This table shows the conjectured optimal spherical codes of N points on the unit sphere in Rn, i.e. those that maximize the minimal angle between any two points. Note that these are not the highest-precision results available in most cases.

Each entry in the table links to a text file with the minimal angle, maximal cosine, and random seed (passed into C++'s srand()) on the first line, followed by the coordinates for a single point on each line.

Discussion on the structure of several of the codes can be found here.

Skip to dimension: 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

NnMaximal Cosine
7 3 0.2101383135622991
8 3 0.2612038754197031
9 3 0.3333333336220768
10 3 0.4043943258686149
11 3 0.4472135964961411
12 3 0.4472135955000456
13 3 0.5426364870425615
14 3 0.5639503007857350
15 3 0.5926059036752986
16 3 0.6122946169232646
17 3 0.6280944153165459
18 3 0.6486958326809263
19 3 0.6731168879760830
20 3 0.6764771383434012
21 3 0.6994984317413468
22 3 0.7103062589554452
23 3 0.7228469854628223
24 3 0.7230784684893239
25 3 0.7473986291161110
26 3 0.7542781775254275
27 3 0.7583892109849939
28 3 0.7732302627999463
29 3 0.7802814165347294
30 3 0.7815518753510448
31 3 0.7911186135594449
32 3 0.7936166151582470
33 3 0.8063976139453374
34 3 0.8109843375890903
35 3 0.8159377719727477
36 3 0.8172481856288693
37 3 0.8248924806044573
38 3 0.8265832596112428
39 3 0.8339913227782089
40 3 0.8371620728277038
41 3 0.8412363433172813
42 3 0.8433315519102305
43 3 0.8472088657779842
44 3 0.8482013785638861
45 3 0.8542494744692499
46 3 0.8575341671833023
47 3 0.8591223620557270
48 3 0.8592922951892934
49 3 0.8666914793212006
50 3 0.8681732092505776
51 3 0.8714843652601374
52 3 0.8729667089801670
53 3 0.8761898042566340
54 3 0.8770043069674769
55 3 0.8807850467794879
56 3 0.8817315786251970
57 3 0.8843637378140830
58 3 0.8865737953384161
59 3 0.8878557374281494
60 3 0.8894735679000609
61 3 0.8920085987955564
62 3 0.8934989775254272
63 3 0.8951931274248627
9 4 0.1620152010052891
10 4 0.1666666666667447
11 4 0.2304055646256036
12 4 0.2500000002375781
13 4 0.3072956555200069
14 4 0.3195185957283529
15 4 0.3509921764166606
16 4 0.3876281780260298
17 4 0.4122663245498179
18 4 0.4228194149170982
19 4 0.4342585465594216
20 4 0.4342585460403926
21 4 0.4713808593522488
22 4 0.4978841314667196
23 4 0.5000000007249005
24 4 0.5000000000002801
25 4 0.5373160576321491
26 4 0.5407896178245443
27 4 0.5575913525798237
28 4 0.5673390459054797
29 4 0.5731485310693875
30 4 0.5842328148813437
31 4 0.5907659050690239
32 4 0.5957201502247524
33 4 0.6007554088142363
34 4 0.6101294922501798
35 4 0.6215239877253954
36 4 0.6283095523962176
37 4 0.6303687232267281
38 4 0.6407338329829083
39 4 0.6469282793064194
40 4 0.6504978020818983
41 4 0.6552796707035409
42 4 0.6584208792719581
43 4 0.6626671994080211
44 4 0.6714703034877149
45 4 0.6769249586912305
46 4 0.6823258671309639
47 4 0.6854009248554835
48 4 0.6898311875943758
49 4 0.6934146470576165
50 4 0.6937261274315910
51 4 0.6937261274315910
52 4 0.6937261274315910
53 4 0.6937261274315910
54 4 0.6937261274315910
55 4 0.6937261274315910
56 4 0.6937261274315910
57 4 0.6937261274315910
58 4 0.6937261274315910
59 4 0.6937261274315910
60 4 0.6937261274315910
61 4 0.7315179737067657
62 4 0.7338855575573391
63 4 0.7387277424294973
64 4 0.7403452551653136
65 4 0.7424833085970075
66 4 0.7470914010235449
67 4 0.7489699590444685
68 4 0.7510444933541366
69 4 0.7526394239723836
70 4 0.7537672789903931
71 4 0.7563760121866121
72 4 0.7580404152407439
11 5 0.1328535444190423
12 5 0.1539316054201190
13 5 0.1872598539367539
14 5 0.2000000008830239
15 5 0.2000000006932102
16 5 0.2000000000000026
17 5 0.2704758369671391
18 5 0.2755017423102548
19 5 0.2918224013818767
20 5 0.2993856934286645
21 5 0.3149169576246721
22 5 0.3549950353243153
23 5 0.3697726985839708
24 5 0.3742329840940252
25 5 0.3796210264667195
26 5 0.3903776934512994
27 5 0.4016678906302231
28 5 0.4081737120095977
29 5 0.4110346162840297
30 5 0.4130297775903973
31 5 0.4339120265917540
32 5 0.4418505355329254
33 5 0.4600046839241236
34 5 0.4704831460267598
35 5 0.4795321446380047
36 5 0.4852488128828129
37 5 0.4944903702863393
38 5 0.4962439576007515
39 5 0.5048985041284060
40 5 0.5075725827033202
41 5 0.5155597225832728
42 5 0.5202395661060080
43 5 0.5262401601201038
44 5 0.5275939169008858
45 5 0.5390871234840878
46 5 0.5443643214132561
47 5 0.5441423411059506
48 5 0.5451194834585578
49 5 0.5508514909889379
50 5 0.5487755488336803
13 6 0.1130797543129282
14 6 0.1324909223260369
15 6 0.1449489752929576
16 6 0.1711476506492790
17 6 0.1832743374711802
18 6 0.1978121910709024
19 6 0.2002260272350757
20 6 0.2142857160216206
21 6 0.2428528487356088
22 6 0.2488696631075335
23 6 0.2500000015658568
24 6 0.2500000011577707
25 6 0.2500000009100448
26 6 0.2500000580022727
27 6 0.2500000000004582
28 6 0.3000000004906118
29 6 0.3292909857465297
30 6 0.3333333597315901
31 6 0.3333334189894869
32 6 0.3333333786437884
33 6 0.3449490682375947
34 6 0.3464514125083021
35 6 0.3582576293470093
36 6 0.3582576006373998
37 6 0.3666236487553832
38 6 0.3676885119636304
39 6 0.3849363873593124
40 6 0.3860968006210172
41 6 0.3894459464987901
42 6 0.3903882674349111
43 6 0.3997739932367496
44 6 0.4056683198313568
45 6 0.4269451127900817
46 6 0.4357341457616315
47 6 0.4417942268833099
48 6 0.4450020443964198
49 6 0.4520999373804058
50 6 0.4558100172608270
15 7 0.0987017787279474
16 7 0.1133208802893402
17 7 0.1248458959995004
18 7 0.1261319839255108
19 7 0.1565973875623423
20 7 0.1695208490074703
21 7 0.1815239631137125
22 7 0.1827439989609300
23 7 0.1827439987494660
24 7 0.1827439981201592
25 7 0.2147372357902891
26 7 0.2313316486130062
27 7 0.2405565426178782
28 7 0.2473671565672048
29 7 0.2489457390982059
30 7 0.2494931206479651
31 7 0.2545077196069866
32 7 0.2655968304188741
33 7 0.2700326742163664
34 7 0.2776967394320963
35 7 0.2897738397119910
36 7 0.3000001958757737
37 7 0.3104851374329737
38 7 0.3185993236661678
39 7 0.3219660757674648
40 7 0.3239398247559443
17 8 0.0877334660293139
18 8 0.0994695744868291
19 8 0.1114099773232373
20 8 0.1194968680811382
21 8 0.1306019391880115
22 8 0.1306019380374326
23 8 0.1574554201530409
24 8 0.1576921456637861
19 9 0.0790671534142696
20 9 0.0870679592012778
21 10 0.0720331423075503
22 10 0.0839571927217079
23 11 0.0662012751714205
24 11 0.0773370777531633
25 12 0.0612819173418897
26 12 0.0718022148473504
27 13 0.0570725266582410
28 13 0.0669529941665468
29 14 0.0534269903418781
30 14 0.0627392311992042
31 15 0.0502371234744619
32 15 0.0589852592674446
33 16 0.0474210320875254
34 16 0.0557239395907531
35 17 0.0449155323263354
36 17 0.0527541425071234
37 18 0.0426710614826203
38 18 0.0501024231733890
39 19 0.0406481718527128
40 19 0.0477139069391241
41 20 0.0388150592486449
42 20 0.0455302778348459