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Spherical code records

A spherical code is a set of N points on the unit sphere in d dimensions, spread out so that no two are close. The aim is to make the largest cosine between two of the points, the cosine of the smallest angle between them, as small as possible. Henry Cohn's table at spherical-codes.org keeps the best code known for every dimension from 3 to 32 and up to 1,024 points.

Thousands of entries in that table are marked as needing more optimization. We start from the table's code, search nearby for a better arrangement by repeated annealing (L-BFGS on a smooth log-sum-exp surrogate of the largest cosine) and polish the result with linear programming. Each new code is checked in 50-digit arithmetic before it is uploaded, and spherical-codes.org recomputes it before listing it.

We upload only gains of at least 10⁻¹⁰ in the largest cosine. Smaller gains, which only finish converging a code, are kept back.

101 listed on the public table.

101 spherical codes

Spherical code records
Entry	Ours	Previous best	Status
N = 109, d = 18	0.167620271519	0.167620272036	listed
N = 110, d = 18	0.168420217286	0.168420354627	listed
N = 111, d = 18	0.169248563774	0.169248610521	listed
N = 77, d = 19	0.112762638197	0.112762647688	listed
N = 86, d = 20	0.112907564559	0.112907857991	listed
N = 87, d = 20	0.112908153470	0.112908491648	listed
N = 88, d = 20	0.112909648241	0.112909650540	listed
N = 89, d = 20	0.112909670887	0.112909693772	listed
N = 90, d = 20	0.112912075424	0.112912295598	listed
N = 54, d = 21	0.066663644239	0.066663853221	listed
N = 55, d = 21	0.066665986715	0.066666055545	listed
N = 58, d = 22	0.066624188773	0.066624214466	listed
N = 59, d = 22	0.066650536244	0.066650782089	listed
N = 60, d = 22	0.066659474283	0.066659847970	listed
N = 63, d = 22	0.076900574526	0.076901515044	listed
N = 64, d = 22	0.076911581577	0.076914997582	listed
N = 65, d = 22	0.076923311915	0.076923314447	listed
N = 54, d = 23	0.047608397767	0.047608618840	listed
N = 60, d = 23	0.066269410579	0.066270318328	listed
N = 61, d = 23	0.066447854856	0.066448344541	listed
N = 62, d = 23	0.066551175474	0.066551990648	listed
N = 67, d = 23	0.076785993943	0.076786919884	listed
N = 68, d = 23	0.076889225196	0.076891127592	listed
N = 73, d = 23	0.081987992074	0.081988022906	listed
N = 76, d = 23	0.083308858656	0.083312231142	listed
N = 77, d = 23	0.083321671336	0.083322942411	listed
N = 78, d = 23	0.083328741241	0.083329842739	listed
N = 79, d = 23	0.083331588958	0.083331899393	listed
N = 84, d = 23	0.089247313946	0.089247322022	listed
N = 71, d = 24	0.076913858840	0.076916261172	listed
N = 109, d = 24	0.110473394569	0.110473427027	listed
N = 65, d = 25	0.064022917420	0.064023059784	listed
N = 66, d = 25	0.065037151687	0.065037684675	listed
N = 67, d = 25	0.066205650071	0.066206034211	listed
N = 68, d = 25	0.066373681489	0.066382704492	listed
N = 71, d = 25	0.071751493696	0.071751494072	listed
N = 72, d = 25	0.073183291145	0.073183293967	listed
N = 79, d = 25	0.078439938518	0.078439939068	listed
N = 82, d = 25	0.079706101579	0.079706316626	listed
N = 94, d = 25	0.089885991544	0.089886926092	listed
N = 109, d = 25	0.095541737033	0.095541930219	listed
N = 55, d = 26	0.039998800127	0.039998823360	listed
N = 68, d = 26	0.063557397602	0.063566535221	listed
N = 69, d = 26	0.064912235089	0.064912312965	listed
N = 76, d = 26	0.074795080392	0.074797298190	listed
N = 78, d = 26	0.076825261390	0.076830183084	listed
N = 79, d = 26	0.076903784658	0.076905277345	listed
N = 80, d = 26	0.076921526628	0.076923144687	listed
N = 86, d = 26	0.079365491814	0.079365497085	listed
N = 96, d = 26	0.085855599552	0.085856265348	listed
N = 97, d = 26	0.088469319016	0.088469338413	listed
N = 70, d = 27	0.062920386869	0.062920512188	listed
N = 72, d = 27	0.064129341354	0.064156764338	listed
N = 74, d = 27	0.066293902946	0.066298537397	listed
N = 75, d = 27	0.067691814168	0.067692400463	listed
N = 77, d = 27	0.072325078729	0.072325256490	listed
N = 80, d = 27	0.075886968557	0.076100396641	listed
N = 81, d = 27	0.076589712124	0.076590294576	listed
N = 82, d = 27	0.076775658093	0.076778465575	listed
N = 83, d = 27	0.076790451383	0.076799847675	listed
N = 84, d = 27	0.076823977220	0.076834258077	listed
N = 91, d = 27	0.079745266281	0.079746855333	listed
N = 95, d = 27	0.080684157650	0.080684181553	listed
N = 99, d = 27	0.084112454286	0.084112497386	listed
N = 75, d = 28	0.064357114567	0.064374628646	listed
N = 76, d = 28	0.064711737596	0.064711751358	listed
N = 77, d = 28	0.065631381324	0.065907662195	listed
N = 84, d = 28	0.075952270167	0.075952483304	listed
N = 85, d = 28	0.076607840234	0.076614614664	listed
N = 87, d = 28	0.076798434182	0.076799993641	listed
N = 88, d = 28	0.076920091356	0.076920728338	listed
N = 94, d = 28	0.079366008823	0.079366041498	listed
N = 95, d = 28	0.079648720231	0.079649162233	listed
N = 96, d = 28	0.079998141117	0.080006188634	listed
N = 99, d = 28	0.080627621938	0.080628081226	listed
N = 104, d = 28	0.084695007185	0.084695086895	listed
N = 106, d = 28	0.085830131449	0.086167575188	listed
N = 82, d = 29	0.068321142664	0.068547694244	listed
N = 83, d = 29	0.070355191635	0.070355413136	listed
N = 89, d = 29	0.076432114627	0.076432617937	listed
N = 91, d = 29	0.076949219854	0.076950752944	listed
N = 102, d = 29	0.080258082868	0.080258137137	listed
N = 103, d = 29	0.080647469962	0.080647606869	listed
N = 104, d = 29	0.080662303818	0.080662618816	listed
N = 107, d = 29	0.082780285200	0.082780285565	listed
N = 80, d = 30	0.062320868659	0.062498581860	listed
N = 83, d = 30	0.064957732986	0.065099117014	listed
N = 97, d = 30	0.077147213685	0.077147356932	listed
N = 100, d = 30	0.077767755807	0.077767758050	listed
N = 103, d = 30	0.078512271579	0.078512280682	listed
N = 106, d = 30	0.080475842847	0.080477209110	listed
N = 107, d = 30	0.080659786287	0.080660373476	listed
N = 108, d = 30	0.080811732539	0.080811777145	listed
N = 111, d = 30	0.082299490556	0.082299565829	listed
N = 86, d = 31	0.064362071287	0.064362086420	listed
N = 89, d = 31	0.068505957449	0.068506081822	listed
N = 90, d = 31	0.069720204888	0.069720206572	listed
N = 99, d = 31	0.076842668064	0.076844264811	listed
N = 84, d = 32	0.058871367237	0.058872175056	listed
N = 87, d = 32	0.061573626214	0.062109454391	listed
N = 110, d = 32	0.077978427249	0.077978641842	listed

Questions

What is a spherical code?

A set of N points on the unit sphere in d dimensions chosen so that the smallest angle between any two of them is as large as possible, equivalently the largest cosine is as small as possible. The best codes matter in coding theory, signal design and the study of sphere packings.

Who keeps the table?

Henry Cohn, at spherical-codes.org. It lists the best known code for each dimension from 3 to 32 and up to 1,024 points, with the source of each.

Are these codes optimal?

Not proved. Optimal spherical codes are known only for a few sizes; these are the best known codes for their sizes, each better than the one listed before it.