New bounds for equiangular line sets
Wei-Hsuan Yu
Michigan State University
Aug 10, 2015
WPI Systems of Line Workshop
Wei-Hsuan Yu
New bounds for equiangular line sets
Definition
If X = {x1 , x2 , · · · , xN } ⊂ Sn−1 (unit sphere in Rn ) and hxi , xj i = a or
b for all i 6= j, then we call X is a spherical two-distance set.
Q : What is the maximum cardinality of a spherical
two-distance set?
Wei-Hsuan Yu
New bounds for equiangular line sets
Figure: The maximum spherical two-distance set in R2 : Pentagon.
Figure: The maximum spherical two-distance set in R3 : Octahedron.
Wei-Hsuan Yu
New bounds for equiangular line sets
Musin’s result
Musin used Delsarte’s linear programming method to prove that
g(n) =
n(n + 1)
2
if 7 ≤ n ≤ 39, n 6= 22, 23
and g(23) = 276 or 277.
Wei-Hsuan Yu
New bounds for equiangular line sets
Barg and Yu 2013
We use the semidefinite programming (SDP) method showing that
g(n) =
n(n + 1)
,
2
7 ≤ n ≤ 93, n 6= 22, 46, 78.
In particular, g(23) = 276.
Wei-Hsuan Yu
New bounds for equiangular line sets
(1)
Definition
A set of lines in Rn is called equiangular if the angle between each pair of
lines is the same.
An equiangular line set can be defined as an unit vectors set
S = {xi }M
i=1 such that |hxi , xj i| = c, 1 ≤ i < j ≤ M for some c > 0.
A equiangular line set can be defined as a spherical two-distance set
with inner product value c and −c.
Question : What is the maximum cardinality of
an equiangular line set in Rn ?
Wei-Hsuan Yu
New bounds for equiangular line sets
Figure: Maximum equiangular lines in R2 : 3 lines through opposite vertices of a
regular hexagon.
Figure: Maximum equiangular lines in R3 : 6 lines through opposite vertices of
the icosahedron.
Wei-Hsuan Yu
New bounds for equiangular line sets
Known results
Let M (n) denote the maximum size of an equiangular line set in Rn
Hanntjes found M (n) for n = 2 and 3 in 1948.
Van Lint and Seidel found the largest number of equiangular lines
for 4 ≤ n ≤ 7 in 1966.
Lemmens and Seidel used linear-algebraic methods to determine
M (n) for most values of n in the region 8 ≤ n ≤ 23 in 1973.
n
2
3
4
5
6
7 ≤ n ≤ 13
14
15
16
M (n)
3
6
6
10
16
28
28-29
36
40-41
1/α
√2
5
√
3; 5
3
3
3
3; 5
5
5
n
17
18
19
20
21
22
23
24 ≤ n ≤ 42
43
M (n)
48-50
48-61
72-76
90-96
126
176
276
≥ 276
≥ 344
Table: Known bounds on M (n) in small dimensions
Wei-Hsuan Yu
New bounds for equiangular line sets
1/α
5
5
5
5
5
5
5
5
7
Our results
Theorem (Barg and Yu 2014)
We use the semidefinite programming (SDP) method to show that
M (n) = 276 for 24 ≤ n ≤ 41 and M (43) = 344.
n
3
4
5
6
7 ≤ n ≤ 13
14
15
16
17
M (n)
6
6
10
16
28
28-29
36
40-41
48-50
SDP bound
6
6
10
16
28
30
36
42
51
n
18
19
20
21
22
23
24 ≤ n ≤ 41
42
43
M (n)
48-61
72-76
90-96
126
176
276
276
≥ 276
344
Table: Bounds on M (n) including new results
Wei-Hsuan Yu
New bounds for equiangular line sets
SDP bound
61
76
96
126
176
276
276
288
344
Gegenbauer polynomials
(n)
Let Gk (t), k = 0, 1, . . . denote the Gegenbauer polynomials of degree k.
(n)
(n)
They are defined recursively as follows: G0 ≡ 1, G1 (t) = t, and
(n)
(n)
Gk (t) =
(n)
(2k + n − 4)tGk−1 (t) − (k − 1)Gk−2 (t)
,
k+n−3
Wei-Hsuan Yu
New bounds for equiangular line sets
k ≥ 2.
Define a matrix Ykn (u, v, t), k ≥ 0
(n−1)
(Ykn (u, v, t))ij = ui v j ((1 − u2 )(1 − v 2 ))k/2 Gk
t − uv
p
(1 − u2 )(1 − v 2 )
and a matrix Skn (u, v, t) by setting
Skn (u, v, t) =
1 X n
Yk (σ(u, v, t)),
6
σ∈S3
S. Bochner (1941) proved that
X
Skn (x · y, x · z, y · z) 0.
(x,y,z)∈C 3
I.J. Schoenberg (1942) proved that
X
Gnk (hx, yi) ≥ 0.
x,y∈C 2
Wei-Hsuan Yu
New bounds for equiangular line sets
(2)
Semidefinite programming
min cT x
m
X
subject to F0 +
Fi xi 0
i=1
m
where c, x ∈ R and Fi is an n by n symmetric matrix ∀i. The sign ””
means that the matrix is positive semidefinite.
Wei-Hsuan Yu
New bounds for equiangular line sets
Theorem (Gerzon, absolute bounds)
If there are M equiangular lines in Rn , then M ≤
n(n+1)
.
2
Gerzon bounds are known to be attained only for n = 2, 3, 7, and 23.
Theorem (Neumann)
If there are M equiangular lines in Rn with angle arccos α and M > 2n,
then 1/α is an odd integer.
Theorem (Lemmens and Seidel)
M1/3 (n) = 2(n − 1) for n ≥ 16, where Mα (n) is the maximum size of an
equiangular line set when the value of the angle is arccos α.
Theorem (Relative bounds)
n(1 − α2 )
1 − nα2
valid for all α such that the denominator is positive.
Mα (n) ≤
Wei-Hsuan Yu
New bounds for equiangular line sets
(3)
SDP bound
Theorem
Let C be an equiangular line set with inner product values either a or −a.
Let p be a positive integer. The cardinality |C| is bounded above by the
solution of the following semi-definite programming problem :
1+
1
max(x1 + x2 )
3
(4)
subject to
1 0 1 0 1 00
+
(x1 + x2 ) +
(x3 + x4 + x5 + x6 ) 0
01
01
3 11
(5)
Skn (1, 1, 1) + Skn (a, a, 1)x1 + Skn (−a, −a, 1)x2 + Skn (a, a, a)x3
+ Skn (a, a, −a)x4 + Skn (a, −a, −a)x5 + Skn (−a, −a, −a)x6 0
3+
(n)
Gk (a)x1
+
(n)
Gk (−a)x2
≥ 0,
(7)
where k = 0, 1, · · · , p and xj ≥ 0, j = 1, · · · , 6.
Wei-Hsuan Yu
(6)
New bounds for equiangular line sets
SDP bound table
n
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
1/5
176
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
279
290
301
313
1/7
39
42
46
50
54
58
64
69
75
82
90
99
108
120
132
148
165
187
213
246
288
344
422
540
736
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1/9
29
31
33
35
37
40
42
44
47
49
52
55
57
60
64
67
70
74
78
82
86
90
95
100
105
110
116
122
129
136
143
151
160
169
179
190
201
214
228
244
261
280
301
1/11
26
28
29
31
32
34
36
37
39
41
43
45
46
48
50
52
54
57
59
61
63
66
68
71
73
76
78
81
84
87
90
93
96
100
103
106
110
114
118
122
126
130
134
Wei-Hsuan Yu
1/13
25
26
27
29
30
31
33
34
36
37
39
40
42
43
45
47
48
50
52
53
55
57
59
60
62
64
66
68
70
72
74
76
78
81
83
85
87
90
92
94
97
99
102
1/15
24
25
26
28
29
30
31
33
34
35
37
38
39
41
42
44
45
46
48
49
51
52
54
56
57
59
60
62
64
65
67
69
70
72
74
76
77
79
81
83
85
87
89
max
176
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
276
288
344
422
540
736
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
Gerzon
253
276
300
325
351
378
406
435
465
496
528
561
595
630
666
703
741
780
820
861
903
946
990
1035
1081
1128
1176
1225
1275
1326
1378
1431
1485
1540
1596
1653
1711
1770
1830
1891
1953
2016
2080
New bounds for equiangular line sets
angle
1/5
1/5
1/5
1/5
1/5
1/5
1/5
1/5
1/5
1/5
1/5
1/5
1/5
1/5
1/5
1/5
1/5
1/5
1/5
1/5
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
SDP bound table (Cont.)
n
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
1/5
326
339
353
367
382
398
416
434
453
473
494
517
542
568
596
626
658
693
731
772
816
866
920
979
1046
1120
1203
1298
1406
1515
1556
1599
1644
1691
1739
1790
1842
1897
1954
2014
2077
2142
2211
1/7
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1/9
325
352
382
418
460
509
568
640
730
845
1000
1216
1540
2080
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
1/11
139
144
148
153
159
164
170
176
182
188
195
202
210
217
225
234
243
252
262
272
283
294
307
320
333
348
364
380
398
417
438
460
485
511
540
571
606
644
686
734
787
848
917
Wei-Hsuan Yu
1/13
105
107
110
113
115
118
121
124
127
130
134
137
140
144
147
151
154
158
162
166
170
174
178
182
186
191
196
200
205
210
215
220
226
231
237
243
249
255
262
268
275
282
289
1/15
91
92
94
97
99
101
103
105
107
109
112
114
116
118
121
123
126
128
130
133
136
138
141
143
146
149
152
154
157
160
163
166
169
172
176
179
182
185
189
192
196
199
203
max
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1216
1540
2080
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
Gerzon
2145
2211
2278
2346
2415
2485
2556
2628
2701
2775
2850
2926
3003
3081
3160
3240
3321
3403
3486
3570
3655
3741
3828
3916
4005
4095
4186
4278
4371
4465
4560
4656
4753
4851
4950
5050
5151
5253
5356
5460
5565
5671
5778
New bounds for equiangular line sets
angle
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/7
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
SDP bound table (Cont.)
n
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
*137
*138
*139
1/5
2282
2358
2437
2521
2609
2702
2800
2904
3015
3132
3257
3390
3532
3684
3848
4024
4214
4419
4643
4887
5153
5447
5770
6130
6531
6982
7493
8075
8747
9528
10450
11553
1/7
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1128
1130
1158
1187
1218
1249
1282
1315
1350
1/9
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
3160
1/11
997
1090
1200
1332
1493
1695
1954
2300
2784
3510
4720
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
Wei-Hsuan Yu
1/13
297
305
313
321
330
339
348
357
367
378
388
399
411
423
436
449
462
477
492
508
524
541
560
579
599
620
643
667
692
719
747
778
1/15
206
210
214
218
222
226
230
234
238
242
247
251
256
260
265
270
275
280
285
290
295
301
306
312
317
323
329
336
342
348
355
362
max
3160
3160
3160
3160
3160
3160
3160
3160
3160
3510
4720
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7140
7493
8075
8747
9528
10450
11553
Gerzon
5886
5995
6105
6216
6328
6441
6555
6670
6786
6903
7021
7140
7260
7381
7503
7626
7750
7875
8001
8128
8256
8385
8515
8646
8778
8911
9045
9180
9316
9453
9591
9730
New bounds for equiangular line sets
angle
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/11
1/11
1/11
1/11
1/11
1/11
1/11
1/11
1/11
1/11
1/11
1/11
1/11
1/11
1/11
1/11
1/11
1/5
1/5
1/5
1/5
1/5
1/5
Our results
Theorem (Barg and Yu)
We use the semidefinite programming method to show that M (n) = 276
for 24 ≤ n ≤ 41 and M (43) = 344 and we get tighter upper bounds for
M (n) when n ≤ 136.
Wei-Hsuan Yu
New bounds for equiangular line sets
Spherical t-design
Definition (Delsarte at el. 77’)
Let t be a natural number. A finite subset X of the unit sphere S n−1 is
called a spherical t-design if, for any polynomial f (x) = f (x1 , x2 , . . . , xn )
of degree at most t, the following equality holds :
Z
1 X
1
f
(x)dσ(x)
=
f (x).
(8)
|S n−1 | S n−1
|X|
x∈X
The set X is a spherical design if
X
f (x) = 0 ∀f (x) ∈ Harmj (Rn ), 1 ≤ j ≤ t.
(9)
x∈X
Delsarte at el 77’ proved that the cardinality of a spherical t-design X is
bounded below by
n+e−1
n+e−2
n+e−1
|X| ≥
+
, |X| ≥ 2
n−1
n−1
n−1
for t = 2e and t = 2e + 1.
The spherical t-design is called tight if any one of these bound is attained.
Wei-Hsuan Yu
New bounds for equiangular line sets
Spherical designs of harmonic index t
Definition
A spherical design of harmonic index t is a finite subset X ⊂ S n−1 such
that
X
f (x) = 0 ∀f (x) ∈ Harmt (Rn ).
(10)
x∈X
1
2
A lower bound of the size of a spherical design of harmonic index t
has been derived by Bannai, Okuda and Tagami 13’.
A lower bound of a spherical design of harmonic index 4 in S n−1 is
(n+1)(n+2)
. If the lower bound on the cardinality is attained, we call
6
it a tight spherical design of harmonic index 4.
Wei-Hsuan Yu
New bounds for equiangular line sets
Tight spherical designs of harmonic index 4
1
2
Bannai, Okuda and Tagami 13’ showd that a tight design of
harmonic index 4 has cardinality (n+1)(n+2)
and give arise to an
6
q
3
n
equiangular line set in R with angle arccos n+4
.
If tight harmonic index 4-designs on S n−1 exist, then n = 2 or n
must be of the form n = 3(2k − 1)2 − 4 = 12k 2 − 12k − 1 for some
integer k ≥ 3, and the configuration is a set of equiangular lines
1
with the angle arccos 2k−1
in the (12k 2 − 12k − 1)-dimensional
Euclidean space.
Wei-Hsuan Yu
New bounds for equiangular line sets
New relative bounds
Theorem (Okuda, Yu 2014)
Let n ≥ 3. Then the following holds:
1
Mα (n) ≤ 2 + (n − 2)
(1 − α)3
α((n − 2)α2 + 6α − 3)
for each α ∈ (0, 1) with
(1 − α)3 (−(n − 2)α2 + 6α + 3) ≥ (1 + α)3 ((n − 2)α2 + 6α − 3) ≥ 0.
2
Mα (n) ≤ 2 + (n − 2)
(1 + α)3
α(−(n − 2)α2 + 6α + 3)
for each α ∈ (0, 1) with
(1 + α)3 ((n − 2)α2 + 6α − 3) ≥ (1 − α)3 (−(n − 2)α2 + 6α + 3) ≥ 0,
where Mα (n) is the maximum size of an equiangular line set in Rn and
the value of the angle is arccos α.
Wei-Hsuan Yu
New bounds for equiangular line sets
Example
Let us consider the cases where nk := 3(2k − 1)2 − 4 and
αk := 1/(2k − 1) for an integer k ≥ 2. Note that in such cases,
Lemmens–Seidel’s relative bound (3) does not work since
1 − nk αk2 = −2(4k 2 − 4k − 1)/(2k − 1)2 < 0. One can compute that
96(k − 1)3 k
,
(2k − 1)5
96(k − 1)k 3
(1 + αk )3 ((nk − 2)αk2 + 6αk − 3) =
,
(2k − 1)5
(1 − αk )3 (−(nk − 2)αk2 + 6αk + 3) =
and hence
(1 + αk )3 ((nk − 2)αk2 + 6αk − 3) ≥ (1 − αk )3 (−(nk − 2)αk2 + 6αk + 3) ≥ 0.
Therefore, by our new relative bounds Theorem, we have
Mαk (nk ) ≤ 2 + (n − 2)
(1 + α)3
α(−(n − 2)α2 + 6α + 3)
= 2(k − 1)(4k 3 − k − 1).
Wei-Hsuan Yu
New bounds for equiangular line sets
Nonexistence of tight spherical designs of harmonic index 4
Observe that
(nk + 1)(nk + 2)
−2(k−1)(4k 3 −k−1) = 2(k−1)(2k−1)(4k 2 −4k−1) > 0
6
when k ≥ 2.
Therefore, we have
(nk +1)(nk +2)
6
> 2(k − 1)(4k 3 − k − 1), when k ≥ 2.
Corollary
For each n ≥ 3, there does not exist a tight harmonic index 4-design on
S n−1 .
Wei-Hsuan Yu
New bounds for equiangular line sets
Spherical designs of harmonic index T
Definition
Let T be a subset of N. A spherical design of harmonic index T is a finite
subset X ⊂ S n−1 such that
X
f (x) = 0 ∀f (x) ∈ Harmt (Rn ), with t ∈ T.
x∈X
Y. Zhu, E. Bannai, E. Bannai, K-T. Kim and W-H. Yu completely discuss
the non-existence of tight spherical designs of harmonic index 6, 8,
asymptotic results for all even numbers, {6,2},{6,4},{8,6},{8,4},{8,2},
{10,6,2},{12,8,4}. The paper is vailable on arxiv:1507.05373.
Wei-Hsuan Yu
New bounds for equiangular line sets
Open problems
1
M (14) = 28 or 29. M (16) = 40 or 41. Can we determine them?
2
The constructions and upper bounds for complex equiangular lines.
3
How to prove that Mα (n) has long stable ranges in general?
1
5
1
7
1
9
1
11
···
23 − 60
276
47 − 131
1128
79 − 227
3160
119 − 349
Wei-Hsuan Yu
7140
New bounds for equiangular line sets