇐⇒ w + w∗ = d
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The Erdos–Ko–Rado Theorem (1961)
Theorem (Erdos–Ko–Rado, 1961; Wilson, 1984)v, d, t ∈ N s.t. v > (t+ 1)(d− t+ 1)
[v] := {1, 2, . . . , v}C ⊂
([v]d
): t-intersecting, i.e., |x ∩ y| ⩾ t (x, y ∈ C)
Then |C| ⩽(v−td−t
). true when v ⩾ (t+ 1)(d− t+ 1)
“=” ⇐⇒ ∃z ∈([v]t
)s.t. C =
{x ∈
([v]d
): z ⊂ x
}∼= J(v − t, d− t)
“the” descendents in J(v, d)
RemarkDelsarte’s LP + BGKM led to the EKR theorems for:
Grassmann schemes (T., 2006);bilinear forms schemes (T., 2006);twisted Grassmann schemes (T., 2012).
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Shannon’s zero error capacity (1956)
X : a finite set = a set of letters
X channel receiver
noise
Define the graph G = (X,R) as follows:
x ∼ ydef⇐⇒ x, y can be “confused” (x, y ∈ X)
Find the independence number α(G) of G !!a good upper bound on
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Shannon’s zero error capacity (1956)
Send words of length n:
Xn channel receiver
noise
x = (x1, . . . , xn),y = (y1, . . . , yn) can be sent without confusion⇐⇒ xi ∼ yi for some i
non-adjacency in the strong product G ⊠ · · · ⊠G = G⊠n
c(G) := supn
n
√α(G⊠n) : the Shannon capacity of G
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Lovasz’s ϑ-function bound for c(G) (1979)
Recall c(G) = supn
n
√α(G⊠n).
Theorem (Lovasz, 1979)Consider the following SDP problem:
all-ones matrixmaximize ϑ = traceMJ
subject to traceM = 1
Mx,y = 0 if x ∼ y
M : positive semidefinite
Then c(G) ⩽ ϑ.
Remarkϑ = ϑ(G) satisfies α(G) ⩽ ϑ(G) and ϑ(G⊠n) = ϑ(G)n.
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Lovasz’s ϑ-function bound for c(G) (1979)
Lemma (Schrijver, 1979)Consider the following SDP problem:
maximize ϑ′ = traceMJ
subject to traceM = 1
Mx,y = 0 if x ∼ y
M : positive semidefinite& non-negative
Then α(G) ⩽ ϑ′(⩽ ϑ).
Theorem (Schrijver, 1979)If G =
(X,
∪i∈D Ri
), then ϑ′ coincides with Delsarte’s LP bound
for D-codes.idea: project M to A !!
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The Terwilliger algebra (1992)
We will consider the hypercube H(d, 2) as an illustrative example!!
X = Fd2 = the power set of [d] := {1, 2, . . . , d}
E∗0 , . . . , E
∗d : the dual idempotents of H(d, 2), i.e.,
(E∗i )x,y =
{1 if x = y ∈
([d]i
)0 otherwise
(x, y ∈ X)
T = ⟨E∗i AjE
∗h : 0 ⩽ i, j, h ⩽ d ⟩ : the Terwilliger algebra
RemarkE∗
i TE∗i = the adjacency algebra of J(d, i) (0 ⩽ i ⩽ d)
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Schrijver’s SDP bound for D-codes (2005)
Recall Schrijver’s problem for G =(X,
∪i∈D Ri
):
maximize ϑ′ = traceMJ
subject to M ∈ A
traceM = 1
Mx,y = 0 if x ∼ y
M : positive semidefinite& non-negative
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Schrijver’s SDP bound for D-codes (2005)
Schrijver (2005) strengthened the problem as follows:
maximize ϑ′+ = traceMJ
subject to M = M1 +M2, M1,M2 ∈ T
traceM = 1
Mx,y = 0 if x ∼ y
M1,M2 : positive semidefinite& non-negative& extra conditions
Theorem (Schrijver, 2005)If C is a D-code, then |C| ⩽ ϑ′
+(⩽ ϑ′).
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Schrijver’s SDP bound for D-codes (2005)
RemarkThe bound ϑ′
+ was formulated for some other cases:J(v, d) (Schrijver, 2005);H(d, q) (Gijswijt–Schrijver–T., 2006);H(d, 2) (Hou–Hou–Gao–Yu, 2018+).
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The primary T-module
χi = χ([d]i ): the characteristic vector of
([d]i
)(0 ⩽ i ⩽ d)
⟨χ0, . . . , χd⟩ : the primary T-module
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Relative t-designs (1977)
C : a t-design def⇐⇒ C : a {1, . . . , t}-design⇐⇒ χTEiχ = 0 (1 ⩽ i ⩽ t)
⇐⇒ Eiχ = 0 (1 ⩽ i ⩽ t)
⊂⟨χ0, . . . , χd⟩
C : a relative t-design def⇐⇒ Eiχ ∈ ⟨Eiχ0⟩ (1 ⩽ i ⩽ t)
Theorem (Delsarte, 1977)C : a relative t-design ⇐⇒ C : a regular t-wise balanced design
RemarkRelative t-designs have been actively studied:
B.–B., 2012; B.–B.–B., 2014; Li–B.–B., 2014; B.–B.–Suda–T., 2015;B.–B.–Zhu, 2015; Yue–Hou–Gao, 2015; Zhu–B.–B., 2016; B.–B.–Zhu, 2017;
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Relative t-designs (1977)
Suppose C is a relative 2ε-design.
S :={i : C ∩
([d]i
)= ∅
}heavily uses T
Theorem (Xiang, 2012; Bannai–Bannai–Suda–T., 2015)If S ⊂ {ε, . . . , d− ε}, then |C| ⩾ mε +mε−1 + · · ·+mε−|S|+1.
C : tight def⇐⇒ “=” holds above
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Relative t-designs (1977)
Recall S ={i : C ∩
([d]i
)= ∅
}.
Theorem (Bannai–Bannai–T.–Zhu, 2018+)Suppose S = {i, j}, where ε ⩽ i < j ⩽ d− ε.If C is tight, i.e., |C| = mε +mε−1, then C induces a coherentconfiguration with two fibers, and the Hahn polynomial
Qε(ξ) = 3F2
(−ξ,−ε, ε− d− 1
j − d,−i
∣∣∣∣ 1) ∈ R[ξ]
has ε simple roots in{0, 1, . . . ,min{i, d− j}
}.
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The cross-intersecting EKR theorem (1986)
Theorem (Pyber, 1986; Matsumoto–Tokushige, 1989)d > 2i, 2j
C ⊂([d]i
), D ⊂
([d]j
): cross-intersecting, i.e., x ∩ y = ∅ (x∈C, y ∈D)
Then |C||D| ⩽(d−1i−1
)(d−1j−1
). true when d ⩾ 2i, 2j
“=” ⇐⇒ ∃ℓ∈ [n] s.t. C ={x∈
([d]i
): ℓ∈ x
}, D=
{y ∈
([d]j
): ℓ∈ y
}
Hajime Tanaka Current progress in the Delsarte theory Algebraic Combinatorics 29 / 35
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The cross-intersecting EKR theorem (1986)
Remark∃SDP-based proof lim
q→1[Suda–T., 2014]:(d
j
)×(di
)all-ones matrix
maximize ϑ′ = traceMi,jJ
subject to M =
[Mi,i Mi,j
Mj,i Mj,j
]∈ (E∗
i +E∗j )T (E∗
i +E∗j )
traceMi,i = traceMj,j = 1
(Mi,j)x,y = 0 if x ∩ y = ∅M : positive semidefinite
& non-negative
This method was explored further (Suda–T.–Tokushige, 2017).
Hajime Tanaka Current progress in the Delsarte theory Algebraic Combinatorics 30 / 35
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Shannon capacity for two channels
G = (X,R) : a finite simple graphX = X+ ⊔X−
α(G) := max{√|C+||C−| : C± ⊂ X±, C+ ∪ C− is independent
}Lemma (Iwabuchi–T., 2018+)
Consider the following SDP problem:|X−| × |X+| all-ones matrix
maximize ϑ = traceM+−J
subject to M =
[M+− M+−M−+ M−−
]traceM++ = traceM−− = 1
Mx,y = 0 if x ∼ y
M : positive semidefinite
Then α(G) ⩽ ϑ.
Hajime Tanaka Current progress in the Delsarte theory Algebraic Combinatorics 31 / 35
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Shannon capacity for two channels
X+, X− : two sets of letters
X+ channel receiver
X− channel receiver
noise
noise
interference
α(G) ←→ the average rate of the two channels
Hajime Tanaka Current progress in the Delsarte theory Algebraic Combinatorics 32 / 35
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Shannon capacity for two channels
Send two sets of words of length n:
(X+)n channel receiver
(X−)n channel receiver
noise
noise
interference
Hajime Tanaka Current progress in the Delsarte theory Algebraic Combinatorics 33 / 35
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Shannon capacity for two channels
Define the graph G⊠n with vertex set (X+)n ⊔ (X−)
n as follows:
sgn = + sgn = −x = (x1, . . . , xn) ∼ y = (y1, . . . , yn)
def⇐⇒
{xi ∼ yi for some i if sgnx = sgny
xi ∼ yi for all i if sgnx = sgny
Lemma (Iwabuchi–T., 2018+)
ϑ = ϑ(G) satisfies ϑ(G⊠n
)= ϑ(G)n.
c(G) := supn
n
√α(G⊠n
)Theorem (Iwabuchi–T., 2018+)
c(G) ⩽ ϑ(G)
Hajime Tanaka Current progress in the Delsarte theory Algebraic Combinatorics 34 / 35
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Shannon capacity for two channels
Theorem (Lovasz, 1979)
c(C5) = ϑ(C5) =√5
Example
c(G) = ϑ(G) =√5
Hajime Tanaka Current progress in the Delsarte theory Algebraic Combinatorics 35 / 35
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Shannon capacity for two channels
Theorem (Lovasz, 1979)
c(C5) = ϑ(C5) =√5
Example
√2√5 ⩽ c(G) ⩽ ϑ(G) =
√5
Hajime Tanaka Current progress in the Delsarte theory Algebraic Combinatorics 35 / 35
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Shannon capacity for two channels
Theorem (Lovasz, 1979)
c(C5) = ϑ(C5) =√5
Example
√2√5 ⩽ c(G) ⩽ ϑ(G) =
√5
Hajime Tanaka Current progress in the Delsarte theory Algebraic Combinatorics 35 / 35
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Shannon capacity for two channels
Theorem (Lovasz, 1979)
c(C5) = ϑ(C5) =√5
Example
√2√5 ⩽ c(G) ⩽ ϑ(G) =
√5
Hajime Tanaka Current progress in the Delsarte theory Algebraic Combinatorics 35 / 35
![Page 39: Current progress in the Delsarte theory › ~htanaka › docs › 2018.11.hefei.pdf · Current progress in the Delsarte theory Hajime Tanaka Research Center for Pure and Applied Mathematics](https://reader033.vdocuments.us/reader033/viewer/2022052519/5f2899bd3dbea4284b00ad63/html5/thumbnails/39.jpg)
Shannon capacity for two channels
Theorem (Lovasz, 1979)
c(C5) = ϑ(C5) =√5
Example
2 ⩽ c(G) ⩽ ϑ(G) =√5
Hajime Tanaka Current progress in the Delsarte theory Algebraic Combinatorics 35 / 35
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Shannon capacity for two channels
Theorem (Lovasz, 1979)
c(C5) = ϑ(C5) =√5
Example
2 ⩽ c(G) ⩽ ϑ(G) =√5
Hajime Tanaka Current progress in the Delsarte theory Algebraic Combinatorics 35 / 35
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Shannon capacity for two channels
Theorem (Lovasz, 1979)
c(C5) = ϑ(C5) =√5
Example
c(G) = ϑ(G) = 2
Hajime Tanaka Current progress in the Delsarte theory Algebraic Combinatorics 35 / 35