Finding and investigating exact spherical codes

Abstract In this pap er w e present the results of computer searches using a v ariatio n of an energy m i nimization algorithm us e d by Kott witz for finding goo d spherical co des. W e pro ve th a t exact cod es exist b y representi ng the inner pro ducts b et w een the v ectors as algebraic n u m b ers. F or selected in teresting cases, we include detaile d discussion of the configurations. Of particular in terest are the 20-point cod e i n R 6 and th e 24-p oin t co de in R 7 , whic h are b oth the union of t w o cross p olytop e s in parallel hyp erplanes. Finally , we catalogue all of the co des w e h a v e found . 1 Finding and In v estigating Exact Spherical Co des Jeffrey W ang No vem b er 10 , 2 018 1 In tro duc tion Giv en N p oin ts that lie on the unit sphere S n − 1 in R n , we wish to determine ho w they should b e placed so that the minimal distance b et w een any tw o p oin ts is maximized. An y set of p oin ts on the unit sphere is called a spherical co de, and the problem of finding the b est co de has b een prop osed many times and in man y contex t s (though usu ally o nly in three dimensions), from pac king circles on a sphere to distributing or ific es on p ollen-grains (e.g. [5], [18]). The only know n optimal solutions in three dimensions are for N ≤ 1 2 and N = 24, while in four dimensions only the cases N ≤ 8, N = 10 [1], and N = 120 ha v e b een pro ve n. The remaining know n optimal co des are for N ≤ 2 n for any n , in which the p oin ts f o rm either a simplex or a subset of the cross p olytope; certain co des deriv ed from the Leec h lattice and the E 8 ro ot system; and lastly an infinite family based on isotropic subspaces [2]. Excluding these few cases, the b est know n co des ha ve b een found either by sp ec ific constructions or, more commonly , b y computer searc hes using v ar ious optimization algorithms. Giv en the difficult y of proving the optimality of ev en v ery small co des, most w o rk related to this pro ble m has b een in finding close approximations to go o d configuratio ns. The most extensiv e table of co des w a s created by N. J. A. Slo a ne , with the collab oration of R. H. Hardin, W. D. Smith and others, and is a v ailable e lectronically [17]. In this w or k w e use the tec hnique of energy minimization to find go o d spherical co des. Lee c h first observ ed the p ossibilit y of suc h an approach , and it ha s b een used in previous w o rk sev eral times [13]. Kott witz gav e a fairly compre hensiv e list of three-dimensional codes for up to 90 p oints [12], 2 whic h w as expanded later b y Buddenhagen and Kottwitz while se a r c hing for co des with m ultiplicit y g r eat er than one (i.e. co des f o r whic h there exist distinct configurations that obta in the same optimal distance) [4]. Nurmela in v estigat ed some global optimization metho ds based on energy minimiza- tion and pro vided n umerical results for co des in up to fiv e dimensions [1 5 ], but the most successful implemen tat io ns ha ve b een based on using a large n um b er of ra nd om starts and a lo cal optimization alg orithm so t h at there is a high probability that a t least one of them con v erges to the global opti- m um. Our alg o rithm has sev era l c hanges to impro ve on old implemen tat ions as w ell, suc h a s reducing exp onen t bias and choosing a differen t lo cal opti- mization metho d. This has r e sulted in three improv ed spherical co des in four dimensions as w ell as new higher dimensional co des , particularly in six and sev en dimensions, that exhibit in teresting symmetries. Section 2 describ e s our algo rithm in detail and Section 3 giv es a brief analysis of the improv ed co des . The remainder of this w ork fo cuses on providin g rigorous analysis of co des , particularly showing that the b est kno wn co des can b e represen ted as exact configurations in terms of algebraic n um b ers as opp osed to close estimates. This is an imp ortan t step tow ard rig orizing computer solutions to the problem and allows us to observ e t r u e equalities and relations b et w een p oin ts a nd edges in the co de. Buddenhagen and Kott witz did similar work while lo oking for three-dimensional co des with m ultiplicity greater than one, and provide d detailed exact descriptions of the tw o distinct optimal 15-p oin t co des [4]. In sev eral cases we w ere able to iden tify a considerable amount of underlying symmetry and structure using the metho ds of Section 4 . F or three particularly in teresting cases w e pro vide brief discuss ion of the config- urations in Section 5. The final section giv es tables o f the exact co des, in dimensions four through eight, based on their minimal p olynomials. 2 Metho dol o gy W e consider an in v erse p o w er la w p oten tial function o n the spherical co de C = { x 1 , x 2 , . . . , x N } ⊂ S n − 1 defined b y E = X 1 ≤ i W and are giv en in terms of u by V = 5 2 u 2 − u − 1 2 , W = − 5 u 2 − 4 u. There are t w o asp e cts of this Gram matrix that a r e particularly intrigu- ing. The first one tha t stands out is that all the edges b et w een V 1 and V 2 corresp ond to an inner pro duct of u or − u , whic h is a result of the p oin ts in V 2 b eing antipo dal. The second is that ev ery elemen t is represen ted as a p olynomial in u of degree tw o or less. The minimal p olynomial, how ev er, is 25 u 4 + 30 u 3 + 24 u 2 + 2 u − 1, whic h has degree four. This is unusual b e- cause in most cases a significant n um b er of elemen ts of the Gram matrix are p olynomials in u of maxim a l degree, i.e. one less than the degree of the minimal p olynomial. While w e do not hav e an explanation for t his, it ma y b e significant to the structure of the co de. 5.2 20 p oin ts in R 6 This configuration, the first especially nice one, is highly symmetric; eac h p oin t in the en t ir e configura t io n is equiv a le nt to ev ery other one. Each p oin t is at the minimal distance with 11 other p oin ts in the co de. F urthermore, since t he minimal p olynomial is 1 4 u − 3, all of the en tries in the Gram matrix are ra tional n umbers, sp ecifically m ultiples of 1 14 . The dot pro ducts corresp onding to the edges that emanate from any p o in t are (in order fr o m largest to smallest)  1 , 3 14 , 0 , − 1 14 , − 5 14 , − 3 7 , − 4 7 , − 9 14  . (5) The only rep etitions are 3 14 , whic h show s up 11 times, and − 9 14 , whic h sho ws up three times. It turns out tha t this co de is the union of t wo fiv e-dimensional cross p olytopes in parallel h yp erplanes. W e can orien t them with resp ect to one another: if w e c ho ose co ordinates to allow one of the cro ss-p olytop es to b e given b y plus or min us the standard orthonor m al basis v ectors, then after pro jection into the equatorial hy p erplane and rescaling, the other cross p olytope is giv en b y 10 ±  2 11 , − 6 11 , − 6 11 , 3 11 , − 6 11  ±  6 11 , 2 11 , 6 11 , 6 11 , − 3 11  ±  − 3 11 , − 6 11 , 2 11 , 6 11 , 6 11  ±  6 11 , 3 11 , − 6 11 , 2 11 , 6 11  ±  6 11 , − 6 11 , 3 11 , − 6 11 , 2 11  The w a y the orien tation is determined is still unkno wn, but there is evi- dence of some sort of underlying structure tha t is worth in ves t igating further. 5.3 24 p oin ts in R 7 This is the other particularly nice configuration. All 24 p oin t s in this co de are symmetric with resp ect to one ano the r and the minimal p olynomial is 19 u 2 + 2 u − 1, so the Gram matrix has v ery few en t r ies and they are all simple. The only entries other than the 1’s along the diagona l are u , − u , − 3 u , and 2 u − 1. Each v ector ha s inner pro duct u with 15 other v ectors, − u with 2 other v ectors, − 3 u with 5 ot he r v ectors, and 2 u − 1 w it h a sin gle other v ector. This configuratio n app ears to b e a very go o d co de a s b oth the 22- and 23- point co des are the same as this with points remo v ed. It is very similar t o the 20-p oin t co de in R 6 as it also consists of tw o cross- p olytopes in parallel hyperplanes. As with the other case, w e can orien t the cross p olytop es relative to eac h other by setting one to b e plus or minu s the standa r d orthonormal basis v ectors. After pro jection in to the equato- rial hyperplane and rescaling, the other cross p olytope is then giv en by the follo wing v ectors: 11 ±  0 , 1 √ 5 , − 1 √ 5 , − 1 √ 5 , 1 √ 5 , 1 √ 5  ±  1 √ 5 , 0 , 1 √ 5 , − 1 √ 5 , − 1 √ 5 , 1 √ 5  ±  − 1 √ 5 , 1 √ 5 , 0 , 1 √ 5 , − 1 √ 5 , 1 √ 5  ±  − 1 √ 5 , − 1 √ 5 , 1 √ 5 , 0 , 1 √ 5 , 1 √ 5  ±  1 √ 5 , − 1 √ 5 , − 1 √ 5 , 1 √ 5 , 0 , 1 √ 5  ±  1 √ 5 , 1 √ 5 , 1 √ 5 , 1 √ 5 , 1 √ 5 , 0  This is a ve ry structured set o f ve ctor s , but also differs from the v ectors in t he six dimensional case. It w ould b e an in teresting question to determine ho w in general the cro ss polytop es should b e arranged relativ e to eac h o t he r to maximiz e the minimal distance. 6 Catalogue of Exact Co n figurations W e presen t sev eral tables of results for newly calculated exact spherical co des in dimensions four through eigh t, sp ecific ally the maximal inner pro duct u and the minimal p olynomial. In eac h dimension, w e attempted to calculate a minimal p olynomial for the maxim um inner pro duct of the configurations for up to 24 p oin ts (27 in four and six dimensions). W e w ere able to do so for quite a few, but there are a still a n umber of co des that w e were unable to do this for. In general w e attempted this calculation using 5 00 decimal places of the maxim um inner pro duct. Moreo v er, for the minimal p olynomials w e calculated, w e w ere able to v erify a ll but t w o of them using the metho ds of Section 4. T hese are 21 p oin ts in five dimensions and 1 4 p oin ts in six dimensions, b oth of whic h hav e p olynomials of high degree. The case of 27 p oin ts in four dimensions should also b e not ed b ecause it has three rattlers, and since their p ositions are not 12 fixed relativ e to the other p oin ts w e did not include t he m in o ur v erification of the minimal p olynomial. Lastly , w e hav e a single new minimal p olynomial in three dimensions, concerning the 22-p oin t co de. The minimal p olynomial w e found is 486 u 18 + 13113 u 17 + 114996 u 16 + 117 476 u 15 + 658 256 u 14 + 378752 u 13 − 34 7056 u 12 − 121388 u 11 + 81 7 24 u 10 − 7 0886 u 9 − 5599 2 u 8 + 12 7 16 u 7 + 65 2 8 u 6 − 2 392 u 5 − 208 u 4 + 284 u 3 + 14 u 2 + 5 u + 4. But as in t h e tw o cases men tioned previous ly , w e ha v e b een unable to ve rif y this through finding an exac t Gram matrix. Note that T ables 1 and 2 are the same co des as recorded in [17] but with minimal p olynomials asso ciated with them. Also, some co des predate references to [8 ], but it is a con v enien t, systematic reference w ork. T able 1: F our-Dimensional Co des N u (20 decimal pla ces) Minimal Polynomial Ref 9 0 . 16201519 961163454918 16 u 3 − 16 u 2 − 4 u + 1 [8] 10 0 . 1666 66666666 6 6 6 6 6667 6 u − 1 [8] 11 0 . 2304 05563594 5 5 5 4 4174 8 u 3 − 12 u 2 − 2 u + 1 12 0 . 2500 00000000 0 0 0 0 0000 4 u − 1 [8] 13 0 . 3072 95653981 0 2 8 8 2233 5 632 u 9 + 94 72 u 8 − 30 72 u 7 − 58 88 u 6 + 54 4 u 5 + 944 u 4 + 15 2 u 3 − 44 u 2 − 14 u − 1 14 0 . 3195 18594212 6 0 3 6 3550 58 u 7 + 17 4 u 6 + 14 0 u 5 − 16 u 4 − 54 u 3 − 10 u 2 + 4 u + 1 15 0 . 3509 92175945 3 4 6 3 0330 36 u 4 − 18 u 3 + 10 u 2 − 1 16 0 . 3876 28177122 5 3 4 2 7776 256 u 10 + 1024 u 9 + 25 6 u 8 − 11 52 u 7 + 16 0 u 6 + 176 u 5 − 14 4 u 4 + 36 u 3 + 37 u 2 − 6 u − 3 17 0 . 4122 66323227 5 5 9 2 5382 Unknown 18 0 . 4228 19414073 0 5 9 3 4403 14 424 u 16 + 42 932 u 15 + 18 232 u 14 − 6210 0 u 13 − 53831 u 12 + 41 528 u 11 + 46 442 u 10 − 18 248 u 9 − 20977 u 8 + 61 80 u 7 + 53 72 u 6 − 15 56 u 5 − 721 u 4 + 24 0 u 3 + 34 u 2 − 16 u + 1 19 0 . 4342 58545910 6 6 4 8 8219 3 u 2 + u − 1 20 0 . 4342 58545910 6 6 4 8 8219 3 u 2 + u − 1 [8] 21 0 . 4713 80858507 3 1 7 9 1682 16 u 8 − 12 8 u 7 − 64 u 6 + 32 u 5 + 72 u 4 + 32 u 3 − 16 u 2 − 8 u + 1 22 0 . 4978 84130843 5 5 2 3 5629 Unknown 23 0 . 5000 00000000 0 0 0 0 0000 2 u − 1 24 0 . 5000 00000000 0 0 0 0 0000 2 u − 1 [8] 25 0 . 5373 16056655 0 7 7 8 7660 Unknown 26 0 . 5407 89617697 5 3 7 0 7673 3392 u 6 + 21 12 u 5 − 49 6 u 4 − 656 u 3 − 13 2 u 2 + 6 u − 1 27 0 . 5575 91351180 1 7 0 1 8253 794 u 5 + 39 3 u 4 − 34 4 u 3 − 82 u 2 + 6 u + 1 13 T able 2: Five-Dimensional Codes N u (20 decimal pla ces) Minimal Polynomial Ref 11 0 . 1328 53542598 5 8 9 9 1809 45 u 3 − 25 u 2 − 5 u + 1 [8] 12 0 . 1539 31605033 0 2 1 2 3095 25 u 4 + 30 u 3 + 24 u 2 + 2 u − 1 13 0 . 1872 59851882 8 5 3 5 8702 17 u 3 − 5 u 2 − 5 u + 1 14 0 . 2000 00000000 0 0 0 0 0000 5 u − 1 15 0 . 2000 00000000 0 0 0 0 0000 5 u − 1 16 0 . 2000 00000000 0 0 0 0 0000 5 u − 1 [8] 17 0 . 2704 75835268 5 7 3 6 2209 9 u 4 − 16 u 3 − 10 u 2 + 1 18 0 . 2755 01741659 8 1 9 2 3839 484 u 5 − 48 8 u 4 + 97 u 3 + 17 u 2 − u − 1 19 0 . 2918 22399024 4 9 0 1 4615 57 u 6 − 38 u 5 − 10 9 u 4 + 32 u 3 + 23 u 2 − 10 u + 1 20 0 . 2993 85692899 1 2 4 7 8230 5 u 3 + 13 u 2 − u − 1 21 0 . 3149 16957175 3 0 3 4 6285 86931 2 u 14 + 87 98656 u 13 − 10 62776 u 12 − 10586 775 u 11 − 96 8269 u 10 + 35 32907 u 9 + 18824 9 u 8 − 65 9974 u 7 − 11 746 u 6 + 72 246 u 5 − 806 u 4 − 52 67 u 3 − 97 u 2 + 20 7 u + 21 22 0 . 3549 95034166 2 5 6 2 0683 Unknown 23 0 . 3697 72696943 0 7 6 3 3377 Unknown 24 0 . 3742 32982465 1 6 7 2 5173 1 620 u 10 + 55 08 u 9 − 57 51 u 8 − 24 06 u 7 + 20 55 u 6 + 276 u 5 + 55 9 u 4 − 21 0 u 3 − 12 7 u 2 + 48 u − 4 7 Ac kno w l edgmen ts I w ould lik e to tha nk m y men tor Henry Cohn fo r suggesting this pro ject and constan tly providing inv aluable assistance along the w ay . I w o uld also lik e t o thank the Microsoft Researc h In ternship and Microsoft High Sc ho ol In ternship programs for arranging m y in ternship and allo wing this all to happ en. La stly , I would like to thank the American Institute of Mathematics for hosting my data online. References [1] C. Bac ho c, F. V allentin. Optimalit y and uniqueness of the (4 , 10 , 1 / 6) spherical co de . Preprin t ( 2007). . [2] B. Ballinger, G . Blekherman, H. Cohn, N. G ians iracusa, E. Kelly , A. Sc h ¨ urmann. Exp erime ntal study of energy-minimizing p oin t configura- tions on sp heres. Preprin t (2007). arXiv:mat h/0611451 . 14 T able 3: Six-D imensional Co des N u (20 decimal pla ces) Minimal Polynomial Ref 13 0 . 1130 79752147 4 4 7 2 1385 96 u 3 − 36 u 2 − 6 u + 1 [8] 14 0 . 1324 90920323 4 7 0 3 1437 2 37299200 u 13 + 46 3738880 u 12 + 36 6062080 u 11 + 14246 2784 u 10 + 23 021376 u 9 − 23 98832 u 8 − 18496 00 u 7 − 34 3428 u 6 − 18 02 u 5 + 11 443 u 4 + 1390 u 3 − 12 8 u 2 − 20 u + 1 15 0 . 1449 48974278 3 1 7 8 0982 20 u 2 + 4 u − 1 16 0 . 1711 47649429 3 9 3 6 5334 40000 00 u 8 + 15 120000 u 7 + 98 96400 u 6 − 24516 00 u 5 − 45 03600 u 4 − 15 11460 u 3 − 87480 u 2 + 43 740 u + 656 1 17 0 . 1832 74337023 1 4 4 8 1858 400 u 4 + 24 0 u 3 + 16 u 2 − 8 u − 1 18 0 . 1978 12188601 9 7 5 4 5241 784 u 8 − 28 0 u 7 − 11 91 u 6 + 12 0 u 5 + 218 u 4 − 62 u 3 − 31 u 2 + 2 u + 1 19 0 . 2002 26025891 2 0 5 4 8304 Unknown 20 0 . 2142 85714285 7 1 4 2 8571 14 u − 3 21 0 . 2428 52848013 6 9 1 7 0251 Unknown 22 0 . 2488 69660788 8 2 4 7 4478 Unknown 23 0 . 2500 00000000 0 0 0 0 0000 4 u − 1 24 0 . 2500 00000000 0 0 0 0 0000 4 u − 1 25 0 . 2500 00000000 0 0 0 0 0000 4 u − 1 26 0 . 2500 00000000 0 0 0 0 0000 4 u − 1 27 0 . 2500 00000000 0 0 0 0 0000 4 u − 1 [8] [3] C. 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Journal of the American Mathematical So ciet y 20 (2007), 99– 148. 15 T able 4: Sev en-Dimensional Co des N u (20 decimal pla ces) Minimal Polynomial Ref 15 0 . 09 87017762 7236447933 175 u 3 − 49 u 2 − 7 u + 1 [8] 16 0 . 11 33208796 0014474125 85823 999 u 12 + 15 3503984 u 11 + 16 3426326 u 10 + 92426 704 u 9 + 29 708081 u 8 + 92 92960 u 7 + 22524 04 u 6 + 41 0976 u 5 + 29 489 u 4 − 3216 u 3 − 74 6 u 2 − 48 u − 1 17 0 . 12 48475898 6639552862 Unknown 18 0 . 12 61319836 2288317392 4 7 u 2 + 2 u − 1 19 0 . 15 65973854 1709551030 1280 u 6 − 35 2 u 4 − 48 u 3 + 37 u 2 + 3 u − 1 20 0 . 16 95208471 9853722593 2 3 u 2 + 2 u − 1 [8] 21 0 . 18 15239608 0041583541 Unknown 22 0 . 18 27439976 3155681015 1 9 u 2 + 2 u − 1 23 0 . 18 27439976 3155681015 1 9 u 2 + 2 u − 1 24 0 . 18 27439976 3155681015 1 9 u 2 + 2 u − 1 T able 5: Eigh t-Dimensional Co des N u (20 decimal places) Minimal Polynomial Ref 17 0 . 08 77334633 2333854567 288 u 3 − 64 u 2 − 8 u + 1 [8] 18 0 . 09 94695727 0878709386 Unknown 19 0 . 11 14099750 2603998543 Unknown 20 0 . 11 94968666 8719356518 244 u 3 + 60 u 2 − 2 u − 1 21 0 . 13 06019374 8187072126 28 u 2 + 4 u − 1 22 0 . 13 06019374 8187072126 28 u 2 + 4 u − 1 [8] 23 0 . 15 74554177 2761612286 Unknown 24 0 . 15 76921449 3936087799 4 u 4 − 4 u 3 − 27 u 2 − 2 u + 1 [8] T. 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