High accuracy semidefinite programming bounds for kissing numbers

HIGH A CCURA CY SEMIDEFINITE PROGRAMMING BOUNDS F OR KISSING NUMBERS HANS D. MITTELMANN AND FRANK V ALLENTIN Abstract. The kissing n umber in n -di mensional Euclidean space i s the max- imal num b er of non-ov erlapping unit s pheres which si multan eously can touch a cen tral unit sphere. Bac ho c and V all en tin developed a m etho d to find upp er bounds for the kissing n umber based on semidefinite programming. This paper is a repor t on high accuracy calculations of these upper b ounds for n ≤ 24. The b ound for n = 16 implies a conjecture of Con wa y and Sloane: There is no 16-dimensional p erio dic sphere pack ing wi th a verage theta series 1 + 7680 q 3 + 4320 q 4 + 276480 q 5 + 61440 q 6 + · · · 1. Introduction In geo metry , the kissing n umb er in n -dimensio nal Euclidea n space is the maximal nu mber of non-overlapping unit spher es which simultaneously can to uch a central unit s phere. The kissing num be r is o nly known in dimensions n = 1 , 2 , 3 , 4 , 8 , 24, and ther e were many a ttempts to find go o d lower and upp er b ounds. W e r efer to Casselman [4] for the history o f this pro blem and to Pfender, Zie gler [14], Elkies [7], and Conw ay , Slo ane [6] for more background information o n spher e pa cking problems. Bachoc and V allentin [1] develop a metho d (Section 2 reca lls it) to find upp er bo unds for the kissing num b er ba sed on s emidefinite pro gramming. T able 1 in Section 3, the main contribution of this pap er, gives the v alues, i.e. the first ten significant digits, of these upper b ounds for all dimensio ns 3 , . . . , 24. In all case s they are the b est known upp er b ounds. Dimension 5 is the first dimension in whic h the kissing n umber is not known. With our computation we could limit the r ange of po ssible v alues fr om { 40 , . . . , 45 } to { 40 , . . . , 44 } . In Section 4 we show that the high accuracy computations for the upp er b ounds in dimension 4 re sult in to a question ab out a p ossible approach to prov e the uniqueness of the kissing configuration in 4 dimensions. Although a c quiring the data for the table is a purely computational task we think that providing this table is v aluable for several rea sons: The kissing num b er is a n impo rtant consta n t in geometry and results can de p end o n goo d upp er b ounds for it. F or ins tance in Section 5 we show that there is no per io dic p oint set in dimension Date : June 26, 2009. 1991 Mathema tics Subje ct Classific ation. 11F11, 52C17, 90C10. Key wor ds and phr ases. kissing num b er, semidefinite programming, a v erage theta series, ex- tremal modular form. The second author was partially supported by the Deutsc he F orsch ungsgemeinsc haft (DF G) under grant SCHU 1503/4. 1 2 H.D. Mi t t e lmann, F. V alle ntin 16 with av erage theta series 1 + 7680 q 3 + 4320 q 4 + 27648 0 q 5 + 61440 q 6 + · · · This prov es a conjecture of Conw ay and Sloane [6, Chapter 7, pag e 190]. F ur- thermore, the actual co mputation of the table was very challenging. Bachoc and V allen tin [1] gav e res ults for dimensio n 3 , . . . , 1 0. How ever, they rep or t on numeri- cal difficulties whic h prev ented them from extending t heir results. No w using new, more sophisticated high accura cy soft w are and faster computers and more compu- tation time we could overcome so me of the numerical difficulties. Section 3 co nt ains details ab out the computations. 2. Not a tion In this section w e set up the notatio n which is needed for our computation. F or more infor mation we refer to [1]. F or natural num b ers d and n ≥ 3 let s d ( n ) be the optimal v a lue of the minimization problem min n 1 + d X k =1 a k + b 11 + h F 0 , S n 0 (1 , 1 , 1) i : a 1 , . . . , a d ∈ R , a 1 , . . . , a d ≥ 0 , b 11 , b 12 , b 22 ∈ R ,  b 11 b 12 b 12 b 22  is p ositive semidefinite, F k ∈ R ( d +1 − k ) × ( d +1 − k ) , F k is po s itive se midefinite, k = 0 , . . . , d, q , q 1 ∈ R [ u ] , deg( p + pq 1 ) ≤ d, p, p 1 sums of squares , r , r 1 , . . . , r 4 ∈ R [ u , v, t ] , deg( r + 4 X i =1 p i r i ) ≤ d, r, r 1 , . . . , r 4 sums of squares , 1 + d X k =1 a k P n k ( u ) + 2 b 12 + b 22 + 3 d X k =0 h F k , S n k ( u, u, 1) i + q ( u ) + p ( u ) q 1 ( u ) = 0 , b 22 + d X k =0 h F k , S n k ( u, v, t ) i + r ( u, v , t ) + 4 X i =1 p i ( u, v, t ) r i ( u, v, t ) = 0  . Here P n k is the normalized Ja c o bi p olynomial of degree k with P n k (1) = 1 and parameters (( n − 3) / 2 , ( n − 3) / 2 ). In gener a l, Jac obi p olynomials with para meters ( α, β ) are or tho gonal p olynomials for the measur e (1 − u ) α (1 + u ) β du on the interv al [ − 1 , 1]. Before we ca n define the ma trices S n k we first define the entry ( i, j ) with i, j ≥ 0 of the (infinite) matrix Y n k containing p olynomials in the v a riables u, v , w by  Y n k  i,j ( u, v, t ) = u i v j · ((1 − u 2 )(1 − v 2 )) k/ 2 P n − 1 k t − uv p (1 − u 2 )(1 − v 2 ) ! . Then we g et S n k by symmetrization: S n k = 1 6 P σ σ Y n k , wher e σ runs throug h a ll per mut ations of the v ariables u, v, t which a cts o n the matrix co efficients in the High acc uracy semide finite programming b ou nds for kissing number s 3 obvious way . T he p olyno mials p , p 1 , . . . , p 4 are given by p ( u ) = − ( u + 1)( u + 1 / 2) , p 1 ( u, v, t ) = p ( u ) , p 2 ( u, v, t ) = p ( v ) , p 3 ( u, v, t ) = p ( t ) , p 4 ( u, v, t ) = 1 + 2 u v t − u 2 − v 2 − t 2 . By h A, B i we denote the inner pro duct b etw een symmetric matrices trac e( AB ). In [1] it is shown that this minimiza tio n problem is a semidefinite pro gram and that every upper b ound on s d ( n ) provides an upper b ound fo r the kissing num ber in dimension n . Cle arly , the n umbers s d ( n ) form a monotonic decreasing sequence in d . 3. Bounds for kissing numbers b est low er b est upper bound SDP n b ound kno wn previously kno wn b ound 3 12 12 s 11 (3) = 12 . 42167009 . . . [16] Sc h ¨ utte, v.d. W aerden, 1953 s 12 (3) = 12 . 40203212 . . . s 13 (3) = 12 . 39266509 . . . s 14 (3) = 12 . 38180947 . . . 4 24 24 s 11 (4) = 24 . 10550859 . . . [11] M usin, 2008 s 12 (4) = 24 . 09098111 . . . s 13 (4) = 24 . 07519774 . . . s 14 (4) = 24 . 06628391 . . . 5 40 45 s 11 (5) = 45 . 06107293 . . . [1] Ba choc, V allentin, 2008 s 12 (5) = 45 . 02353644 . . . s 13 (5) = 45 . 00650838 . . . s 14 (5) = 44 . 99899685 . . . 6 72 78 s 11 (6) = 78 . 58344077 . . . [1] Ba choc, V allentin, 2008 s 12 (6) = 78 . 35518719 . . . s 13 (6) = 78 . 29404232 . . . s 14 (6) = 78 . 24061272 . . . 7 126 135 s 11 (7) = 134 . 8824614 . . . [1] Ba choc, V allentin, 2008 s 12 (7) = 134 . 7319671 . . . s 13 (7) = 134 . 5730609 . . . s 14 (7) = 134 . 4488169 . . . 8 240 240 s 11 (8) = 240 . 0000000 . . . [12] Odlyzk o, S loane, 19 79 [9] Lev enshtein, 1979 9 306 366 s 11 (9) = 365 . 3229274 . . . [1] Ba choc, V allentin, 2008 s 12 (9) = 364 . 7282746 . . . s 13 (9) = 364 . 3980087 . . . s 14 (9) = 364 . 0919287 . . . 10 500 567 s 11 (10) = 558 . 144281 3 . . . [1] Ba choc, V allentin, 2008 s 12 (10) = 556 . 284073 6 . . . s 13 (10) = 555 . 239902 4 . . . s 14 (10) = 554 . 5075418 . . . 11 582 915 s 11 (11) = 878 . 615804 4 . . . [12] Odlyzk o, S loane, 19 79 s 12 (11) = 873 . 379009 4 . . . s 13 (11) = 871 . 971853 3 . . . s 14 (11) = 870 . 8831157 . . . 4 H.D. Mi t t e lmann, F. V alle ntin 12 840 1416 s 11 (12) = 1364 . 68381 0 . . . [12] Odlyzk o, Sloane, 1979 s 12 (12) = 1362 . 20029 7 . . . s 13 (12) = 1359 . 28383 4 . . . s 14 (12) = 1357 . 889300 . . . 13 1130 2233 s 11 (13) = 2089 . 11633 1 . . . [12] Odlyzk o, Sloane, 1979 s 12 (13) = 2080 . 63151 8 . . . s 13 (13) = 2073 . 07479 6 . . . s 14 (13) = 2069 . 587585 . . . 14 1582 3492 s 11 (14) = 3224 . 95075 1 . . . [12] Odlyzk o, Sloane, 1979 s 12 (14) = 3202 . 44890 2 . . . s 13 (14) = 3189 . 12764 4 . . . s 14 (14) = 3183 . 133169 . . . 15 2564 5431 s 11 (15) = 4949 . 65043 1 . . . [12] Odlyzk o, Sloane, 1979 s 12 (15) = 4893 . 47944 6 . . . s 13 (15) = 4876 . 03722 9 . . . s 14 (15) = 4866 . 245659 . . . 16 4320 8312 s 11 (16) = 7515 . 95264 4 . . . [13] Pfender, 2007 s 12 (16) = 7432 . 72071 8 . . . s 13 (16) = 7374 . 09374 2 . . . s 14 (16) = 7355 . 809036 . . . 17 5346 12210 s 11 (17) = 11568 . 4167 4 . . . [13] Pfender, 2007 s 12 (17) = 11333 . 8426 5 . . . s 13 (17) = 11128 . 2622 7 . . . s 14 (17) = 11072 . 37543 . . . 18 7398 17877 s 11 (18) = 17473 . 4801 6 . . . [12] Odlyzk o, Sloane, 1979 s 12 (18) = 17034 . 3248 8 . . . s 13 (18) = 16686 . 2890 8 . . . s 14 (18) = 16572 . 26478 . . . 19 10668 25900 s 11 (19) = 26397 . 3479 4 . . . [3] Boyv alenk ov, 1994 s 12 (19) = 25636 . 9895 8 . . . s 13 (19) = 25029 . 8743 2 . . . s 14 (19) = 24812 . 30254 . . . 20 17400 37974 s 11 (20) = 39045 . 3276 1 . . . [12] Odlyzk o, Sloane, 1979 s 12 (20) = 37844 . 1038 0 . . . s 13 (20) = 37067 . 1896 6 . . . s 14 (20) = 36764 . 40138 . . . 21 27720 56851 s 11 (21) = 58087 . 0385 7 . . . [3] Boyv alenk ov, 1994 s 12 (21) = 56079 . 2168 5 . . . s 13 (21) = 55170 . 0344 9 . . . s 14 (21) = 54584 . 76757 . . . 22 49896 86537 s 11 (22) = 87209 . 0626 1 . . . [12] Odlyzk o, Sloane, 1979 s 12 (22) = 84922 . 0910 1 . . . s 13 (22) = 84117 . 9210 3 . . . s 14 (22) = 82340 . 08003 . . . 23 93150 128095 s 11 (23) = 128360 . 796 9 . . . [3] Boyv alenk ov, 1994 s 12 (23) = 127323 . 709 5 . . . s 13 (23) = 125978 . 765 5 . . . s 14 (23) = 124416 . 9796 . . . 24 1965 60 196560 s 11 (24) = 196560 . 000 0 . . . [12] Odlyzk o, Sloane, 1979 [9] Lev enshtein, 1979 T able 1. New u pp er b ounds f or the k issing n umber (b est known v alues are underlined). High acc uracy semide finite programming b ou nds for kissing number s 5 Finding the solution of the semidefinite pr ogram defined in Section 2 is a com- putational challenge. It tur ns out that the ma jor obsta cle is n umerica l instability and not the pro ble m size. When d is fi xed, t hen the size of t he input matrices stays constant with n ; when n is fixed, then it g rows rather mo der ately with d . There is a n umber of av aila ble softw are pack ages for so lving semidefinite pro- grams. Mittelmann compares many existing pa ck a ges in [10]. F or our pur po se first o rder, g radient-based methods such as SDPL R are far to o inaccur ate, and sec- ond o rder, primal-dual interior p oint m etho ds are more suitable. Here increasingly ill-conditioned linear systems hav e to b e solved even if the underlying problem is well-conditioned. This happ e ns in the final phase of the algor ithm when o ne ap- proaches an optimal so lution. O ur pro ble ms are not well-conditioned a nd even the most robust so lver SeDu Mi which uses partial qua druple ar ithmetic in the final phase do es not pro duce reliable r e s ults for d > 10 . W e thus had to fall back on the implementation SDPA-GM P [8] whic h is muc h slow er but muc h more accurate than other softw are pack ages b ecause it uses the GNU Multiple Precisio n Arithmetic Library . W e worked with 200–3 00 binary digits and r elative s topping criteria of 1 0 − 30 . The ten sig nificant digits listed in the table are thus guar a nteed to b e co rrect. One problem was the conv ergence. Ev en with the control para meter settings recommended by the author s of SDPA -GMP for “ slow but stable ” co mputations, the algor ithm faile d to converge in several instances. How ever, we found parameter choices which worked for all cas es: W e v aried the parameter la mbdaSt ar b etw een 10 0 and 10000 depending on the cas e while the other par ameters could b e c hosen a t or nea r the v alues recommended for “ slow but stable ” p er formance. The computations were done on Intel Cor e 2 platforms with one and tw o Q uad pro cessor s. T he computation time v aried b etw een five and ten weeks p er ca se fo r d = 12 . An a ccuracy of 128 bits in S DPA-GM P did yield s ufficient accur acy but did not yie ld a reduction in computing time. After the computatio ns for the ca ses d = 1 1 and d = 1 2 were finished new 12 8 - bit versions (quadr uple precision) of S DPA a nd CSDP b ecame av ailable; pa r tly with our assistance. These new versions do not rely on the GNU Multiple Precisio n Arithmetic Library . So the co mputation time fo r the cases d = 13 and d = 14 were reasona ble: fr om one week to tw o and a half weeks. 4. Question about the optimality of the D 4 root sy stem Lo oking at the v alues s d (4) in T able 1 one is led to the following question: Question 4 .1. Is lim d →∞ s d (4) = 24 ? If the answer to this question is yes (whic h a t the moment appea rs unlikely bec ause we computed s 15 (4) = 23 . 06274 835 . . . ), then it would hav e tw o noteworthy consequences ab out optimality prop er ties of the ro ot s y stem D 4 . The ro ot system D 4 defines (up to orthog onal transforma tions) a po int config - uration on the unit s phere S 3 = { x ∈ R 4 : x · x = 1 } co nsisting o f 24 p oints; it is the sa me p oint configur ation as the o ne co ming from the vertices of the r egular 24 cell. This p oint configur ation has the prop erty that the spheric a l distance o f every t wo distinct p oints is at least a rccos 1 / 2. Hence, these po ints can b e the maximal 24 touching p oints of unit spheres kissing the central unit s phere S 3 . 6 H.D. Mi t t e lmann, F. V alle ntin If lim d →∞ s d (4) = 24, then this would pr ov e that the r o ot s ystem D 4 is the unique optimal po in t configuratio n of cardinality 24. Here optimality means that one ca nnot distribute 24 p oints o n S 3 so tha t the minimal spherical dista nc e b e- t ween tw o distinct p oints exceeds a r ccos 1 / 2. Thus, the r o ot system D 4 would b e characterized by its kissing pr op erty . This is generally believed to b e tr ue but so far no pr o of c ould b e g iven. Another cons equence would b e that there is no universally optimal p oint con- figuration of 24 p oints in S 3 as conjectured b y Co hn, Co nw a y , Elkies, Kumar [5]. Univ ers ally optimal po int config urations minimize every a bsolutely monotonic p o- ten tial function. The co njecture would follow if the a ns wer to our question is yes: Every universally optimal po int configura tion is automatically optimal and Co hn, Conw ay , Elkies, Kumar [5] show that the ro ot s ystem D 4 is not universally optimal. 5. Nonexistence of a sphere p acking Our new upp er b ound of 7355 for the kissing num b er in dimension 16 implies that there is no p erio dic p oint set in dimension 16 whos e av era ge theta series equals (1) 1 + 7680 q 3 + 4320 q 4 + 27648 0 q 5 + 61440 q 6 + · · · . This settles a conjecture of Conw ay a nd Slo ane [6 , Chapter 7, page 190]. In this section we explain this result. W e refer to Conw ay , Sloa ne [6 ], E lkies [7], a nd to Bow ert [2] for more infor ma tion. An n -dimensional p erio dic p oint set Λ is a finite union o f translates o f an n - dimensional la ttice, i.e. one can wr ite Λ a s Λ = ( A Z n + v 1 ) ∪ . . . ∪ ( A Z n + v N ), with v 1 , . . . , v N ∈ R n , and A : R n → R n is a linea r isomorphis m. The aver age theta series of Λ is Θ Λ ( z ) = 1 N N X i =1 N X j =1 X v ∈ Z n q k Av − v i + v j k 2 , with q = e π iz . This is a holomorphic function defined on the complex upp er half-plane. A holo - morphic function f whic h is defined o n the complex upper half-plane, which is meromorphic for z → i ∞ , a nd which s atisfies the tra nsformation laws f ( − 1 /z ) = z 8 f ( z ) , a nd f ( z + 2) = f ( z ) for all z ∈ C with ℑ z > 0 , is called a mo dular forms of weigh t 8 for the Heck e g r oup G (2). The expr e s sion (1) defines the unique mo dular form of weigh t 8 for the Heck e gr oup G (2) which starts off with 1 + 0 q 1 + 0 q 2 . It is also calle d an extr emal mo dular form , see Scharlau and Sch ulze-Pillot [15] If there would be a 16-dimensio nal p erio dic po int set whose av erag e theta ser ies coincides with (1) then this p erio dic p oint set would define the spher e centers of a spher e pa cking with extra ordinar y high density (see [6, Chapter 7, page 190]). A t the sa me time the existence o f such a p erio dic po in t s et would show that the kissing num b er in dimensio n 16 is a t least 7680 . This is not the ca se. Ackno wledgements W e thank Etienne de Klerk and Renata So tirov for initiating our colla bo ration and we thank F rank Bow ert a nd Rudolf Sc har lau for bringing the conjecture of Conw ay and Sloane to our attention. High acc uracy semide finite programming b ou nds for kissing number s 7 References [1] C. Bachoc, F. V allentin, New upp er b ounds for kissing numb ers fr om semidefinite pr o g r am- ming , J. Amer. Math. So c. 21 (2008), 909–924. [2] F. 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Soc. 51 (2004), 873–883. [15] R. Sc harlau, R. Sch ulze-Pillot, Ext rema l lattic es , p. 139–170 i n A lgorithmic alge bra and n umber theory (Heidelb erg, 1997) , Spr inger, 1999. [16] K. Sc h ¨ utt e, B.L. v an der W aerden, Das Pr oblem der dr eizehn Kugeln , M ath. A nn. 125 (1953) 325–334 . H.D. Mittelmann, School of Ma thema tical and St a tistical Sciences, Arizona St a te University, Tempe, AZ 852 87-1804, USA E-mail addr ess : mittelmann@ asu.edu F. V allentin, Delft Institute of Applied Ma thema tics, Technical University of Delft, P.O. Box 50 31, 2600 GA Delft, The Netherla nds E-mail addr ess : f.vallentin @tudelft.nl