New bounds for spherical two-distance sets and equiangular lines Michigan State University Oct 8-31, 016 Anhui University
Definition If X = {x 1, x,, x N } S n 1 (unit sphere in R n ) and x i, x j = a or b for all i j, then we call X is a spherical two-distance set. Q : What is the maximum cardinality of a spherical two-distance set?
Figure: The maximum spherical two-distance set in R : Pentagon. Figure: The maximum spherical two-distance set in R 3 : Octahedron.
Known results Let g(n) denote the maximum size of spherical two-distance set in R n 1 Let e 1,..., e n+1 be the standard basis in R n+1. The points e i + e j, i j form a spherical two-distance set in the plane x 1 + + x n+1 =, g(n) n(n + 1), n. (1)
Known results Let g(n) denote the maximum size of spherical two-distance set in R n 1 Let e 1,..., e n+1 be the standard basis in R n+1. The points e i + e j, i j form a spherical two-distance set in the plane x 1 + + x n+1 =, g(n) n(n + 1), n. (1) Delsarte, Goethals, and Seidel in 1977 proved so-called harmonic bound : n(n + 3) g(n). () They also showed that this bound is tight for n =, 6,.
Known results Let g(n) denote the maximum size of spherical two-distance set in R n 1 Let e 1,..., e n+1 be the standard basis in R n+1. The points e i + e j, i j form a spherical two-distance set in the plane x 1 + + x n+1 =, g(n) n(n + 1), n. (1) Delsarte, Goethals, and Seidel in 1977 proved so-called harmonic bound : n(n + 3) g(n). () They also showed that this bound is tight for n =, 6,. 3 Therefore, n(n+1) g(n) n(n+3).
Musin s result Musin used Delsarte s linear programming method to prove that g(n) = and g(3) = 76 or 77. n(n + 1) if 7 n 39, n, 3
Barg and Yu 014 We use the semidefinite programming (SDP) method showing that g(n) = In particular, g(3) = 76. n(n + 1), 7 n 93, n, 46, 78.
Maximum spherical two-distance sets Theorem (Yu 016+) n(n + 1) g(n) =, 7 n 417, n, 46, 78, 118, 166,, 86, 358 which are (k + 1) 3 for k =,, 9. For a + b 0, Oleg Musin proved the upper bounds are n(n+1). For a + b < 0, the upper bounds can be obtained from the bounds of equiangular lines in one higher dimension. Lemma If there are at most n(n 1) equiangular lines in R n, then maximum size of a spherical two-distance set in R n 1 is n(n 1). Musin Conjecture: g(n) = n(n + 1) n N, except n + 3 = (k + 1), k N.
Maximum spherical two-distance sets Theorem (Glazyrin and Yu 016+) g(n) = n(n + 1) n N, except n = (k + 1) 3, k N.
Definition A set of lines in R n 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 = {x i } M i=1 such that x i, x j = 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 R n?
Figure: Maximum equiangular lines in R : 3 lines through opposite vertices of a regular hexagon. Figure: Maximum equiangular lines in R 3 : 6 lines through opposite vertices of the icosahedron.
Known results Let M(n) denote the maximum size of an equiangular line set in R n Hanntjes found M(n) for n = 3 and 4 in 1948. Van Lint and Seidel found the largest number of equiangular lines for 5 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 3 in 1973. n M(n) 1/α n M(n) 1/α 3 17 48-50 5 3 6 5 18 48-61 5 4 6 3; 5 19 7-76 5 5 10 3 0 90-96 5 6 16 3 1 16 5 7 n 13 8 3 176 5 14 8-9 3; 5 3 76 5 15 36 5 4 n 4 76 5 16 40-41 5 43 344 7 Table: Known bounds on M(n) in small dimensions
Our results Theorem (Barg and Yu 014) We use the semidefinite programming (SDP) method to show that M(n) = 76 for 4 n 41 and M(43) = 344. n M(n) SDP bound n M(n) SDP bound 3 6 6 18 48-61 61 4 6 6 19 7-76 76 5 10 10 0 90-96 96 6 16 16 1 16 16 7 n 13 8 8 176 176 14 8-9 30 3 76 76 15 36 36 4 n 41 76 76 16 40-41 4 4 76 88 17 48-50 51 43 344 344 Table: Bounds on M(n) including new results Remark : Recently, we can show no 76 equiangular lines in R 19.
Gegenbauer polynomials Let G (n) k (t), k = 0, 1,... denote the Gegenbauer polynomials of degree k. They are defined recursively as follows: G (n) 0 1, G (n) 1 (t) = t, and G (n) k (k + n 4)tG(n) (t) = k 1 (t) (k 1)G(n) k + n 3 k (t), k.
Define a matrix Yk n (u, v, t), k 0 ( (Yk n (u, v, t)) ij = u i v j ((1 u )(1 v )) k/ G (n 1) t uv ) k (1 u )(1 v ) and a matrix Sk n (u, v, t) by setting S n k (u, v, t) = 1 6 σ S 3 Y n k (σ(u, v, t)), (3) I.J. Schoenberg (194) proved that if C is the finite set in S n 1, then (x,y) C G n k( x, y ) 0. (x,y,z) C 3 S n k (x y, x z, y z) 0.
Semidefinite programming min c T x m subject to F 0 + F i x i 0 where c, x R m and F i is an n by n symmetric matrix i. The sign means that the matrix is positive semidefinite. i=1 CVX (MATLAB toolkit) can solve an SDP in a second.
Theorem (Gerzon, absolute bounds) If there are M equiangular lines in R n, then M n(n+1). Gerzon bounds are known to be attained only for n =, 3, 7, and 3. Theorem (Neumann) If there are M equiangular lines in R n with angle arccos α and M > n, then 1/α is an odd integer. Theorem (Lemmens and Seidel) M 1/3 (n) = (n 1) for n 16, where M α (n) is the maximum size of an equiangular line set when the value of the angle is arccos α. Theorem (Relative bounds) M α (n) n(1 α ) 1 nα (4) valid for all α such that the denominator is positive.
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 : subject to 1 + 1 3 max(x 1 + x ) (5) ( 1 0 ) + 1 ( 0 1 ) ( 0 0 ) (x 0 1 3 1 1 1 + x ) + (x 0 1 3 + x 4 + x 5 + x 6 ) 0 (6) S n k (1, 1, 1) + S n k (a, a, 1)x 1 + S n k ( a, a, 1)x + S n k (a, a, a)x 3 + S n k (a, a, a)x 4 + S n k (a, a, a)x 5 + S n k ( a, a, a)x 6 0 (7) 3 + G (n) k (a)x 1 + G (n) k ( a)x 0, (8) where k = 0, 1,, p and x j 0, j = 1,, 6.
SDP bound table n 1/5 1/7 1/9 1/11 1/13 1/15 max Gerzon angle 176 39 9 6 5 4 176 53 1/5 3 76 4 31 8 6 5 76 76 1/5 4 76 46 33 9 7 6 76 300 1/5 5 76 50 35 31 9 8 76 35 1/5 6 76 54 37 3 30 9 76 351 1/5 7 76 58 40 34 31 30 76 378 1/5 8 76 64 4 36 33 31 76 406 1/5 9 76 69 44 37 34 33 76 435 1/5 30 76 75 47 39 36 34 76 465 1/5 31 76 8 49 41 37 35 76 496 1/5 3 76 90 5 43 39 37 76 58 1/5 33 76 99 55 45 40 38 76 561 1/5 34 76 108 57 46 4 39 76 595 1/5 35 76 10 60 48 43 41 76 630 1/5 36 76 13 64 50 45 4 76 666 1/5 37 76 148 67 5 47 44 76 703 1/5 38 76 165 70 54 48 45 76 741 1/5 39 76 187 74 57 50 46 76 780 1/5 40 76 13 78 59 5 48 76 80 1/5 41 76 46 8 61 53 49 76 861 1/5 4 76 88 86 63 55 51 88 903 1/7 43 76 344 90 66 57 5 344 946 1/7 44 76 4 95 68 59 54 4 990 1/7 45 76 540 100 71 60 56 540 1035 1/7 46 76 736 105 73 6 57 736 1081 1/7 47 76 118 110 76 64 59 118 118 1/7 48 76 118 116 78 66 60 118 1176 1/7 49 76 118 1 81 68 6 118 15 1/7 50 76 118 19 84 70 64 118 175 1/7 51 76 118 136 87 7 65 118 136 1/7 5 76 118 143 90 74 67 118 1378 1/7 53 76 118 151 93 76 69 118 1431 1/7 54 76 118 160 96 78 70 118 1485 1/7 55 76 118 169 100 81 7 118 1540 1/7 56 76 118 179 103 83 74 118 1596 1/7 57 76 118 190 106 85 76 118 1653 1/7 58 76 118 01 110 87 77 118 1711 1/7 59 76 118 14 114 90 79 118 1770 1/7 60 76 118 8 118 9 81 118 1830 1/7 61 79 118 44 1 94 83 118 1891 1/7 6 90 118 61 16 97 85 118 1953 1/7 63 301 118 80 130 99 87 118 016 1/7 64 313 118 301 134 10 89 118 080 1/7
SDP bound table (Cont.) n 1/5 1/7 1/9 1/11 1/13 1/15 max Gerzon angle 65 36 118 35 139 105 91 118 145 1/7 66 339 118 35 144 107 9 118 11 1/7 67 353 118 38 148 110 94 118 78 1/7 68 367 118 418 153 113 97 118 346 1/7 69 38 118 460 159 115 99 118 415 1/7 70 398 118 509 164 118 101 118 485 1/7 71 416 118 568 170 11 103 118 556 1/7 7 434 118 640 176 14 105 118 68 1/7 73 453 118 730 18 17 107 118 701 1/7 74 473 118 845 188 130 109 118 775 1/7 75 494 118 1000 195 134 11 118 850 1/7 76 517 118 116 0 137 114 116 96 1/9 77 54 118 1540 10 140 116 1540 3003 1/9 78 568 118 080 17 144 118 080 3081 1/9 79 596 118 3160 5 147 11 3160 3160 1/9 80 66 118 3160 34 151 13 3160 340 1/9 81 658 118 3160 43 154 16 3160 331 1/9 8 693 118 3160 5 158 18 3160 3403 1/9 83 731 118 3160 6 16 130 3160 3486 1/9 84 77 118 3160 7 166 133 3160 3570 1/9 85 816 118 3160 83 170 136 3160 3655 1/9 86 866 118 3160 94 174 138 3160 3741 1/9 87 90 118 3160 307 178 141 3160 388 1/9 88 979 118 3160 30 18 143 3160 3916 1/9 89 1046 118 3160 333 186 146 3160 4005 1/9 90 110 118 3160 348 191 149 3160 4095 1/9 91 103 118 3160 364 196 15 3160 4186 1/9 9 198 118 3160 380 00 154 3160 478 1/9 93 1406 118 3160 398 05 157 3160 4371 1/9 94 1515 118 3160 417 10 160 3160 4465 1/9 95 1556 118 3160 438 15 163 3160 4560 1/9 96 1599 118 3160 460 0 166 3160 4656 1/9 97 1644 118 3160 485 6 169 3160 4753 1/9 98 1691 118 3160 511 31 17 3160 4851 1/9 99 1739 118 3160 540 37 176 3160 4950 1/9 100 1790 118 3160 571 43 179 3160 5050 1/9 101 184 118 3160 606 49 18 3160 5151 1/9 10 1897 118 3160 644 55 185 3160 553 1/9 103 1954 118 3160 686 6 189 3160 5356 1/9 104 014 118 3160 734 68 19 3160 5460 1/9 105 077 118 3160 787 75 196 3160 5565 1/9 106 14 118 3160 848 8 199 3160 5671 1/9 107 11 118 3160 917 89 03 3160 5778 1/9
SDP bound table (Cont.) n 1/5 1/7 1/9 1/11 1/13 1/15 max Gerzon angle 108 8 118 3160 997 97 06 3160 5886 1/9 109 358 118 3160 1090 305 10 3160 5995 1/9 110 437 118 3160 100 313 14 3160 6105 1/9 111 51 118 3160 133 31 18 3160 616 1/9 11 609 118 3160 1493 330 3160 638 1/9 113 70 118 3160 1695 339 6 3160 6441 1/9 114 800 118 3160 1954 348 30 3160 6555 1/9 115 904 118 3160 300 357 34 3160 6670 1/9 116 3015 118 3160 784 367 38 3160 6786 1/9 117 313 118 3160 3510 378 4 3510 6903 1/11 118 357 118 3160 470 388 47 470 701 1/11 119 3390 118 3160 7140 399 51 7140 7140 1/11 10 353 118 3160 7140 411 56 7140 760 1/11 11 3684 118 3160 7140 43 60 7140 7381 1/11 1 3848 118 3160 7140 436 65 7140 7503 1/11 13 404 118 3160 7140 449 70 7140 766 1/11 14 414 118 3160 7140 46 75 7140 7750 1/11 15 4419 118 3160 7140 477 80 7140 7875 1/11 16 4643 118 3160 7140 49 85 7140 8001 1/11 17 4887 118 3160 7140 508 90 7140 818 1/11 18 5153 118 3160 7140 54 95 7140 856 1/11 19 5447 118 3160 7140 541 301 7140 8385 1/11 130 5770 118 3160 7140 560 306 7140 8515 1/11 131 6130 118 3160 7140 579 31 7140 8646 1/11 13 6531 1130 3160 7140 599 317 7140 8778 1/11 133 698 1158 3160 7140 60 33 7140 8911 1/11 134 7493 1187 3160 7140 643 39 7493 9045 1/5 135 8075 118 3160 7140 667 336 8075 9180 1/5 136 8747 149 3160 7140 69 34 8747 9316 1/5 *137 958 18 3160 7140 719 348 958 9453 1/5 *138 10450 1315 3160 7140 747 355 10450 9591 1/5 *139 11553 1350 3160 7140 778 36 11553 9730 1/5
Our results Theorem (Barg and Yu) We use the semidefinite programming method to show that M(n) = 76 for 4 n 41 and M(43) = 344 and we get tighter upper bounds for M(n) when n 136.
Observation angle dim bounds 1 3 4 3 59 76 = 5 1 47 48 47 131 118 = 7 1 79 80 79 7 3160 = 9 1 119 10 119 349 7140 = 11 Theorem (Yu 016+) We prove M a (n) 1 ( 1 a 1)( 1 a ) for all a N and for all 1 a n 3 a 16
Relaxation semidefinite programming problems 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 : subject to 1 + 1 3 max(x 1 + x ) (9) ( 1 0 ) + 1 ( 0 1 ) ( 0 0 ) (x 0 1 3 1 1 1 + x ) + (x 0 1 3 + x 4 + x 5 + x 6 ) 0 (10) S n k (1, 1, 1) + S n k (a, a, 1)x 1 + S n k ( a, a, 1)x + S n k (a, a, a)x 3 + S n k (a, a, a)x 4 + S n k (a, a, a)x 5 + S n k ( a, a, a)x 6 0 (11) 3 + G (n) k (a)x 1 + G (n) k ( a)x 0, (1) where k = 0, 1,, p and x j 0, j = 1,, 6. We find that above constraints only S n 1 0, S n 3 0 and (9) are crucial.
Symbolic semidefinite programming problems Theorem The solution of following optimization problem is 1 ( 1 a )( 1 a 1). max(1 + A) subject to a 4 (3a + 1) A + (6a 1)(a + 1) 3 ) B + a 4 (3a 1) (6a 1)(a 1) 3 ) C 0 (13) A + a 1 + a B + a a 1 C 0 (14) A(A 1) B + C (15) proof: We choose suitable t, where t = 16a 6 (6a 1)(a+1) (a 1) such that t a 1+a + a4 (3a+1) (6a 1)(a+1) = t a 3 a 1 + If we calculate t(14) + (13), we will get a4 (3a 1) (6a 1)(a 1) 3 a (t + 1)A + (t 1 + a + a 4 (3a + 1) (6a )(B + C) 0 1)(a + 1) 3 10a6 + 13a 4 8a + 1 (6a 1)(a 1) (a + 1) A a 4 (5a 1) (6a 1)(a 1) (B + C) 0 (a + 1)
notice that a 4 (5a 1) (6a 1)(a 1) 0 and we plug in (15). (a + 1) 10a6 + 13a 4 8a + 1 (6a 1)(a 1) (a + 1) A a 4 (5a 1) (6a 1)(a 1) A(A 1) 0 (a + 1) 10a6 + 13a 4 8a + 1 (6a 1)(a 1) (a + 1) a 4 (5a 1) (6a 1)(a 1) (A 1) (a + 1) 1 3a a 4 a 4 A 1 A 1 3a a 4 Then, it is not hard to see that A + 1 1 ( 1 a )( 1 a 1).
New bounds for equiangular lines Theorem (Glazyrin and Yu 016+) M 1 (n) n( a 3 a + 4 7 ) +, for all a 3 and for all n N. Theorem If n 359, then M(n) (a )(a 1), where a is the unique positive odd integer such that a n (a + ) 3. Notice that if n = a, then we just obtain Gerzon bound M(n) n(n+1). For other cases, we can prove that M(n) n(n 1).
Maximum spherical two-distance sets Theorem (Glazyrin and Yu 016+) g(n) = n(n + 1) n N, except n = (k + 1) 3, k N.
Future work 1 M(14) = 8 or 9. M(16) = 40 or 41. Can we determine them? The constructions and upper bounds for complex equiangular lines. 3 The maximum spherical 3-distance sets in R n. (only n =, 3, 8 and are known) 4 How much do we know the Maximum Separation Codes on sphere? If we have M points on S n 1, then max x i, x j M n n(m 1). = attained iff equiangular tight frames.