Spherical Codes

Based on joint work with R. H. Hardin, W. D. Smith and Others

A library of putatively optimal spherical codes, together with other arrangements which may not be optimal but are especially interesting for some reason.

Keywords: points on a sphere, spherical codes, spherical designs, spherical coverings, 4-dimensional arrangements, 5-dimensional arrangements, 6-dimensional arrangements, 7-dimensional arrangements, 8-dimensional arrangements, 9-dimensional arrangements, 10-dimensional arrangements, 11-dimensional arrangements, 12-dimensional arrangements, 13-dimensional arrangements, 14-dimensional arrangements, 15-dimensional arrangements, 16-dimensional arrangements, 17-dimensional arrangements, 18-dimensional arrangements, 19-dimensional arrangements, 20-dimensional arrangements, 21-dimensional arrangements, 22-dimensional arrangements, 23-dimensional arrangements, 24-dimensional arrangements, higher-dimensional arrangements, Archimedian solids, Barnes-Wall lattice, Coxeter-Todd lattice, cube, cuboctahedron, dodecahedron, E_8 lattice, Gosset lattice, hypercube, hypersphere, icosahedron, Leech lattice, Nordstrom-Robinson code, octahedron, generalized octahedron, orthoplex, Platonic solids, rhombic dodecahedron, simplex, 600-cell, snub cube, improved snub cube, tetrahedron, truncated cube, truncated dodecahedron, truncated icosahedron, truncated octahedron, truncated tetrahedron, 24-cell, twisted cube

The Problem Place n points on a sphere in d dimensions so as to maximize the minimal distance (or equivalently the minimal angle) between them.

This file has two parts

1. Part 1. Tables of putatively optimal packings in 3, 4 and 5 dimensions with n = 4, ..., 130 points.
2. Part 2. Tables of other good (though not always optimal) packings in dimensions 3 through 24 with various numbers of points.

Remarks

• If you use any of these arrangements, please acknowledge this source. Such an acknowledgement might say something like:
N. J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith and others, Tables of Spherical Codes, published electronically at www.research.att.com/~njas/packings/
• Contributions of additional arrangements of points will be welcomed, as well as links to other sites on the Web where similar arrangements can be found. Please send all correspondence to NJAS at the following address: njas@research.att.com All such contributions will be gratefully acknowledged.
• Note that there are a large number of tables of other types of arrangements of points on spheres (minimal energy arrangements, coverings, etc.) on NJAS's home page,

```
```

Part 1, Dimension 3 through 5, With Up To 130 Points

• The first three tables give putatively optimal packings in 3, 4 and 5 dimensions, for n = 4, ..., 130 points.
• Most of these were found (or for small numbers of points, rediscovered) by our programs. Correspondents have sent us improvements to a few entries in three dimensions. All such improvements will be gratefully acknowledged. Please send them to NJAS at the following address: njas@research.att.com
• Go to library of 3-d packings | library of 4-d packings | library of 5-d packings

Summary of results:

Packings of points on a sphere

Arrangements for which no other source is listed are copyright R. H. Hardin, N. J. A. Sloane & W. D. Smith, 1994-1996.

```
dim npts min separation (degrees)

3    4  109.4712206
3    5   90.0000000
3    6   90.0000000
3    7   77.8695421
3    8   74.8584922
3    9   70.5287794
3   10   66.1468220
3   11   63.4349488
3   12   63.4349488
3   13   57.1367031
3   14   55.6705700
3   15   53.6578501
3   16   52.2443957
3   17   51.0903285
3   18   49.5566548
3   19   47.6919141
3   20   47.4310362
3   21   45.6132231	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
3   22   44.7401612
3   23   43.7099642
3   24   43.6907671
3   25   41.6344612
3   26   41.0376616
3   27   40.6776007	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
3   28   39.3551436
3   29   38.7136512
3   30   38.5971159
3   31   37.7098291
3   32   37.4752140
3   33   36.2545530	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
3   34   35.8077844
3   35   35.3198076
3   36   35.1897322
3   37   34.4224080
3   38   34.2506607
3   39   33.4890466
3   40   33.1583563
3   41   32.7290944
3   42   32.5063863
3   43   32.0906244
3   44   31.9834230
3   45   31.3230814
3   46   30.9591635
3   47   30.7818159
3   48   30.7627855
3   49   29.9235851
3   50   29.7529564
3   51   29.3684069
3   52   29.1947579
3   53   28.8138972
3   54   28.7169205	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
3   55   28.2627914
3   56   28.1480466
3   57   27.8266759
3   58   27.5564159
3   59   27.3949757
3   60   27.1928300
3   61   26.8732779
3   62   26.6839970
3   63   26.4869225
3   64   26.2350433
3   65   26.0698299
3   66   25.9474437	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
3   67   25.6839813
3   68   25.4638245
3   69   25.3336364	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3   70   25.1709200
3   71   24.9879381
3   72   24.9264861
3   73   24.5537792	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3   74   24.4209398	D.A. Kottwitz, private communication
3   75   24.3017225
3   76   24.1281944
3   77   24.0012837
3   78   23.9310254
3   79   23.6239917
3   80   23.5530672	Jim Buddenhagen (jbuddenh(AT)earthlink.net) private communication
3   81   23.3476377	D.A. Kottwitz, private communication
3   82   23.1946074
3   83   23.0829976
3   84   23.0517306
3   85   22.7791621
3   86   22.6743694	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
3   87   22.5466574
3   88   22.4678810
3   89   22.3166023
3   90   22.1540232
3   91   22.0517963
3   92   22.0275815
3   93   21.8103801
3   94   21.7237135	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3   95   21.5945501
3   96   21.5206099
3   97   21.4006197
3   98   21.3710607
3   99   21.1359674	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  100   21.0312020
3  101   20.9286834
3  102   20.8556887	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  103   20.7382700	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  104   20.6566210
3  105   20.5388524
3  106   20.4394089
3  107   20.3612035	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  108   20.3044447	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  109   20.1493196	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  110   20.1113276	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  111   19.9824769	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  112   19.8913044	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  113   19.8056013
3  114   19.7450093
3  115   19.6239931	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  116   19.5497969	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  117   19.4612911
3  118   19.3893497	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  119   19.3257514	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  120   19.3240201
3  121   19.1357298	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  122   19.0700369	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  123   19.0063891	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  124   18.9539116	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  125   18.8448151	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  126   18.7815856
3  127   18.6900568	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  128   18.6349726
3  129   18.5634726	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
3  130   18.5103522
4    5  104.4775121
4    6   90.0000000
4    7   90.0000000
4    8   90.0000000
4    9   80.6761150
4   10   80.4059318
4   11   76.6790498
4   12   75.5224878
4   13   72.1036706
4   14   71.3661859
4   15   69.4519870
4   16   67.1930021
4   17   65.6531563
4   18   64.9872827
4   19   64.2618766
4   20   64.2618766
4   21   61.8760316
4   22   60.1398863
4   23   60.0000000
4   24   60.0000000
4   25   57.4988826
4   26   57.2625923
4   27   56.1106132
4   28   55.4351398
4   29   55.0299245
4   30   54.2511897
4   31   53.7886221
4   32   53.4360130
4   33   53.0759811
4   34   52.4011334
4   35   51.5724903
4   36   51.0745199
4   37   50.9226685
4   38   50.1535207
4   39   49.6895967
4   40   49.4208547
4   41   49.0594727
4   42   48.8204491
4   43   48.4963928
4   44   48.0251062
4   45   47.3961868
4   46   46.9743371
4   47   46.7328543
4   48   46.3832527
4   49   46.0990591
4   50   46.0742073
4   51   45.4625355
4   52   45.2812081
4   53   44.9542243
4   54   44.7883187
4   55   44.3025299
4   56   44.0114102
4   57   43.7495816
4   58   43.5341667
4   59   43.3907742
4   60   43.2167495
4   61   42.9862214
4   62   42.7870922
4   63   42.3770430
4   64   42.3062196
4   65   42.0566173
4   66   41.7550028
4   67   41.4987735
4   68   41.3146370
4   69   41.1813742
4   70   41.0851193
4   71   40.8538202
4   72   40.7098566
4   73   40.6070686
4   74   40.5754396
4   75   40.4865078
4   76   40.4418535
4   77   40.0749825
4   78   39.7568300
4   79   39.3704881
4   80   39.1321954
4   81   38.9404678
4   82   38.7616584
4   83   38.6411550
4   84   38.5216197
4   85   38.3971687
4   86   38.2834796
4   87   38.2316167
4   88   38.1934424
4   89   37.9525738
4   90   37.7779155
4   91   37.6840939
4   92   37.6073513
4   93   37.5288195
4   94   37.4811292
4   95   37.1570600
4   96   37.0622214
4   97   36.9191853
4   98   36.7966891
4   99   36.6983263
4  100   36.5636318
4  101   36.4956325
4  102   36.4186254
4  103   36.3807276
4  104   36.2615258
4  105   36.1859494
4  106   36.1333759
4  107   36.0959131
4  108   36.0509369
4  109   36.0309659
4  110   36.0076295
4  111   36.0013484
4  112   36.0000058
4  113   36.0000000
4  114   36.0000000
4  115   36.0000000
4  116   36.0000000
4  117   36.0000000
4  118   36.0000000
4  119   36.0000000
4  120   36.0000000
4  121   34.4465422
4  122   34.2716955
4  123   34.1743519
4  124   33.9853008
4  125   33.8729785
4  126   33.7679953
4  127   33.6685430
4  128   33.5726406
4  129   33.5033828
4  130   33.4075174
4  600   19.8433177
5    6  101.5369590
5    7   90.0000000
5    8   90.0000000
5    9   90.0000000
5   10   90.0000000
5   11   82.3654810
5   12   81.1451619
5   13   79.2070847
5   14   78.4630410
5   15   78.4630410
5   16   78.4630410
5   17   74.3074162
5   18   74.0080831
5   19   73.0329080
5   20   72.5792898
5   21   71.6441994
5   22   69.2068599
5   23   68.2984004
5   24   68.0230856
5   25   67.6897899
5   26   67.0305252
5   27   66.3175208
5   28   65.9100901
5   29   65.7301670
5   30   65.6046967
5   31   64.2839257
5   32   63.7779969
5   33   62.6125998
5   34   61.9343521
5   35   61.4165697
5   36   61.0530896
5   37   60.4074688
5   38   60.2482021
5   39   60.0017727
5   40   60.0000000
5   41   59.0029024
5   42   58.7856545
5   43   58.3513041
5   44   58.1661569
5   45   57.6514221
5   46   57.4326036
5   47   57.1800994
5   48   56.9676480
5   49   56.8441582
5   50   56.8441582
5   51   56.3071628
5   52   55.9343732
5   53   55.4667168
5   54   55.1747097
5   55   54.9279561
5   56   54.7453222
5   57   54.7356103
5   58   54.4462372
5   59   54.0539784
5   60   53.6051457
5   61   53.3979587
5   62   53.2010567
5   63   52.8751126
5   64   52.7108774
5   65   52.5214037
5   66   52.4752830
5   67   52.2729583
5   68   52.0286818
5   69   51.9176291
5   70   51.8973507
5   71   51.8317602
5   72   51.8317602
5   73   51.8272924
5   74   51.8272924
5   75   51.3428135
5   76   50.8459281
5   77   50.4692075
5   78   50.1905273
5   79   49.9276863
5   80   49.6870598
5   81   49.5416542
5   82   49.3488655
5   83   49.0496962
5   84   48.8819546
5   85   48.7570741
5   86   48.6290153
5   87   48.5000839
5   88   48.3577476
5   89   48.2141763
5   90   48.1896851
5   91   47.9008875
5   92   47.7716828
5   93   47.6364239
5   94   47.5270717
5   95   47.4198451
5   96   47.3535100
5   97   47.2151568
5   98   47.1352000
5   99   46.9889294
5  100   46.9094506
5  101   46.7465353
5  102   46.5719302
5  103   46.4564803
5  104   46.3923772
5  105   46.3118627
5  106   46.1395418
5  107   46.0632232
5  108   45.9430913
5  109   45.8583450
5  110   45.7248275
5  111   45.6276320
5  112   45.5373551
5  113   45.4294817
5  114   45.3304185
5  115   45.2457325
5  116   45.1419275
5  117   45.0181994
5  118   44.9252672
5  119   44.8177246
5  120   44.7301784
5  121   44.6404435
5  122   44.5817814
5  123   44.4453109
5  124   44.3486865
5  125   44.2250403
5  126   44.1807177
5  127   44.0598726
5  128   43.9739131
5  129   43.8672061
5  130   43.8024944

```

```
```

Part 2, Other Nice Arrangements of Points on Spheres

Remarks

• These are arrangements that I have assembled over the past 20 years. They have been obtained from many sources, including:
• computer searches
• constructions based on codes and lattices
• constructions based on polytopes

• Many but not all of these arrangements were constructed with the goal of getting good packings, that is, minimizing the minimal angle between the points (or equivalently maximizing the secant of the minimal angle between the points). However, others are good spherical t-designs, or good coverings, etc. At some point I plan to go through this list and annotate it with further information about minimal angle, covering radius, index as a t-design, order of automorphism group, etc.
• Format: Two different formats have been used here: either one point per line or one coordinate per line. Occasionally other formats (for example Maple-readable coordinates) are also given.
• It is sometimes much easier to use more coordinates than are strictly necessary, in order to get nicer coordinates or to display the symmetry of the arrangement. For example, arrangements based on the A_n lattice are simpler when described using n+1 coordinates that add to 0. A Fortran program for converting from n+1 coordinates adding to 0 to ordinary n-dimensional coordinates is available here.
• There is some overlap with the tables of packings in dimensions 3, 4 and 5 mentioned in Part 1 of this file, and with the tables of spherical coverings, spherical t-designs, minimal energy arrangements, minimal volume arrangements, etc. listed elsewhere on my home page.
• There are a large number of other arrangements in my files while I am gradually adding to this page. If there is a particular dimension and number of points that you are interested in which is not given here, let me know. This page is under construction.

Dimension 10

• 10 dims, 24 points (secant 10; found by wds and njas April 1990)
• 10 dims, 40 points (best packing; found by wds, beatified by jhc and njas)
• 10 dims, 500 points (points achieving highest kissing number known in 10-D, hence minimal angle 60 degrees. Ref. [SPLAG], Chaps. 1 and 5)