This article was downloaded by: [Nipissing University]On: 08 October 2014, At: 14:24Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Experimental MathematicsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/uexm20

New Bounds for Spherical Two-Distance SetsAlexander Barg a & Wei-Hsuan Yu ba Department of Electrical and Computer Engineering and Institute for Systems Research ,University of Maryland , College Park , MD , 20742b Department of Mathematics and Institute for Systems Research , University of Maryland ,College Park , MD , 20742Published online: 25 Apr 2013.

To cite this article: Alexander Barg & Wei-Hsuan Yu (2013) New Bounds for Spherical Two-Distance Sets, ExperimentalMathematics, 22:2, 187-194, DOI: 10.1080/10586458.2013.767725

To link to this article: http://dx.doi.org/10.1080/10586458.2013.767725

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Experimental Mathematics, 22(2):187–194, 2013Copyright C© Taylor & Francis Group, LLCISSN: 1058-6458 print / 1944-950X onlineDOI: 10.1080/10586458.2013.767725

New Bounds for Spherical Two-Distance SetsAlexander Barg and Wei-Hsuan Yu

CONTENTS

1. Introduction2. Positive Definite Matrices and SDP Bounds3. The BoundsAcknowledgmentsReferences

2000 AMS Subject Classification: 52C35, 94B75Keywords: two-distance sets, positive definite matrices,semidefinite programming

A spherical two-distance set is a finite collection of unit vectors inR n such that the distance between two distinct vectors assumesone of only two values. We use the semidefinite programmingmethod to compute improved estimates of the maximum sizeof spherical two-distance sets. Exact answers are found for di-mensions n = 23 and 40 ≤ n ≤ 93 (n �= 46, 78), where previousresults gave divergent bounds.

1. INTRODUCTION

This paper is devoted to the application of the semidefi-nite programming method to estimates of the size of thelargest possible two-distance set on the sphere Sn−1(R ).A spherical two-distance set is a finite collection C of unitvectors in R n such that the set of distances between twodistinct vectors in C has cardinality two. Estimating themaximum size g(n) of such a set is a classical problemin distance geometry that has been studied for severaldecades.

We begin with an overview of known results. A lowerbound on g(n) is obtained as follows. Let e1 , . . . , en+1

be the standard basis in R n+1. The points ei + ej , i �= j,form a spherical two-distance set in the plane x1 + · · · +xn+1 = 2 (after scaling), and therefore,

g(n) ≥ n(n + 1)2

, n ≥ 2. (1–1)

The first major result for upper bounds was obtainedin [Delsarte et al. 77]. The authors proved that irrespec-tive of the actual values of the distances, the following“harmonic” bound holds:

g(n) ≤ n(n + 3)2

. (1–2)

They also showed that this bound is tight for dimen-sions n = 2, 6, 22, in which cases it is related to setsof equiangular lines in dimension n + 1. Moreover,the results of [Delsarte et al. 77, Bannai et al. 04,Nebe and Venkov 12] imply that g(n) can attain theharmonic bound only if n = (2m + 1)2 − 3, m ≥ 1, withthe exception of an infinite sequence of values of m that

187

Dow

nloa

ded

by [

Nip

issi

ng U

nive

rsity

] at

14:

24 0

8 O

ctob

er 2

014

188 Experimental Mathematics, Vol. 22 (2013), No. 2

begins with m = 3, 4, 6, 10, 12, 22, 38, 30, 34, 42, 46.Therefore, unless n is of the above form, we haveg(n) ≤ n(n + 3)/2 − 1. These results are proved usingthe link between two-distance sets and tight spherical4-designs established in [Delsarte et al. 77].

Another advance in estimating the function g(n) wasmade in [Musin 09]. Let C = {z1 , z2 , . . . } and supposethat zi · zj ∈ {a, b}, i �= j, where 2 − 2a, 2 − 2b are thevalues of the squared distances between the points. Musinproved that

|C| ≤ n(n + 1)2

if a + b ≥ 0. (1–3)

He then used Delsarte’s linear programming method toprove that g(n) = n(n + 1)/2 if 7 ≤ n ≤ 39, n �= 22, 23.

Here we make another step for spherical two-distancesets, extending the range of dimensions in which thebound (1–1) is tight. The state of the art for g(n) canbe summarized as follows.

Theorem 1.1. We have g(2) = 5, g(3) = 6, g(4) = 10,g(5) = 16, g(6) = 27, g(22) = 275, as well as

n(n + 1)2

≤ g(n) ≤ n(n + 3)2

− 1, for n = 46, 78,

g(n) =n(n + 1)

2, for 7 ≤ n ≤ 93, n �= 22, 46, 78,

(1–4)

and 4465 ≤ g(94) ≤ 4492. If n ≥ 95, then

g(n) ≤ n(n + 3)2

orn(n + 3)

2− 1,

as detailed in the remarks after (1–2) above.

The part of this theorem that is established in thepresent paper relates to dimensions n = 23 and 40 ≤ n ≤94, n �= 46, 78. Our results are computational in natureand are obtained using the semidefinite programmingmethod. The other parts of this theorem follow fromresults in [Delsarte et al. 77, Bannai et al. 04, Musin 09,Nebe and Venkov 12].

As far as actual constructions of spherical two-distancesets are concerned, rather little is known beyond theset of midpoints of the edges of a regular simplex,as mentioned above. Another way of constructing suchsets is to start with a set of equiangular lines in R n

[Lemmens and Seidel 73]. If the angle between each pairof lines is α, then taking one point from each pair ofpoints on Sn−1 defined by the line, we obtain a two-distance set with a = α, b = −α. The largest possiblenumber of equiangular lines in R n is n(n + 1)/2 (this

result is due to Gerzon; see [Lemmens and Seidel 73]).This bound is attained for n = 3, 7, 23. For instance, forn = 3, the set of six lines is obtained from six diago-nals of the icosahedron, which gives many ways of con-structing inequivalent spherical two-distance sets of car-dinality 6. The only three instances in which the knownspherical two-distance sets are of cardinality greater thann(n + 1)/2 occur in dimensions n = 2, 6, 22.

2. POSITIVE DEFINITE MATRICES AND SDPBOUNDS

A semidefinite program (SDP) is an optimization prob-lem of the form

max{ 〈X,C〉 | X 0, 〈X,Ai〉 = bi, i = 1, . . . ,m

},

(2–1)

where X is an n × n variable matrix, A1 , . . . , Am andC are given Hermitian matrices, (b1 , . . . , bm ) is a givenvector, and 〈X,Y 〉 = trace(Y ∗X) is the inner product oftwo matrices. Semidefinite programming is an extensionof linear programming that has found a range of applica-tions in combinatorial optimization, control theory, dis-tance geometry, and coding theory. A general introduc-tion to semidefinite programming is given, for instance,in [Ben-Tal and Nemirovski 01].

The main problem addressed by the SDP method indistance geometry is related to deriving bounds on thecardinality of point sets in a metric space X with a givenset of properties such as a given minimum separationbetween distinct points in the set. The SDP method hasits roots in harmonic analysis of the isometry group ofthe metric space in question. It is broadly applicable inboth finite and compact infinite spaces.

Examples of the former include the Hamming andJohnson spaces, their q-analogues, other metric spaces onthe set of n-strings over a finite alphabet, and the finiteprojective space. The main example in the infinite caseis given by real and complex spheres, although the SDPmethod is also applicable in other compact homogeneousspaces.

Working out the details in each example is a nontriv-ial task that includes analysis of irreducible modules inthe space of functions f : X → C under the action of theisometry group G of X . The zonal matrices that arisein this analysis initially have large size that can be re-duced by relying on symmetries arising from the groupaction. This gives rise to an SDP optimization problemthat is solved by computer for a given set of dimensions

Dow

nloa

ded

by [

Nip

issi

ng U

nive

rsity

] at

14:

24 0

8 O

ctob

er 2

014

Barg and Yu: New Bounds for Spherical Two-Distance Sets 189

(the numerical part is also not straightforward and rathertime-consuming).

Foundations and analysis of particular cases have beenthe subject of a considerable number of research andoverview publications in the last decade; see in particu-lar the recent surveys [Bachoc et al. 12, Bachoc 09] andreferences therein.

The origins of the SDP method and the discussedapplications can be traced back to [Delsarte 73], whichintroduced the machinery of association schemes in theanalysis of point configurations (codes) in finite spaces.Delsarte derived linear programming (LP) bounds on thecardinality of a set of points in the space under conditionson the minimum separation of distinct points in the set.Delsarte’s results were linked to harmonic analysis andgroup representations in [Delsarte et al. 77] (for the caseSn−1) and [Kabatyansky and Levenshtein 78] (for gen-eral compact symmetric spaces).

From now on, we focus on the case X = Sn−1 . LetG

(n)k (t), k = 0, 1, . . . , denote the Gegenbauer polynomi-

als of degree k. They are defined recursively as follows:G

(n)0 ≡ 1, G

(n)1 (t) = t, and

G(n)k (t) =

(2k + n − 4)tG(n)k−1(t) − (k − 1)G(n)

k−2(t)k + n − 3

,

for k ≥ 2. It is shown in [Delsarte et al. 77] that for everyfinite set of points C ⊂ Sn−1 ,∑

(x,y )∈C2

G(n)k (x · y) ≥ 0, k = 1, 2, . . . . (2–2)

The proof of this inequality in [Delsarte et al. 77] usedthe addition formula for spherical harmonics. An ear-lier, geometric, proof of (2–2) had been given in[Schoenberg 42], although that work was not known toresearchers in the area discussed until at least the 1990s.

Positivity conditions (2–2) give rise to the LP boundon the cardinality of spherical two-distance sets.

Theorem 2.1. [Delsarte et al. 77] Let C ⊂ Sn−1 be a finiteset and suppose that x · y ∈ {a, b} for all x, y ∈ C. Then

|C| ≤ max{1 + α1 + α2 | 1 + α1G

(n)i (a) + α2G

(n)i (b) ≥ 0,

i = 0, 1, . . . , p;αj ≥ 0, j = 1, 2}.

In this theorem, α1 , α2 are the optimization variablesthat refer to the number of ordered pairs of points in Cwith inner products a and b, respectively. For instance,

α1 = |C|−1�{

(z1 , z2) ∈ C2 | z1 · z2 = a},

This theorem is a specialization of a more generalLP bound on spherical codes of [Delsarte et al. 77,Kabatyansky and Levenshtein 78].

Applications of semidefinite programming in codingtheory and distance geometry gained momentum afterthe pioneering work [Schrijver 05], which derived SDPbounds on codes in the Hamming and Johnson spaces.Schijver’s approach was based on the so-called Terwilligeralgebra of the association scheme and formed a far-reaching generalization of [Delsarte 73].

Elements of the groundwork for SDP bounds in theHamming space were laid in [Dunkl 76], although thisconnection was also made somewhat later [Vallentin 09].We refer to [Martin and Tanaka 09] for a detailed gen-eral survey of the approach via association schemes andfurther references.

SDP bounds for the real sphere were derived in[Bachoc and Vallentin 08] in the context of the kissingnumber problem. The kissing number k(n) is the max-imum number of unit spheres that can touch a unitsphere without overlapping, i.e., the maximum numberof points on the sphere such that the angular sepa-ration between any pair of them is at least π/3. Fol-lowing [Bachoc and Vallentin 08], define a (p − k + 1) ×(p − k + 1) matrix Y n

k (u, v, t), k ≥ 0, by setting

(Y nk (u, v, t))ij = uivj ((1 − u2)(1 − v2))k/2

× G(n−1)k

(t − uv√

(1 − u2)(1 − v2)

),

where p is a positive integer, and a matrix Snk (u, v, t) by

setting

Snk (u, v, t) =

16

∑σ

Y nk

(σ(u, v, t)

), (2–3)

where the sum is over all permutations on three elements.Note that (Sn

k (1, 1, 1))ij = 0 for all i, j and all k ≥ 1. Oneof the main results of [Bachoc and Vallentin 08] is thatfor every finite set of points C ⊂ Sn−1 ,∑

(x,y ,z )∈C3

Snk (x · y, x · z, y · z) 0. (2–4)

The matrices Snk play the role of the constraints Ai in

the general SDP problem (2–1).Positivity constraints (2–3) give rise to a general SDP

bound on the cardinality of the point sets obtained in[Bachoc and Vallentin 08], where it was used to improveupper bounds on k(n) in small dimensions. In the nextsection, we state a specialization of this bound for thecase of two-distance sets.

Dow

nloa

ded

by [

Nip

issi

ng U

nive

rsity

] at

14:

24 0

8 O

ctob

er 2

014

190 Experimental Mathematics, Vol. 22 (2013), No. 2

As a final remark, we note that constraints (2–2) arisefrom the unrestricted action of G on Sn−1 . Constraints(2–3) are obtained by considering only actions that fixan arbitrary given point on the sphere. Further SDPbounds can be obtained by considering zonal matricesthat arise from actions that fix any given number ofpoints; however, even for two points, actual evaluationof the bounds requires significant computational effort[Mittelman and Vallentin 10].

3. THE BOUNDS

The general SDP bound on the spherical codes of[Bachoc and Vallentin 08] specializes to our case as fol-lows.

Theorem 3.1. Let C be a spherical two-distance set withinner products a and b. Let p be a positive integer. Thecardinality |C| is bounded above by the solution of thefollowing semidefinite programming problem:

1 +13

max(x1 + x2) (3–1)

subject to (1 00 0

)+

13

(0 11 1

)(x1 + x2) (3–2)

+

(0 00 1

)(x3 + x4 + x5 + x6) 0,

3 + G(n)i (a)x1 + G

(n)i (b)x2 ≥ 0, i = 1, 2, . . . , p, (3–3)

Sni (1, 1, 1) + Sn

i (a, a, 1)x1 + Sni (b, b, 1)x2 + Sn

i (a, a, a)x3

+ Sni (a, a, b)x4 + Sn

i (a, b, b)x5 + Sni (b, b, b)x6 0,

(3–4)

i = 0, 1, . . . , p, xj ≥ 0, j = 1, . . . , 6, where Si(·, ·, ·) are the(p − i + 1) × (p − i + 1) matrices defined in (2–3).

In this theorem, the variables x1 , x2 refer to the num-bers of ordered pairs of vectors in C with inner producta and b respectively; namely, we have xi = 3αi , i = 1, 2.

We note that the SDP problem seeks to optimizethe same linear form as the LP problem, but it addsmore constraints on the configuration. Because of this,Theorem 3.1 usually gives tighter bounds than The-orem 2.1. This fact is evident from Table 1 and isalso known from the calculation of kissing numbers in[Bachoc and Vallentin 08].

3.1. Calculation of the Bound

Several remarks are in order. First, implementation ofSDP for two-distance sets differs from earlier compu-tations in [Bachoc and Vallentin 08, Mittelman andVallentin 10] in that in our case, there are no limitson the minimum separation of the points. Next, we re-strict our calculations to the case p ≤ 5, since no improve-ment is observed for larger values. Finally, by a result in[Larman et al. 77], if |C| > 2n + 3, then the inner prod-ucts a, b are related by

b = bk (a) =ka − 1k − 1

,

where k ∈ {2, . . . , �(1 +√

2n)/2�} is an integer. Thus weobtain a family of SDP bounds parameterized by a. Sincebk (a) ≥ −1, a + bk (a) < 0, we get that

a ∈ Ik :=[0,

12k − 1

).

Moreover, if −1 ≤ b < a ≤ 0, then |C| cannot be large bythe Rankin bounds [Rankin 55], and if a + b ≥ 0, then|C| is bounded by (1–3). We conclude as follows.

Theorem 3.2. Let SDP(a) be the solution of the SDP prob-lem (3–1)–(3–4), where b = bk (a). Let C be a sphericaltwo-distance set with inner products a, b. Then

|C| ≤

⎧⎪⎨⎪⎩

n(n + 1)/2, if a + b ≥ 0,SDP(a), if a ∈ Ik ,n + 1, if −1 ≤ b < a < 0.

For instance, for n = 23, k = 3, we obtain Ik = [0, 0.2).Partitioning Ik into a number of small segments, we plotthe value SDP(a) as a function of a evaluated at thenodes of the partition. The result is shown in Figure 1(a).A part of the segment around the maximum appears inFigure 1(b). This computation gives an indication of theanswer, but in principle, the value SDP(a) could oscil-late between the nodes of the partition. Ruling this outrequires perturbation analysis of the SDP problem, whichis not immediate.

3.1.1. The Dual Problem

The dual problem of (3–1)–(3–4) has the following form:

1 + min

{p∑

i=1

αi + β11 + 〈F0 , Sn0 (1, 1, 1)〉

}

subject to (β11 β22

β12 β22

) 0, (3–5)

Dow

nloa

ded

by [

Nip

issi

ng U

nive

rsity

] at

14:

24 0

8 O

ctob

er 2

014

Barg and Yu: New Bounds for Spherical Two-Distance Sets 191

n LP Bound SDP Bound n(n + 1)/2 k n SDP Bound n(n + 1)/2 k

7 28 28 28 2 52 1128 1378 48 31 28 36 2 53 1128 1431 49 34 29 45 2 54 1128 1485 410 37 29 55 2 55 1128 1540 411 40 29 66 2 56 1128 1596 412 44 28 78 2 57 1162 1653 213 47 29 91 3 58 1200 1711 214 52 35 105 2 59 1240 1770 215 56 41 120 3 60 1282 1830 216 61 50 136 3 61 1324 1891 217 66 60 153 3 62 1372 1953 218 76 75 171 3 63 1428 2016 219 96 95 190 3 64 1482 2080 220 126 125 210 3 65 1540 2145 221 176 175 231 3 66 1604 2211 222 275 275 253 3 67 1672 2278 223 277 276 276 3 68 1745 2346 224 280 276 300 3 69 1822 2415 225 284 276 325 3 70 1907 2485 226 288 276 351 3 71 1999 2556 227 294 276 378 3 72 2097 2628 228 299 276 406 3 73 2206 2701 229 305 276 435 3 74 2325 2775 230 312 276 465 3 75 2394 2850 231 319 276 496 3 76 2468 2926 232 327 276 528 3 77 2542 3003 233 334 276 561 3 78∗ 3159 3081 234 342 276 595 3 79 3160 3160 435 360 276 630 2 80 3160 3240 436 416 276 666 2 81 3160 3321 437 488 276 703 2 82 3160 3403 438 584 276 741 2 83 3160 3486 439 721 292 780 2 84 3185 3570 440 928 315 820 2 85 3294 3655 441 341 861 2 86 3408 3741 442 370 903 2 87 3522 3828 443 422 946 2 88 3645 3916 444 540 990 2 89 3749 4005 445 736 1035 2 90 3905 4095 446∗ 1127 1081 2 91 4038 4186 447 1128 1128 2 92 4171 4278 448 1128 1176 2 93 4335 4371 449 1128 1225 2 94∗ 4492 4465 450 1128 1275 4 95∗ 4668 4560 451 1128 1326 4 96∗ 4828 4656 4

TABLE 1. Bounds on two-distance sets. The starred rows correspond to dimensions for which the value of g(n) is not knownexactly.

and

2β12 + β22 +p∑

i=1

(αiG(n)i (a) + 3〈Fi, S

ni (a, a, 1)〉) ≤ −1

(3–6)

2β12 + β22 +p∑

i=1

(αiG(n)i (b) + 3〈Fi, S

ni (b, b, 1)〉) ≤ −1

(3–7)

and

β22 +p∑

i=0

〈Fi, Sni (y1 , y2 , y3)〉 ≤ 0, (3–8)

where

(y1 , y2 , y3) ∈ {(a, a, a), (a, a, b), (a, b, b), (b, b, b)},αi ≥ 0, Fi 0, i = 1, . . . , p.

Dow

nloa

ded

by [

Nip

issi

ng U

nive

rsity

] at

14:

24 0

8 O

ctob

er 2

014

192 Experimental Mathematics, Vol. 22 (2013), No. 2

0 0.05 0.1 0.15 0.20

50

100

150

200

250

300

a

SD

P(a

)

n=23, k=3, a ∈ [0,0.2]

(a) The value SDP(a) for n = 23

0.165 0.17 0.175 0.18 0.185 0.19 0.195 0.2260

262

264

266

268

270

272

274

276

a

SD

P(a

)

n=23, k=3, a ∈ [0.165,0.2]

(b) The neighborhood of the maximum

FIGURE 1. Evaluation of the SDP bound on g(23) (color figure available online).

We need to estimate from above the maximum valueof this problem over a ∈ Ik = [a1 , a2 ]. Accounting for acontinuous value set of the parameter in SDP problemsis a challenging task. We approach it by employing thesum-of-squares method. Constraints (3–6)–(3–8) imposepositivity conditions on some univariate polynomials ofa for a ∈ Ik . The following sequence of steps transformsthe constraints to semidefinite conditions. Observe thata polynomial f(a) of degree at most m satisfies f(a) ≥ 0for a ∈ Ik if and only if the polynomial

f+(a) =(1 + a2)m f

(a1 + a2a

2

1 + a2

)

of degree at most 2m is nonnegative for all a ∈ R .Next, a polynomial nonnegative on the entire real axis

can be written as a sum of squares, f(x) =∑

i r2i (x),

where the ri are polynomials. Further, by a result of[Nesterov 00], a polynomial f(x) of degree 2m is asum of squares if and only if there exists a positivesemidefinite matrix Q such that f = XQXt , where X =(1, x, x2 , . . . , xm ). Thus, constraints (3–6)–(3–8) can betransformed to semidefinite conditions.

As a result, we obtain an SDP problem that can besolved by computer. We solved the resulting problem for7 ≤ n ≤ 96 using the Matlab toolbox SOSTOOLS in the

YALMIP environment.1 An advantage in SOSTOOLS is thatit accepts a as an SDP variable, thereby accounting forall the values of a in the segment. Thus, we obtain thevalue max SDP(a), a ∈ Ik .

However, this may impose excessive constraints onthe value of the SDP problem, because all the condi-tions for different values of a are involved at the sametime. To work around this accumulation, we use a sub-partitioning of the segment Ik into smaller segments.For each of them, SOSTOOLS outputs the largest valueof the minimum of the SDP problem over all a in thesegment.

It turns out that in many cases, the maximum of thesesolutions is smaller than max SDP(a), a ∈ Ik , computeddirectly by the package. The estimates of the answercomputed from the primal problem serve as guidance forthe needed step length of the partition. The solution ofthe sum-of-squares SDP optimization problem providesa rigorous proof for the estimates obtained by discretiz-ing the primal problem (3–1)–(3–4). For instance, forn = 23, we partition I3 into 20 subsegments, obtaining276.5 as the maximum value of the dual SDP problem fora ∈ I3 , and similarly for other dimensions.

1SOSTOOLS is available at http://www.cds.caltech.edu/sostools/,YALMIP at http://users.isy.liu.se/johanl/yalmip/.

Dow

nloa

ded

by [

Nip

issi

ng U

nive

rsity

] at

14:

24 0

8 O

ctob

er 2

014

Barg and Yu: New Bounds for Spherical Two-Distance Sets 193

3.1.2. Results

The results of the calculation are summarized in Ta-ble 1. The part of the table for 7 ≤ n ≤ 40, except forthe values of the SDP bound, is from [Musin 09]. The im-provement provided by Theorem 3.1 over the LP boundis quite substantial even for relatively small dimensions.The LP bound is above n(n + 1)/2 for n ≥ 40 and is notincluded starting with n = 41. The cases n = 46, 78 andn ≥ 94 are not resolved by SDP, although for n = 94,we still obtain an improvement over the harmonic bound(1–2). The value of k shown in the table accounts forthe largest value of the SDP problem among the possi-ble choices of k. This guarantees that the value SDP(a)is less than or equal to the number in the table for allthe possible values of the inner products a, b in the pointset.

Notice that for n = 46, 78, the SDP bound coin-cides with the bound (1–2). For n = 23, the resultsof [Musin 09] leave two possibilities: g(n) = 276 andg(n) = 277. The SDP bound resolves this for the for-mer, establishing the corresponding part of the claimin (1–4). As is seen from Figure 1(b), the largest valueof SDP(a) is attained for a = 0.2 and is equal to 276.This case corresponds to 276 equiangular lines in R23

with angle arccos 0.2, which can be constructed usingeither strongly regular graphs or the Leech lattice (see[Lemmens and Seidel 73] for details).

ACKNOWLEDGMENTS

We are grateful to Chao-Wei Chen and Johan Lofbergfor their significant help with the Matlab implementation.Research supported in part by NSF grants DMS1101697,CCF0916919, and CCF1217894, and NSA grant H78230-12-1-0260.

REFERENCES

[Bachoc 09] C. Bachoc. “Semidefinite Programming, Har-monic Analysis and Coding Theory.” arXiv:0909.4767,2009.

[Bachoc and Vallentin 08] C. Bachoc and F. Vallentin. “NewUpper Bounds for Kissing Numbers from SemidefiniteProgramming.” J. Amer. Math. Soc. 21 (2008), 909–924.

[Bachoc et al. 12] C. Bachoc, D.C. Gijswijt, A. Schrijver,and F. Vallentin. “Invariant Semidefinite Programs.” InHandbook on Semidefinite, Conic and Polynomial Opti-mization,, Internat. Ser. Oper. Res. Management Sci. 166,pp. 219–269. Springer, 2012.

[Bannai et al. 04] E. Bannai, A. Munemasa, and B. Venkov.“The Nonexistence of Certain Tight Spherical Designs.”St. Petersburg Math. J. 16 (2005), 609–625.

[Ben-Tal and Nemirovski 01] A. Ben-Tal and A. Nemirovski.Lectures on Modern Convex Optimization. Analysis, Al-gorithms, and Engineering Applications. SIAM, 2001.

[Delsarte 73] P. Delsarte, “An Algebraic Approach to theAssociation Schemes of Coding Theory.” Philips ResearchRepts Suppl. 10 (1973), 1–97.

[Delsarte et al. 77] P. Delsarte, J. M. Goethals, and J. J. Sei-del. “Spherical Codes and Designs.” Geometriae Dedicata6 (1977), 363–388.

[Dunkl 76] C. Dunkl. “A Krawtchouk Polynomial AdditionTheorem and Wreath Products of Symmetric Groups.”Indiana Univ. Math. J. 25 (1976), 335–358.

[Kabatyansky and Levenshtein 78] G. A. Kabatyansky andV. I. Levenshtein. “Bounds for Packings on the Sphereand in the Space.” Problems of Information Transmission14 (1978), 3–25.

[Larman et al. 77] D. G. Larman, C. A. Rogers, and J. J.Seidel. “On Two-Distance Sets in Euclidean Space.” Bull.London Math. Soc. 9 (1977), 261–267.

[Lemmens and Seidel 73] P. W. H. Lemmens and J. J. Seidel.“Equiangular Lines.” Journal of Algebra 24 (1973), 494–512.

[Martin and Tanaka 09] W. J. Martin and H. Tanaka. “Com-mutative Association Schemes.” European J. Combin. 30(2009), 1497–1525.

[Mittelman and Vallentin 10] H. D. Mittelman and F. Val-lentin. “High Accuracy Semidefinite ProgrammingBounds for Kissing Numbers.” Experimental Mathemat-ics 19 (2010), 174–178.

[Musin 09] O. R. Musin. “Spherical Two-Distance Sets.”J. Combin. Theory Ser. A 116 (2009), 988–995.

[Nebe and Venkov 12] G. Nebe and B. Venkov. “On TightSpherical Designs.” arXiv:1201.1830, 2012.

[Nesterov 00] Yu. Nesterov, “Squared Functional Systemsand Optimization Problems.” In High Performance Op-timization, edited by H. Frenk, K. Roos, T. Terlaky, andS. Zhanget, pp. 405–440. Kluwer, 2000.

[Rankin 55] R. A. Rankin. “The Closest Packing of SphericalCaps in n Dimensions.” Proc. Glasgow Math. Assoc. 2(1955), 139–144.

[Schoenberg 42] I. J. Schoenberg. “Positive Definite Func-tions on Spheres.” Duke Math. J. 9 (1942), 96–107.

Dow

nloa

ded

by [

Nip

issi

ng U

nive

rsity

] at

14:

24 0

8 O

ctob

er 2

014

194 Experimental Mathematics, Vol. 22 (2013), No. 2

[Schrijver 05] A. Schrijver. “New Code Upper Bounds fromthe Terwilliger Algebra and Semidefinite Programming.”IEEE Trans. Inform. Theory 51 (2005), 2859–2866.

[Vallentin 09] F. Vallentin. “Symmetry in SemidefinitePrograms.” Linear Algebra Appl. 430 (2009), 360–369.

Alexander Barg, Department of Electrical and Computer Engineering and Institute for Systems Research, University ofMaryland, College Park, MD 20742 (abarg@umd.edu)

Wei-Hsuan Yu, Department of Mathematics and Institute for Systems Research, University of Maryland, College Park, MD20742 (mathyu@math.umd.edu)

Dow

nloa

ded

by [

Nip

issi

ng U

nive

rsity

] at

14:

24 0

8 O

ctob

er 2

014