Game of Sloanes

A game to find putatively optimal packings in complex projective space with the goal of proving optimality of as many packings as possible. Formerly hosted here.

This project is licensed under the terms of the Creative Commons Attribution 4.0. Please cite Game of Sloanes; https://github.com/gnikylime/GameofSloanes/; Authors: John Jasper, Emily J. King, and Dustin G. Mixon

This repository concerns the problem of packing $n$ lines, represented by unit vectors, through the origin of $C^d$ such that the angles between the lines are as large as possible. The problem has applications in fields such as compressed sensing, digital fingerprinting, quantum state tomography, and multiple description coding.

Let $\Phi=\left(\varphi_j\right)_{j=1}^n$ be a set of unit norm vectors. The coherence $\mu$ of $\Phi$ is defined to be
$$\mu\left(\Phi\right) = \max_{1\leq j < k\leq n} \vert\langle \varphi_j, \varphi_k \rangle\vert.$$

We call $\Phi$ a Grassmannian frame or (optimal) projective packing if
$$\Phi \in \text{argmin}\lbrace \mu(\Psi) \text{ } : \text{ } \Psi \text{ } n \text{ unit norm vectors in }C^d \rbrace.$$

For a review of results about complex Grassmannian frames, see the paper [JKM19].

Submissions

If you have a collection of $n$ vectors in $C^d$ which has a better coherence (to at least the eighth decimal point) than what is listed above, we welcome you to submit it. We will accept sporadic packings for $d$ and $n$ not listed above if there is evidence that the packing is truly optimal.
Submissions must be in the following format. A .txt file consisting of a newline-separated list of the $2 d n$ entries of the (unit-norm) vectors, starting with the real parts of the components of the first vector, then the real parts of the components of the second vector, and so on before working through the imaginary parts of each vector.
The filenames must be of the form `dxn_init.txt`, where `d` is the dimension, `n` is the number of vectors and `init` is a three- or four-character (i.e., only alphabet characters) string to list in the Creator column.
Please make a branch of the repository and put the new packings in the Packings directory. Then do a pull request. If you would like to send additional information, write to our Gmail account asongofvectorsandangles.

Open Problems

• Conjecture: The frames of $8$ vectors in $C^3$ that result from removing a single vector from any of the known equiangular tight frames of $9$ vectors in $C^3$ are Grassmannian frames.
  Sources: [JKM19]
  (independently conjectured by Henry Cohn)
• Conjecture: Assume that an equiangular tight frame of $d^2$ vectors in $C^d$ exists. Then removing one vector from that equiangular tight frame results in a Grassmannian frame of $d^2-1$ vectors in $C^d$.
  Source: Conjectured by Henry Cohn.
• Conjecture: The following is a Grassmannian frame of $5$ vectors in $C^3$:

  $$\left(\matrix{ a & b & b & c & c \cr b & a & b & c w & cw^2 \cr b & b & a & cw^2 & cw}\right),$$
  $$a = \frac{\sqrt{13}+\sqrt{2+\sqrt{13}}-1}{3\sqrt{3}}, \quad b = \sqrt{\frac{1-a^2}{2}}, \quad c = \frac{1}{\sqrt{3}},\quad w=e^{2\pi i/3}.$$ Source: [JKM19]

• Conjecture: The configuration of $85$ vectors in $C^5$ listed numerically in the leader board and explicitly constructed as the union of the root vectors corresponding to the $45$ $2$-reflections which generate the unitary reflection group $W(K_5)$ and $40$ vectors representing each antipodal pair of minimal vectors of the Coxeter-Todd lattice $O_{10}$ is a Grassmannian frame.
  Source: [BGM+22]
• Open Problem: Generalize the Levenstein and Bukh-Cox bounds to general subspace packings.
  Source: [JKM19]

Leader Board

Packings for all $2 \leq d \leq 7$ and $d+2 \leq n \leq 49$, plus sporadic packings.

Legend and Further Notes

• The Grassmannian frames for $d = n$ are orthonormal bases and for $d+1 = n$ are regular simplices, hence we do not list them.
• Best Coherence gives the best coherence of the packings which have been submitted for that $d$ and $n$. A new packing is considered better if it beats the previous packing in at least the eighth decimal place.
• Lower Bound gives the maximum of whichever of the Bukh-Cox, Welch-Rankin, orthoplex, and/or Levenstein bounds are valid and largest for that $d$ and $n$.
• Creator gives a 3–4 character string indicating the creator of the packing.
  • etf: A known equiangular tight frame. See, e.g., [FM16] and the references therein.
  • orth: The maximal known set of vectors in $`C^d`$ which saturate the orthoplex bound. When $`d`$ is a prime power, this collection is a maximal set of mutually unbiased bases consisting of $`d^2 + d`$ vectors. See, e.g., [KlRo04] and the reference therein.
  • Lev: A set of vectors which saturate the (second) Levenstein bound.
  • B-C: A set of vectors which saturate the Bukh-Cox bound.
  • AUTO: Removing a vector from the best known packing for $d$ and $n+1$ results in a better coherence than any of the submitted packings for $d$ and $n$.
  • njas: Grassmannian frames in $C^2$ are equivalent to optimal spherical codes in $S^2$. The configurations labeled with njas are best known spherical codes in $S^2$ downloaded from [Sloane1].
  • dgm, ejk, and JJ: Dustin G. Mixon, Emily J. King, and John Jasper.
  • hlc: Henry Cohn.
  • gKmR: Gene Kopp and Michael Ren.
  • BGM+22: From the paper [BGM+22].
  • CID+21: From the paper [CID+21]. (Has since been knocked off the leader board.)
• Text File contains a link to a .txt file with the best submitted packing formated as a newline-separated list of the $2 d n$ entries of the vectors, starting with the real parts of the first vector, then the real parts of the second vector, and so on until the imaginary parts of the last vector.
• Notes contain the following information:
  • ○ indicates that the configuration is provably optimal.
  • △ indicates that the configuration is conjectured to be optimal.
  • When pertinent, a reference to a paper or website is listed.
• For a listing of numerically approximated and explicitly defined optimal packings when $n=d^2$, see, e.g., [FSDH], [Grassl], and [Flamm].

d	n	Best Coherence	Lower Bound	Creator	Text File	Notes
2	4	0.57735027	0.57735027	etf	File	○
2	5	0.70710678	0.70710678	AUTO	File	○
2	6	0.70710678	0.70710678	orth	File	○
2	7	0.77786191	0.73029674	njas	File	○
2	8	0.79410449	0.75000000	njas	File	○
2	9	0.81649658	0.77777778	njas	File	○
2	10	0.83797205	0.80000000	njas	File	○
2	11	0.85065081	0.81818182	njas	File	○
2	12	0.85065081	0.83333333	njas	File	○
2	13	0.87824726	0.84615385	njas	File	○
2	14	0.88429359	0.85714286	njas	File	○
2	15	0.89235808	0.86666667	njas	File
2	16	0.89785706	0.87500000	njas	File
2	17	0.90224565	0.88235294	njas	File
2	18	0.90793607	0.88888889	njas	File
2	19	0.91463569	0.89473684	njas	File
2	20	0.91555370	0.90000000	njas	File
2	21	0.92181843	0.90476190	njas	File
2	22	0.92474490	0.90909091	njas	File
2	23	0.92812903	0.91304348	njas	File
2	24	0.92819138	0.91666667	njas	File	○
2	25	0.93471884	0.92000000	njas	File
2	26	0.93655704	0.92307692	njas	File
2	27	0.93765378	0.92592593	njas	File
2	28	0.94160243	0.92857143	njas	File
2	29	0.94347269	0.93103448	njas	File
2	30	0.94380927	0.93333333	njas	File
2	31	0.94633995	0.93548387	njas	File
2	32	0.94699963	0.93750000	njas	File
2	33	0.95036772	0.93939394	njas	File
2	34	0.95157352	0.94117647	njas	File
2	35	0.95287401	0.94285714	njas	File
2	36	0.95321776	0.94444444	njas	File
2	37	0.95522052	0.94594595	njas	File
2	38	0.95566293	0.94736842	njas	File
2	39	0.95759890	0.94871795	njas	File
2	40	0.95842633	0.95000000	njas	File
2	41	0.95948849	0.95121951	njas	File
2	42	0.96003426	0.95238095	njas	File
2	43	0.96104341	0.95348837	njas	File
2	44	0.96130156	0.95454545	njas	File
2	45	0.96287317	0.95555556	njas	File
2	46	0.96372563	0.95652174	njas	File
2	47	0.96413753	0.95744681	njas	File
2	48	0.96418159	0.95833333	njas	File
2	49	0.96609820	0.95918367	njas	File
3	5	0.43425855	0.41406779	dgm	File	△
3	6	0.44721360	0.44721360	etf	File	○
3	7	0.47140452	0.47140452	etf	File	○
3	8	0.50000000	0.48795004	AUTO	File	△
3	9	0.50000000	0.50000000	etf	File	○
3	10	0.57735027	0.57735027	AUTO	File	○
3	11	0.57735027	0.57735027	AUTO	File	○
3	12	0.57735027	0.57735027	orth	File	○
3	13	0.62214387	0.59160798	dgm	File
3	14	0.63763053	0.60302269	JJ	File
3	15	0.64038820	0.61237244	JJ	File
3	16	0.64775448	0.62017367	hlc	File
3	17	0.65436636	0.62678317	JJ	File
3	18	0.66295789	0.63245553	hlc	File
3	19	0.67647327	0.63737744	hlc	File
3	20	0.68646389	0.64168895	hlc	File
3	21	0.68797170	0.64549722	hlc	File
3	22	0.70357473	0.64888568	hlc	File
3	23	0.70964840	0.65192024	hlc	File
3	24	0.71921927	0.65465367	hlc	File
3	25	0.72566532	0.65712874	hlc	File
3	26	0.72973896	0.65938047	hlc	File
3	27	0.73417690	0.66143783	hlc	File
3	28	0.73682183	0.66332496	hlc	File
3	29	0.74150196	0.66506217	hlc	File
3	30	0.74480938	0.66666667	hlc	File
3	31	0.74532713	0.66815310	hlc	File
3	32	0.74813017	0.66953406	hlc	File
3	33	0.75301857	0.67082039	hlc	File
3	34	0.75686171	0.67202151	hlc	File
3	35	0.75965558	0.67314560	hlc	File
3	36	0.76088362	0.67419986	hlc	File
3	37	0.76619123	0.67519060	hlc	File
3	38	0.77222053	0.67612340	hlc	File
3	39	0.77507569	0.67974369	hlc	File
3	40	0.77793911	0.68377223	hlc	File
3	41	0.77848807	0.68765248	hlc	File
3	42	0.77853167	0.69139330	hlc	File
3	43	0.78370830	0.69500286	hlc	File
3	44	0.78689437	0.69848866	AUTO	File
3	45	0.78689437	0.70185760	JJ	File
3	46	0.79438661	0.70511609	hlc	File
3	47	0.79703767	0.70827002	hlc	File
3	48	0.79945875	0.71132487	hlc	File
3	49	0.80156708	0.71428571	hlc	File
4	6	0.32732684	0.32278096	dgm	File
4	7	0.35355339	0.35355339	etf	File	○
4	8	0.37796447	0.37796447	etf	File	○
4	9	0.40185012	0.39528471	hlc	File
4	10	0.41077812	0.40824829	hlc	File
4	11	0.42543778	0.41833001	hlc	File
4	12	0.42774232	0.42640143	JJ	File
4	13	0.43301270	0.43301270	etf	File	○
4	14	0.44721360	0.43852901	AUTO	File
4	15	0.44721360	0.44320263	AUTO	File	△
4	16	0.44721360	0.44721360	etf	File	○
4	17	0.50000000	0.50000000	AUTO	File	○
4	18	0.50000000	0.50000000	AUTO	File	○
4	19	0.50000000	0.50000000	AUTO	File	○
4	20	0.50000000	0.50000000	orth	File	○
4	21	0.53721579	0.50874702	hlc	File
4	22	0.54543820	0.51639778	hlc	File
4	23	0.55046239	0.52314836	hlc	File
4	24	0.55701605	0.52915026	hlc	File
4	25	0.56649311	0.53452248	hlc	File
4	26	0.57175264	0.53935989	hlc	File
4	27	0.57499858	0.54373907	hlc	File
4	28	0.57718288	0.54772256	hlc	File
4	29	0.57733909	0.55136195	hlc	File
4	30	0.57735019	0.55470020	hlc	File
4	31	0.57735027	0.55777335	AUTO	File
4	32	0.57735027	0.56061191	AUTO	File
4	33	0.57735027	0.56324185	AUTO	File
4	34	0.57735027	0.56568542	AUTO	File
4	35	0.57735027	0.56796183	AUTO	File
4	36	0.57735027	0.57008771	AUTO	File
4	37	0.57735027	0.57207755	AUTO	File
4	38	0.57735027	0.57394404	AUTO	File
4	39	0.57735027	0.57569833	AUTO	File
4	40	0.57735027	0.57735027	Lev	File	○
4	41	0.62181158	0.57890857	hlc	File
4	42	0.62418264	0.58038100	hlc	File
4	43	0.62691117	0.58177447	hlc	File
4	44	0.63028266	0.58309519	hlc	File
4	45	0.63207330	0.58434871	hlc	File
4	46	0.63583562	0.58554004	hlc	File
4	47	0.63981695	0.58667371	hlc	File
4	48	0.64342772	0.58775381	hlc	File
4	49	0.64632889	0.58878406	hlc	File
4	64	0.69749229	0.60000000	ejk	File
5	7	0.26635681	0.26447408	dgm	File
5	8	0.29520808	0.29277002	hlc	File
5	9	0.32011714	0.31622777	hlc	File
5	10	0.33333333	0.33333333	etf	File	○
5	11	0.34641016	0.34641016	etf	File	○
5	12	0.35738925	0.35675303	hlc	File
5	13	0.36760232	0.36514837	JJ	File
5	14	0.37567476	0.37210420	hlc	File
5	15	0.37993678	0.37796447	hlc	File
5	16	0.38809284	0.38297084	hlc	File
5	17	0.39353551	0.38729833	hlc	File
5	18	0.39763025	0.39107694	dgm	File
5	19	0.39763025	0.39440532	AUTO	File
5	20	0.39763025	0.39735971	JJ	File
5	21	0.40000000	0.40000000	etf	File	○
5	22	0.40824829	0.40237391	AUTO	File
5	23	0.40824829	0.40451992	AUTO	File
5	24	0.40824829	0.40646942	AUTO	File	△
5	25	0.40824829	0.40824829	etf	File	○
5	26	0.44721360	0.44721360	AUTO	File	○
5	27	0.44721360	0.44721360	AUTO	File	○
5	28	0.44721360	0.44721360	AUTO	File	○
5	29	0.44721360	0.44721360	AUTO	File	○
5	30	0.44721360	0.44721360	orth	File	○
5	31	0.48159575	0.45291081	hlc	File
5	32	0.48688165	0.45812285	hlc	File
5	33	0.49102490	0.46291005	hlc	File
5	34	0.49759420	0.46732302	hlc	File
5	35	0.50000000	0.47140452	AUTO	File
5	36	0.50000000	0.47519096	AUTO	File
5	37	0.50000000	0.47871355	AUTO	File
5	38	0.50000000	0.48199920	AUTO	File
5	39	0.50000000	0.48507125	AUTO	File
5	40	0.50000000	0.48795004	AUTO	File
5	41	0.50000000	0.49065338	AUTO	File
5	42	0.50000000	0.49319696	AUTO	File
5	43	0.50000000	0.49559463	AUTO	File
5	44	0.50000000	0.49785866	AUTO	File
5	45	0.50000000	0.50000000	Lev	File	○
5	46	0.52760056	0.50202841	hlc	File
5	47	0.53731871	0.50395263	hlc	File
5	48	0.54098230	0.50578054	hlc	File
5	49	0.54351100	0.50751922	hlc	File
5	50	0.56191071	0.50917508	JJ	File
5	85	0.57735027	0.54006172	BGMP	File	△[BGM+22]
6	8	0.22400924	0.22400924	B-C	File	○
6	9	0.25000000	0.25000000	etf	File	○
6	10	0.27220315	0.27216553	hlc	File
6	11	0.28867513	0.28867513	etf	File	○
6	12	0.30151134	0.30151134	etf	File	○
6	13	0.31182456	0.31180478	hlc	File
6	14	0.32031664	0.32025631	hlc	File
6	15	0.32739651	0.32732684	hlc	File
6	16	0.33333333	0.33333333	etf	File	○
6	17	0.33951091	0.33850160	hlc	File
6	18	0.34498346	0.34299717	hlc	File
6	19	0.34925830	0.34694433	hlc	File
6	20	0.35236257	0.35043832	hlc	File
6	21	0.35644714	0.35355339	hlc	File
6	22	0.35955441	0.35634832	hlc	File
6	23	0.36537169	0.35887028	hlc	File
6	24	0.36998252	0.36115756	hlc	File
6	25	0.37267800	0.36324158	AUTO	File
6	26	0.37267800	0.36514837	AUTO	File
6	27	0.37267800	0.36689969	AUTO	File
6	28	0.37267800	0.36851387	AUTO	File
6	29	0.37267800	0.37000643	AUTO	File
6	30	0.37267800	0.37139068	AUTO	File
6	31	0.37267800	0.37267800	etf	File	○
6	32	0.37796447	0.37387825	AUTO	File
6	33	0.37796447	0.37500000	AUTO	File
6	34	0.37796447	0.37605072	AUTO	File
6	35	0.37796447	0.37703695	AUTO	File	△
6	36	0.37796447	0.37796447	etf	File	○
6	37	0.40824829	0.40824829	orth	File	○
6	38	0.41684076	0.40824829	hlc	File
6	39	0.42301555	0.40824829	hlc	File
6	40	0.42843380	0.40824829	hlc	File
6	41	0.43384916	0.40824829	hlc	File
6	42	0.43731303	0.40824829	hlc	File
6	43	0.44210040	0.41217007	hlc	File
6	44	0.44751763	0.41585133	hlc	File
6	45	0.45132461	0.41931393	hlc	File
6	46	0.45581416	0.42257713	hlc	File
6	47	0.45969656	0.42565792	hlc	File
6	48	0.46279728	0.42857143	hlc	File
6	49	0.46609460	0.43133109	hlc	File
7	9	0.19601846	0.19428362	JJ	File
7	10	0.22157245	0.21951220	hlc	File
7	11	0.23931889	0.23904572	hlc	File
7	12	0.25510428	0.25482360	hlc	File
7	13	0.26727220	0.26726124	hlc	File
7	14	0.27735010	0.27735010	etf	File	○
7	15	0.28571429	0.28571429	etf	File	○
7	16	0.29279313	0.29277002	hlc	File
7	17	0.29885987	0.29880715	hlc	File
7	18	0.30417036	0.30403450	hlc	File
7	19	0.30884718	0.30860670	hlc	File
7	20	0.31334596	0.31264095	hlc	File
7	21	0.31733742	0.31622777	gKmR	File
7	22	0.32172912	0.31943828	hlc	File
7	23	0.32539307	0.32232919	hlc	File
7	24	0.32960269	0.32494624	hlc	File
7	25	0.33248007	0.32732684	hlc	File
7	26	0.33318270	0.32950179	hlc	File
7	27	0.33333333	0.33149677	AUTO	File
7	28	0.33333333	0.33333333	etf	File	○
7	29	0.34081228	0.33502970	hlc	File
7	30	0.34639121	0.33660139	hlc	File
7	31	0.34990310	0.33806170	hlc	File
7	32	0.35260718	0.33942212	hlc	File
7	33	0.35355339	0.34069257	AUTO	File
7	34	0.35355339	0.34188173	AUTO	File
7	35	0.35355339	0.34299717	AUTO	File
7	36	0.35355339	0.34404556	AUTO	File
7	37	0.35355339	0.34503278	AUTO	File
7	38	0.35355339	0.34596404	AUTO	File
7	39	0.35355339	0.34684399	AUTO	File
7	40	0.35355339	0.34767675	AUTO	File
7	41	0.35355339	0.34846603	AUTO	File
7	42	0.35355339	0.34921515	AUTO	File
7	43	0.35355339	0.34992711	AUTO	File
7	44	0.35355339	0.35060460	AUTO	File
7	45	0.35355339	0.35125009	AUTO	File
7	46	0.35355339	0.35186578	AUTO	File
7	47	0.35355339	0.35245369	AUTO	File
7	48	0.35355339	0.35301567	AUTO	File	△
7	49	0.35355339	0.35355339	etf	File	○
8	15	0.25000000	0.25000000	etf	File	○
8	57	0.33071891	0.33071891	etf	File	○
8	64	0.33333333	0.33333333	etf	File	○
9	13	0.19245009	0.19245009	etf	File	○
9	19	0.24845200	0.24845200	etf	File	○
9	37	0.29397237	0.29397237	etf	File	○
9	73	0.31426968	0.31426968	etf	File	○
9	81	0.31622777	0.31622777	etf	File	○
10	19	0.22360680	0.22360680	etf	File	○
10	25	0.25000000	0.25000000	etf	File	○
10	91	0.30000000	0.30000000	etf	File	○
10	100	0.30151134	0.30151134	etf	File	○
11	23	0.22268089	0.22268089	etf	File	○
11	121	0.28867513	0.28867513	etf	File	○
12	16	0.14907120	0.14907120	etf	File	○
12	23	0.20412415	0.20412415	etf	File	○
12	45	0.25000000	0.25000000	etf	File	○
16	21	0.12500000	0.12500000	etf	File	○
16	80	0.25000000	0.22501758	JJ	File

Last updated: 2025-06-04.

Code

See the readme file in the Matlab directory. We also encourage submission of code in other languages. Please create a branch and add a directory named `LanguageTools` where `Language` is the language the code is written in.

Papers

Review Paper

[JKM19] John Jasper, Emily J. King, and Dustin G. Mixon: “Game of Sloanes: Best known packings in complex projective space.” Wavelets and Sparsity XVIII, SPIE Proceedings 11138, 416–425 [arXiv]

Further Literature and Websites

[BGM+22] Dmitriy Bilyk, Alexey Glazyrin, Ryan Matzke, Josiah Park, and Oleksandr Vlasiuk: “Optimal measures for $p$-frame energies on spheres.” Rev. Mat. Iberoam. 38(4), 1129–1160 (2022)
[CID+21] Diego Cuevas, Carlos Beltrán, Ignacio Santamaria, Vít Tuček, and Gunnar Peters: “A Fast Algorithm for Designing Grassmannian Constellations.” (2021) https://gtas.unican.es/files/pub/non_coherent_wsa21_full_paper.pdf, Conference: WSA 2021 - 25th International ITG Workshop on Smart Antennas
[FM16] Matt Fickus and Dustin G. Mixon: “Tables of the existence of equiangular tight frames.” (2016) [arXiv]
[Flamm] Steve Flammia: “Exact SIC fiducial vectors.” http://www.physics.usyd.edu.au/~sflammia/SIC/
[FSDH] Christopher Fuchs, Blake Stacey, John DeBrota, and Michael Hoang: “QBism: Quantum Theory as a Hero's Handbook: SIC-POVM Solutions.” http://www.physics.umb.edu/Research/QBism/solutions.html
[Grassl] Markus Grassl and Andrew J. Scott: “SIC-POVMs” http://sicpovm.markus-grassl.de/
[KlRo04] Andreas Klappenecker and Martin Rötteler: “Constructions of mutually unbiased bases.” (2004) in Finite fields and applications, Lecture Notes in Comput. Sci. 2948, 137–144, Springer, Berlin. [arXiv]
[MeDa14a] Ahmed Medra and Timothy N. Davidson: “Flexible codebook design for limited feedback systems via sequential smooth optimization on the Grassmannian manifold.” IEEE Trans. Signal Process. 62(5), 1305–1318 (2014)
[MeDa14b] Ahmed Medra and Timothy N. Davidson: “Flexible codebook design for limited feedback systems.” http://www.ece.mcmaster.ca/~davidson/pubs/Flexible_codebook_design.html
[Sloane1] Neil J. A. Sloane: “Spherical codes.” http://neilsloane.com/packings/
[Sloane2] Neil J. A. Sloane: “How to pack lines, planes, $3$-spaces, etc.” http://neilsloane.com/grass/
[TDHS05] Joel A. Tropp, Inderjit S. Dhillon, Robert W. Heath, Jr., and Thomas Strohmer: “Designing structured tight frames via alternating projection.” IEEE Trans. Info. Theory 51(1), 188–209 (2005)