The kissing number in n-dimensional Euclidean space is the maximal number of
non-overlapping unit spheres which simultaneously can touch a central unit
sphere. Bachoc and Vallentin developed a method to find upper bounds for the
kissing number based on semidefinite programming. This paper is a report on
high accuracy calculations of these upper bounds for n <= 24. The bound for n =
16 implies a conjecture of Conway and Sloane: There is no 16-dimensional
periodic point set with average theta series 1 + 7680q^3 + 4320q^4 + 276480q^5
+ 61440q^6 + ...Comment: 7 pages (v3) new numerical result in Section 4, to appear in
  Experiment. Mat

Mittelmann, Hans D.

Vallentin, Frank

English

arXiv

The kissing number in n-dimensional Euclidean space is the maximal number of non-overlapping unit spheres which simultaneously can touch a central unit sphere. Bachoc and Vallentin developed a method to find upper bounds for the kissing number based on semidefinite programming. This paper is a report on high accuracy calculations of these upper bounds for n <= 24. The bound for n = 16 implies a conjecture of Conway and Sloane: There is no 16-dimensional periodic point set with average theta series 1 + 7680q^3 + 4320q^4 + 276480q^5 + 61440q^6 + ..

Mittelmann, H.D.

Vallentin, F. (Frank)

CWI's Institutional Repository

High-Accuracy Semidefinite Programming Boundsfor Kissing NumbersHans D. Mittelmann and Frank VallentinCONTENTS1. Introduction2. Notation3. Bounds for Kissing Numbers4. Question about the Optimality of theD4 Root System5. Nonexistence of a Sphere PackingAcknowledgmentsReferences2000 AMS Subject Classification: 11F11, 52C17, 90C10Keywords: kissing number, semidefinite programming,average theta series, extremal modular formThe kissing number in n-dimensional Euclidean space is themaximal number of nonoverlapping unit spheres that simulta-neously can touch a central unit sphere. Bachoc and Vallentindeveloped a method to find upper bounds for the kissing num-ber based on semidefinite programming. This paper is a re-port on high-accuracy calculations of these upper bounds forn ≤ 24. The bound for n = 16 implies a conjecture of Conwayand Sloane: there is no 16-dimensional periodic sphere pack-ing with average theta series 1 + 7680q3 + 4320q4 + 276480q5+ 61440q6 + · · · .1. INTRODUCTIONIn geometry, the kissing number in n-dimensional Eu-clidean space is the maximal number of nonoverlappingunit spheres that simultaneously can touch a central unitsphere. The kissing number is known only in dimensionsn = 1, 2, 3, 4, 8, 24, and there have been many attemptsto find good lower and upper bounds. We refer to [Cas-selman 04] for the history of this problem and to [Pfenderand Ziegler 04, Elkies 00, Conway and Sloane 99] for morebackground information on sphere-packing problems.In [Bachoc and Vallentin 08] a method is developed(Section 2 recalls it) to find upper bounds for the kissingnumber based on semidefinite programming. Table 1,the main contribution of this paper, gives the values—the first ten significant digits—of these upper bounds forall dimensions 3, . . . , 24. In all cases they are the bestknown upper bounds. Dimension 5 is the first dimensionin which the kissing number is not known.With our computation we could limit the range of pos-sible values from {40, . . . , 45} to {40, . . . , 44}. In Sec-tion 4 we show that the high-accuracy computations forthe upper bounds in dimension 4 lead to a question abouta possible approach to proving the uniqueness of the kiss-ing configuration in four dimensions.c© A K Peters, Ltd.1058-6458/2010 $ 0.50 per pageExperimental Mathematics 19:2, page 175176 Experimental Mathematics, Vol. 19 (2010), No. 2Although acquiring the data for the table is a purelycomputational task, we think that providing this tableis valuable for several reasons: The kissing number is animportant constant in geometry and results can dependon good upper bounds for it. For instance, in Section 5 weshow that there is no periodic point set in dimension 16with average theta series1 + 7680q3 + 4320q4 + 276480q5 + 61440q6 + · · · .This proves a conjecture from [Conway and Sloane 99,p. 190]. Furthermore, the actual computation of the ta-ble was very challenging. Results are given in [Bachocand Vallentin 08] for dimensions 3, . . . , 10. However, theauthors report on numerical difficulties that preventedthem from extending their results. Now using new, moresophisticated, high-accuracy software and faster comput-ers and more computation time, we were able to overcomesome of the numerical difficulties. Section 3 contains de-tails about the computations.2. NOTATIONIn this section we set up the notation that is needed forour computation. For more information we refer to [Ba-choc and Vallentin 08]. For natural numbers d and n ≥ 3let sd(n) be the optimal value of the following minimiza-tion problem:Minimize 1 +d∑k=1ak + b11 + 〈F0, Sn0 (1, 1, 1)〉subject to the following conditions:a1, . . . , ad ∈ R, a1, . . . , ad ≥ 0,b11, b12, b22 ∈ R,(b11 b12b12 b22)is positive semidefinite,Fk ∈ R(d+1−k)×(d+1−k), Fk is positive semidefinite,k = 0, . . . , d,q, q1 ∈ R[u], deg(p + pq1) ≤ d, p, p1 sums of squares,r, r1, . . . , r4 ∈ R[u, v, t], deg(r +4∑i=1piri)≤ d,r, r1, . . . , r4 sums of squares,1 +d∑k=1akPnk (u) + 2b12 + b22 + 3d∑k=0〈Fk, Snk (u, u, 1)〉+ q(u) + p(u)q1(u) = 0,b22 +d∑k=0〈Fk, Snk (u, v, t)〉+ r(u, v, t)+4∑i=1pi(u, v, t)ri(u, v, t) = 0.Here Pnk is the normalized Jacobi polynomial of degreek with Pnk (1) = 1 and parameters ((n− 3)/2, (n− 3)/2).In general, Jacobi polynomials with parameters (α, β)are orthogonal polynomials for the measure (1−u)α(1+u)β du on the interval [−1, 1]. Before we can define thematrices Snk , we first define the entry (i, j) with i, j ≥ 0of the (infinite) matrix Y nk containing polynomials in thevariables u, v, w by(Y nk)i,j(u, v, t) = uivj((1− u2)(1− v2))k/2Pn−1k×(t− uv√(1− u2)(1− v2)).Then we get Snk by symmetrization: Snk =16∑σ σYnk ,where σ runs through all permutations of the variablesu, v, t, which acts on the matrix coefficients in the obviousway. The polynomials p, p1, . . . , p4 are given byp(u) = −(u + 1)(u + 1/2),p1(u, v, t) = p(u), p2(u, v, t) = p(v), p3(u, v, t) = p(t),p4(u, v, t) = 1 + 2uvt− u2 − v2 − t2.By 〈A,B〉 we denote the inner product between symmet-ric matrices trace(AB).In [Bachoc and Vallentin 08] it is shown that this min-imization problem is a semidefinite program and that ev-ery upper bound on sd(n) provides an upper bound forthe kissing number in dimension n. Clearly, the numberssd(n) form a monotonic decreasing sequence in d.3. BOUNDS FOR KISSING NUMBERSFinding the solution of the semidefinite program definedin Section 2 is a computational challenge. It turns outthat the major obstacle is numerical instability and notthe problem size. When d is fixed, the size of the inputmatrices stays constant with n; when n is fixed, it growsrather moderately with d.There are several software packages available for solv-ing semidefinite programs. Many existing packages arecompared in [Mittelmann 03]. For our purpose, first-order gradient-based methods such as SDPLR are fartoo inaccurate, and second-order primal-dual interior-point methods are more suitable. Here increasingly ill-conditioned linear systems have to be solved even if theunderlying problem is well conditioned. This happensin the final phase of the algorithm when one approachesan optimal solution. Our problems are not well condi-tioned, and even the most robust solver, SeDuMi, whichMittelmann and Vallentin: High-Accuracy Semidefinite Programming Bounds for Kissing Numbers 177Best Lower Best Upper Bound SDPn Bound Known Previously Known Bound3 12 12 s11(3) = 12.42167009 . . . , s12(3) = 12.40203212 . . .[Schu¨tte and van der Waerden 53] s13(3) = 12.39266509 . . . , s14(3) = 12.38180947 . . .4 24 24 s11(4) = 24.10550859 . . . , s12(4) = 24.09098111 . . .[Musin 08] s13(4) = 24.07519774 . . . , s14(4) = 24.06628391 . . .5 40 45 s11(5) = 45.06107293 . . . , s12(5) = 45.02353644 . . .[Bachoc and Vallentin 08] s13(5) = 45.00650838 . . . , s14(5) = 44.99899685 . . .6 72 78 s11(6) = 78.58344077 . . . , s12(6) = 78.35518719 . . .[Bachoc and Vallentin 08] s13(6) = 78.29404232 . . . , s14(6) = 78.24061272 . . .7 126 135 s11(7) = 134.8824614 . . . , s12(7) = 134.7319671 . . .[Bachoc and Vallentin 08] s13(7) = 134.5730609 . . . , s14(7) = 134.4488169 . . .8 240 240 s11(8) = 240.0000000 . . .[Odlyzko 79][Levenshtein 79]9 306 366 s11(9) = 365.3229274 . . . , s12(9) = 364.7282746 . . .[Bachoc and Vallentin 08] s13(9) = 364.3980087 . . . , s14(9) = 364.0919287 . . .10 500 567 s11(10) = 558.1442813 . . . , s12(10) = 556.2840736 . . .[Bachoc and Vallentin 08] s13(10) = 555.2399024 . . . , s14(10) = 554.5075418 . . .11 582 915 s11(11) = 878.6158044 . . . , s12(11) = 873.3790094 . . .[Odlyzko 79] s13(11) = 871.9718533 . . . , s14(11) = 870.8831157 . . .12 840 1416 s11(12) = 1364.683810 . . . , s12(12) = 1362.200297 . . .[Odlyzko 79] s13(12) = 1359.283834 . . . , s14(12) = 1357.889300 . . .13 1154 2233 s11(13) = 2089.116331 . . . , s12(13) = 2080.631518 . . .[Odlyzko 79] s13(13) = 2073.074796 . . . , s14(13) = 2069.587585 . . .14 1606 3492 s11(14) = 3224.950751 . . . , s12(14) = 3202.448902 . . .[Odlyzko 79] s13(14) = 3189.127644 . . . , s14(14) = 3183.133169 . . .15 2564 5431 s11(15) = 4949.650431 . . . , s12(15) = 4893.479446 . . .[Odlyzko 79] s13(15) = 4876.037229 . . . , s14(15) = 4866.245659 . . .16 4320 8312 s11(16) = 7515.952644 . . . , s12(16) = 7432.720718 . . .[Pfender 07] s13(16) = 7374.093742 . . . , s14(16) = 7355.809036 . . .17 5346 12210 s11(17) = 11568.41674 . . . , s12(17) = 11333.84265 . . .[Pfender 07] s13(17) = 11128.26227 . . . , s14(17) = 11072.37543 . . .18 7398 17877 s11(18) = 17473.48016 . . . , s12(18) = 17034.32488 . . .[Odlyzko 79] s13(18) = 16686.28908 . . . , s14(18) = 16572.26478 . . .19 10668 25900 s11(19) = 26397.34794 . . . , s12(19) = 25636.98958 . . .[Boyvalenkov 94] s13(19) = 25029.87432 . . . , s14(19) = 24812.30254 . . .20 17400 37974 s11(20) = 39045.32761 . . . , s12(20) = 37844.10380 . . .[Odlyzko 79] s13(20) = 37067.18966 . . . , s14(20) = 36764.40138 . . .21 27720 56851 s11(21) = 58087.03857 . . . , s12(21) = 56079.21685 . . .[Boyvalenkov 94] s13(21) = 55170.03449 . . . , s14(21) = 54584.76757 . . .22 49896 86537 s11(22) = 87209.06261 . . . , s12(22) = 84922.09101 . . .[Odlyzko 79] s13(22) = 84117.92103 . . . , s14(22) = 82340.08003 . . .23 93150 128095 s11(23) = 128360.7969 . . . , s12(23) = 127323.7095 . . .[Boyvalenkov 94] s13(23) = 125978.7655 . . . , s14(23) = 124416.9796 . . .24 196560 196560 s11(24) = 196560.0000 . . .[Odlyzko 79][Levenshtein 79]TABLE 1. New upper bounds for the kissing number (best known values are underlined).uses partial quadruple arithmetic in the final phase, doesnot produce reliable results for d > 10.We thus had to fall back on the implementationSDPA-GMP [Fujisawa et al. 08], which is much slower butmuch more accurate than other software packages be-cause it uses the GNU Multiple Precision Arithmetic Li-brary. We worked with 200 to 300 binary digits andrelative stopping criteria of 10−30. The ten significantdigits listed in the table are thus guaranteed to be cor-rect. One problem was convergence. Even with the178 Experimental Mathematics, Vol. 19 (2010), No. 2control parameter settings recommended by the authorsof SDPA-GMP for “slow but stable” computations, thealgorithm failed to converge in several instances. How-ever, we found parameter choices that worked for allcases: We varied the parameter lambdaStar between 100and 10,000 depending on the case, while the other param-eters could be chosen at or near the values recommendedfor “slow but stable” performance.The computations were done on Intel Core 2 plat-forms with one and two Quad processors. The compu-tation time varied between five and ten weeks per casefor d = 12. An accuracy of 128 bits in SDPA-GMP didyield sufficient accuracy but did not yield a reduction incomputing time.After the computations for the cases d = 11 andd = 12 were finished, new 128-bit versions (quadrupleprecision) of SDPA and CSDP became available, partlywith our assistance. These new versions do not rely onthe GNU Multiple Precision Arithmetic Library. So thecomputation times for the cases d = 13 and d = 14 werereasonable: from one week to two and a half weeks.4. QUESTION ABOUT THE OPTIMALITY OF THED4 ROOT SYSTEMLooking at the values sd(4) in Table 1, one is led to thefollowing question:Question 4.1. Is limd→∞ sd(4) = 24?If the answer to this question is yes (which at the mo-ment appears unlikely because we computed s15(4) =23.06274835 . . .), then it would have two noteworthy con-sequences about optimality properties of the root sys-tem D4.The root system D4 defines (up to orthogonal trans-formations) a point configuration on the unit sphereS3 = {x ∈ R4 : x · x = 1} consisting of 24 points; it isthe same point configuration as the one coming from thevertices of the regular 24 cell. This point configurationhas the property that the spherical distance of every twodistinct points is at least arccos 12 . Hence, these pointscan be the maximal 24 touching points of unit sphereskissing the central unit sphere S3.If limd→∞ sd(4) = 24, then this would prove that theroot system D4 is the unique optimal point configurationof cardinality 24. Here optimality means that one can-not distribute 24 points on S3 in such a way that theminimal spherical distance between two distinct pointsexceeds arccos 12 . Thus, the root system D4 would becharacterized by its kissing property. This is generallybelieved to be true, but so far, no proof has been given.Another consequence would be that there is no uni-versally optimal point configuration of 24 points in S3 asconjectured in [Cohn et al. 07]. Universally optimal pointconfigurations minimize every absolutely monotonic po-tential function. The conjecture will follow if the answerto our question is yes: every universally optimal pointconfiguration is automatically optimal, and it is shownin [Cohn et al. 07] that the root system D4 is not univer-sally optimal.5. NONEXISTENCE OF A SPHERE PACKINGOur new upper bound of 7355 for the kissing number indimension 16 implies that there is no periodic point setin dimension 16 whose average theta series equals1 + 7680q3 +4320q4 +276480q5 +61440q6 + · · · . (5–1)This settles a conjecture from [Conway and Sloane 99,p. 190]. In this section we explain this result. We referto [Conway and Sloane 99, Elkies 00, Bowert 04] for moreinformation.An n-dimensional periodic point set Λ is a finite unionof translates of an n-dimensional lattice, i.e., one canwrite Λ as Λ = (AZn + v1) ∪ · · · ∪ (AZn + vN ), withv1, . . . , vN ∈ Rn, and A : Rn → Rn is a linear isomor-phism. The average theta series of Λ isΘΛ(z) =1NN∑i=1N∑j=1∑v∈Znq‖Av−vi+vj‖2, with q = eπiz.This is a holomorphic function defined on the complexupper half-plane. A holomorphic function f that is de-fined on the complex upper half-plane is meromorphic forz → i∞ and satisfies the transformation lawsf(−1z)= z8f(z),f(z + 2) = f(z) for all z ∈ C with z > 0,is called a modular form of weight 8 for the Hecke groupG(2). The expression (5–1) defines the unique modularform of weight 8 for the Hecke group G(2) that begins1+0q1 +0q2. It is also called an extremal modular form;see [Scharlau and Schulze-Pillot 99].If there were a 16-dimensional periodic point set whoseaverage theta series coincided with (5–1), then this peri-odic point set would define the sphere centers of a spherepacking with extraordinarily high density (see [Conwayand Sloane 99, p. 190]). At the same time, the existenceMittelmann and Vallentin: High-Accuracy Semidefinite Programming Bounds for Kissing Numbers 179of such a periodic point set would show that the kissingnumber in dimension 16 was at least 7680. This is notthe case.ACKNOWLEDGMENTSWe thank Etienne de Klerk and Renata Sotirov for initiat-ing our collaboration. We thank Frank Bowert and RudolfScharlau for bringing the conjecture of Conway and Sloaneto our attention. We thank Maho Nakata for developing andsupporting SDPA-GMP, and we thank Peter Boyvalenkov for hisfeedback on the paper.The second author was partially supported by theDeutsche Forschungsgemeinschaft (DFG) under grant SCHU1503/4.REFERENCES[Bachoc and Vallentin 08] C. Bachoc and F. Vallentin. “NewUpper Bounds for Kissing Numbers from Semidefinite Pro-gramming.” J. Amer. Math. Soc. 21 (2008), 909–924.[Bowert 04] F. Bowert. “Gewichtsza¨hler und Distanzza¨hlervon Codes und Kugelpackungen.” PhD thesis, Universityof Dortmund, 2004.[Boyvalenkov 94] P. Boyvalenkov. “Small Improvements ofthe Upper Bounds of the Kissing Numbers in Dimensions19, 21 and 23.” Atti Sem. Mat. Fis. Univ. Modena 42(1994), 159–163.[Casselman 04] B. Casselman, “The Difficulties of Kissing inThree Dimensions.” Notices Amer. Math. Soc. 51 (2004),884–885.[Cohn et al. 07] H. Cohn, J. H. Conway, N. D. Elkies, andA. Kumar. “The D4 Root System Is Not Universally Opti-mal.” Exp. Math. 16 (2007), 313–320.[Conway and Sloane 99] J. H. Conway and N. J. A. Sloane.Sphere Packings, Lattices and Groups, third edition. NewYork: Springer, 1999.[Elkies 00] N. D. Elkies. “Lattices, Linear Codes, and In-variants, Parts I/II.” Notices Amer. Math. Soc. 47 (2000),1238–1245, 1382–1391.[Fujisawa et al. 08] K. Fujisawa, M. Fukuda, K. Kobayashi,M. Kojima, K. Nakata, M. Nakata, and M. Ya-mashita. “SDPA (SemiDefinite Programming Algorithm)and SDPA-GMP User’s Manual—version 7.1.1.” ResearchReport B-448, Department of Mathematical and Comput-ing Sciences, Tokyo Institute of Technology, June 2008.[Levenshtein 79] V. I. Levenshtein. “On Bounds for Packingin n-Dimensional Euclidean Space.” Soviet Math. Dokl. 20(1979), 417–421.[Mittelmann 03] H. D. Mittelmann. “An Independent Bench-marking of SDP and SOCP Solvers.”Math. Prog. 95 (2003),407–430.[Musin 08] O. R. Musin. “The Kissing Number in Four Di-mensions.” Ann. of Math. 168 (2008), 1–32.[Odlyzko 79] A. M. Odlyzko and N. J. A. Sloane. “NewBounds on the Number of Unit Spheres That Can Touch aUnit Sphere in n Dimensions.” J. Combin. Theory Ser. A26 (1979), 210–214.[Pfender 07] F. Pfender. “Improved Delsarte Bounds forSpherical Codes in Small Dimensions.” J. Combin. TheorySer. A 14 (2007), 1133–1147.[Pfender and Ziegler 04] F. Pfender and G. M. Ziegler. “Kiss-ing Numbers, Sphere Packings and Some UnexpectedProofs.” Notices Amer. Math. Soc. 51 (2004), 873–883.[Scharlau and Schulze-Pillot 99] R. Scharlau and R. Schulze-Pillot. “Extremal Lattices.” In Algorithmic Algebra andNumber Theory (Heidelberg, 1997), pp. 139–170. New York:Springer, 1999.[Schu¨tte and van der Waerden 53] K. Schu¨tte and B. L. vander Waerden. “Das Problem der dreizehn Kugeln.” Math.Ann. 125 (1953) 325–334.Hans D. Mittelmann, School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA(mittelmann@asu.edu)Frank Vallentin, Delft Institute of Applied Mathematics, Technical University of Delft, P.O. Box 5031, 2600 GA Delft,The Netherlands (f.vallentin@tudelft.nl)Received February 10, 2009; accepted June 24, 2009.

High accuracy semidefinite programming bounds for kissing numbers

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