Table of Grassmannian Packings

N. J. A. Sloane: Table of Best Grassmannian Packings

Gives best packings known of N n-spaces in m-space.

Based on joint work with A. R. Calderbank, J. H. Conway, R. H. Hardin, E. M. Rains, P. W. Shor and others.

Grassmannian packings home page

The following table gives the best packing known to us of N n-dimensional subspaces in ordinary real Euclidean m-space.

The case n=1, m=3 thus corresponds to the best way to separate N laser beams passing through the center of an ordinary sphere.

The entry gives the largest known minimal squared distance D.

For n=1 this is defined to be sin(theta)^2, where theta is the minimal angle between any two of the beams.

In general the squared distance between two n-spaces is the sum of the squares of the sines of the n principal angles between the two spaces.

The table is in two sections. The first section gives packings found (mostly) by computer, the second gives packings constructed (mostly) by hand.

In the lower dimensions the packings labeled with file names are the best found after very extensive computer searches. But for the higher-dimensional packings and for larger values of N only a few runs were made, in order to have a rough standard by which to judge other constructions.

Any improvements to this table should be sent to
Neil J. A. Sloane, AT&T Research, Room 2C-376, 600 Mountain Ave, Murray Hill, NJ 07974 USA
Email address: njasloane@gmail.com; Fax: 908 582 2379; Home page
All contributions will be acknowledged in later versions of the table.

If you have a nice construction that gives packings that only match ours, rather than giving new records, we would also like to hear about it, since the majority of the these packings were found by computer search, and the results have not been beautified (except in the most symmetrical cases).

Last modified Jan 20 1997

Sources and References

CHS96: J. H. Conway, R. H. Hardin and N. J. A. Sloane, Packing Lines, Planes, etc., Packings in Grassmannian Spaces, Experimental Mathematics, Vol. 5 (1996), 139-159 [postscript, pdf]. Many of these packings are available through the Grassmannian packings home page

SS96: P. W. Shor and N. J. A. Sloane, A family of optimal packings in Grassmannian manifolds, J. Algebraic Combinatorics, to appear (1997). [postscript, pdf].

CCKS96: Calderbank-Cameron-Kantor-Seidel (1996), Z4-Kerdock sets, orthogonal spreads, and extremal Euclidean line-sets, (to appear in J. London Math. Soc.)

grass3: A. R. Calderbank, R. H. Hardin, E. M. Rains, P. W. Shor and N. J. A. Sloane, "A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces" [postscript, pdf].

For further references see the Grassmannian packings home page

The Table of Grassmannian Packings

The entries give the parent (or mother) dimension m, the dimension n of the subspaces being packed, the number N of subspaces in the packing, the minimal squared distance D achieved, and then a file name, reference or other comments.

m n N D name

3 1 3 1.000000000000000 /usr/rhh/grass/best.3.1.3 3 1 4 0.888888888888889 /usr/rhh/grass/best.3.1.4 3 1 5 0.799999999999999 /usr/rhh/grass/best.3.1.5 3 1 6 0.800000000000000 /usr/rhh/grass/best.3.1.6 3 1 7 0.666666666666667 /usr/rhh/grass/best.3.1.7 3 1 8 0.580628514621758 /usr/rhh/grass/best.3.1.8 3 1 9 0.551954085526548 /usr/rhh/grass/best.3.1.9 3 1 10 0.529210992451761 /usr/rhh/grass/OFFICIAL/best.3.1.10 3 1 11 0.489583348932409 /usr/rhh/grass/best.3.1.11 3 1 12 0.445688721477877 /usr/rhh/grass/best.3.1.12 3 1 13 0.409964971108470 /usr/rhh/grass/best.3.1.13 3 1 14 0.390628992130893 /usr/rhh/grass/best.3.1.14 3 1 15 0.381325614173148 /usr/rhh/grass/best.3.1.15 3 1 16 0.368524269666694 /usr/rhh/grass/best.3.1.16 3 1 17 0.332854698888097 /usr/rhh/grass/best.3.1.17 3 1 18 0.319330521178142 /usr/rhh/grass/best.3.1.18 3 1 19 0.300009387775088 /usr/rhh/grass/best.3.1.19 3 1 20 0.291972234676310 /usr/rhh/grass/best.3.1.20 3 1 21 0.284209946852422 /usr/rhh/grass/best.3.1.21 3 1 22 0.279189472772918 /usr/rhh/grass/best.3.1.22 3 1 23 0.257690403520805 /usr/rhh/grass/best.3.1.23 3 1 24 0.252464036071819 /usr/rhh/grass/best.3.1.24 3 1 25 0.238729837953039 /usr/rhh/grass/best.3.1.25 3 1 26 0.230800083465611 /usr/rhh/grass/best.3.1.26 3 1 27 0.224023578236251 /usr/rhh/grass/best.3.1.27 3 1 28 0.218197811785817 /usr/rhh/grass/best.3.1.28 3 1 29 0.213560845613763 /usr/rhh/grass/best.3.1.29 3 1 30 0.206083829524788 /usr/rhh/grass/best.3.1.30 3 1 31 0.199074255890140 /usr/rhh/grass/best.3.1.31 3 1 32 0.191478238089024 /usr/rhh/grass/best.3.1.32 3 1 33 0.186355763864167 /usr/rhh/grass/best.3.1.33 3 1 34 0.182051475639965 /usr/rhh/grass/best.3.1.34 3 1 35 0.176873541869869 /usr/rhh/grass/best.3.1.35 3 1 36 0.172969977394539 /usr/rhh/grass/best.3.1.36 3 1 37 0.169159004330758 /usr/rhh/grass/best.3.1.37 3 1 38 0.166585408506718 /usr/rhh/grass/best.3.1.38 3 1 39 0.163407377655453 /usr/rhh/grass/best.3.1.39 3 1 40 0.156827559568762 /usr/rhh/grass/best.3.1.40 3 1 41 0.152564470649171 /usr/rhh/grass/best.3.1.41 3 1 42 0.149016165933882 /usr/rhh/grass/best.3.1.42 3 1 43 0.146919917606886 /usr/rhh/grass/best.3.1.43 3 1 44 0.142778750069872 /usr/rhh/grass/best.3.1.44 3 1 45 0.140913249295026 /usr/rhh/grass/best.3.1.45 3 1 46 0.138426779180331 /usr/rhh/grass/best.3.1.46 3 1 47 0.136244207001001 /usr/rhh/grass/best.3.1.47 3 1 48 0.133922710127452 /usr/rhh/grass/best.3.1.48 3 1 49 0.130314072572911 /usr/rhh/grass/best.3.1.49 3 1 50 0.127171795359092 /usr/rhh/grass/best.3.1.50 3 1 51 0.124832568762863 /usr/rhh/grass/best.3.1.51 3 1 52 0.122547134450475 /usr/rhh/grass/best.3.1.52 3 1 53 0.120006865673398 /usr/rhh/grass/best.3.1.53 3 1 54 0.118727643427711 /usr/rhh/grass/best.3.1.54 3 1 55 0.118140805167447 /usr/rhh/grass/best.3.1.55 3 1 56 0.115388134484211 /usr/rhh/grass/best.3.1.56 3 1 57 0.113009404837890 /usr/rhh/grass/best.3.1.57 3 1 58 0.111296389735414 /usr/rhh/grass/best.3.1.58 3 1 59 0.109420282880937 /usr/rhh/grass/best.3.1.59 3 1 60 0.107978607847378 /usr/rhh/grass/best.3.1.60 3 1 61 0.105833705290657 /usr/rhh/grass/best.3.1.61 3 1 62 0.104062838945961 /usr/rhh/grass/best.3.1.62 3 1 63 0.102659683328511 /usr/rhh/grass/best.3.1.63 3 1 64 0.100763118268079 /usr/rhh/grass/best.3.1.64 3 1 65 0.099555285967510 /usr/rhh/grass/best.3.1.65 3 1 66 0.098873014014444 /usr/rhh/grass/best.3.1.66 3 1 67 0.097379846763750 /usr/rhh/grass/best.3.1.67 3 1 68 0.095730715402984 /usr/rhh/grass/best.3.1.68 3 1 69 0.094462004445983 /usr/rhh/grass/best.3.1.69 3 1 70 0.093184210872874 /usr/rhh/grass/best.3.1.70 3 1 71 0.092340029732400 /usr/rhh/grass/best.3.1.71 3 1 72 0.090772540025765 /usr/rhh/grass/best.3.1.72 3 1 73 0.089432553170031 /usr/rhh/grass/best.3.1.73 3 1 74 0.088109008390916 /usr/rhh/grass/best.3.1.74 3 1 75 0.087363675809667 /usr/rhh/grass/best.3.1.75 3 1 76 0.085801930750855 /usr/rhh/grass/best.3.1.76 3 1 77 0.084783708232720 /usr/rhh/grass/best.3.1.77 3 1 78 0.083881651421912 /usr/rhh/grass/best.3.1.78 3 1 79 0.082942435486681 /usr/rhh/grass/best.3.1.79 3 1 80 0.082061980721853 /usr/rhh/grass/best.3.1.80 3 1 81 0.081619715138100 /usr/rhh/grass/best.3.1.81 3 1 82 0.080090700083675 /usr/rhh/grass/best.3.1.82 3 1 83 0.079128078721218 /usr/rhh/grass/best.3.1.83 3 1 84 0.078258923540841 /usr/rhh/grass/best.3.1.84 3 1 85 0.077213946247774 /usr/rhh/grass/best.3.1.85 3 1 86 0.076281287948203 /usr/rhh/grass/best.3.1.86 3 1 87 0.075388084983349 /usr/rhh/grass/best.3.1.87 3 1 88 0.074598157508994 /usr/rhh/grass/best.3.1.88 3 1 89 0.073747510789452 /usr/rhh/grass/best.3.1.89 3 1 90 0.072919720948541 /usr/rhh/grass/best.3.1.90 3 1 91 0.072130629031819 /usr/rhh/grass/best.3.1.91 3 1 92 0.071468599736074 /usr/rhh/grass/best.3.1.92 3 1 93 0.070863040752315 /usr/rhh/grass/best.3.1.93 3 1 94 0.070514708899570 /usr/rhh/grass/best.3.1.94 3 1 95 0.069719023292807 /usr/rhh/grass/best.3.1.95 3 1 96 0.069381997576942 /usr/rhh/grass/best.3.1.96 3 1 97 0.068301654258732 /usr/rhh/grass/best.3.1.97 3 1 98 0.067550138551399 /usr/rhh/grass/best.3.1.98 3 1 99 0.066860735744301 /usr/rhh/grass/best.3.1.99 3 1 100 0.066033482100068 /usr/rhh/grass/best.3.1.100 4 1 3 1.000000000000000 /usr/rhh/grass/best.4.1.3 4 1 4 1.000000000000000 /usr/rhh/grass/best.4.1.4 4 1 5 0.937500000000000 /usr/rhh/grass/best.4.1.5 4 1 6 0.888888888888889 /usr/rhh/grass/best.4.1.6 4 1 7 0.847597050800551 /usr/rhh/grass/best.4.1.7 4 1 8 0.828427124746189 /usr/rhh/grass/best.4.1.8 4 1 9 0.811419515303501 /usr/rhh/grass/best.4.1.9 4 1 10 0.811419515303554 /usr/rhh/grass/best.4.1.10 4 1 11 0.749999999999998 /usr/rhh/grass/best.4.1.11 4 1 12 0.750000000000000 /usr/rhh/grass/best.4.1.12 4 1 13 0.678607075867782 /usr/rhh/grass/best.4.1.13 4 1 14 0.651809706125386 /usr/rhh/grass/best.4.1.14 4 1 15 0.629436328666762 /usr/rhh/grass/best.4.1.15 4 1 16 0.618033988749893 /usr/rhh/grass/best.4.1.16 4 1 17 0.602026118389278 /usr/rhh/grass/best.4.1.17 4 1 18 0.594680204918101 /usr/rhh/grass/best.4.1.18 4 1 19 0.581845195281235 /usr/rhh/grass/best.4.1.19 4 1 20 0.573609548666094 /usr/rhh/grass/best.4.1.20 4 1 21 0.561764947362636 /usr/rhh/grass/best.4.1.21 4 1 22 0.548089696226266 /usr/rhh/grass/best.4.1.22 4 1 23 0.526348543164106 /usr/rhh/grass/best.4.1.23 4 1 24 0.518284221171048 /usr/rhh/grass/best.4.1.24 4 1 25 0.499076625451418 /usr/rhh/grass/best.4.1.25 4 1 26 0.488718539171565 /usr/rhh/grass/best.4.1.26 4 1 27 0.474756670266025 /usr/rhh/grass/best.4.1.27 4 1 28 0.467848201812152 /usr/rhh/grass/best.4.1.28 4 1 29 0.459334948855484 /usr/rhh/grass/best.4.1.29 4 1 30 0.452339411515362 /usr/rhh/grass/best.4.1.30 4 1 31 0.448062826203093 /usr/rhh/grass/best.4.1.31 4 1 32 0.446961067787606 /usr/rhh/grass/best.4.1.32 4 1 33 0.438333260834811 /usr/rhh/grass/best.4.1.33 4 1 34 0.429422967079093 /usr/rhh/grass/best.4.1.34 4 1 35 0.425813386436871 /usr/rhh/grass/best.4.1.35 4 1 36 0.424067611172926 /usr/rhh/grass/best.4.1.36 4 1 37 0.420897337156133 /usr/rhh/grass/best.4.1.37 4 1 38 0.420780576046273 /usr/rhh/grass/best.4.1.38 4 1 39 0.407675863868527 /usr/rhh/grass/best.4.1.39 4 1 40 0.396446609406726 /usr/rhh/grass/OFFICIAL/best.4.1.40 4 1 41 0.388112925398326 /usr/rhh/grass/best.4.1.41 4 1 42 0.384285379993366 /usr/rhh/grass/best.4.1.42 4 1 43 0.375373141179475 /usr/rhh/grass/best.4.1.43 4 1 44 0.368019855106968 /usr/rhh/grass/best.4.1.44 4 1 45 0.364192528549638 /usr/rhh/grass/best.4.1.45 4 1 46 0.362176444508696 /usr/rhh/grass/best.4.1.46 4 1 47 0.355404521399127 /usr/rhh/grass/best.4.1.47 4 1 48 0.351454047462951 /usr/rhh/grass/best.4.1.48 4 1 49 0.347764862913922 /usr/rhh/grass/best.4.1.49 4 1 50 0.346744163946879 /usr/rhh/grass/best.4.1.50 4 1 51 0.345595783734413 /usr/rhh/grass/best.4.1.51 4 1 52 0.345493613916129 /usr/rhh/grass/best.4.1.52 4 1 53 0.345491502976725 /usr/rhh/grass/best.4.1.53 4 1 54 0.345491502812525 /usr/rhh/grass/best.4.1.54 4 1 55 0.345491502812463 /usr/rhh/grass/best.4.1.55 4 1 56 0.345491502812490 /usr/rhh/grass/best.4.1.56 4 1 57 0.345491502812507 /usr/rhh/grass/best.4.1.57 4 1 58 0.345491502812500 /usr/rhh/grass/best.4.1.58 4 1 59 0.345491502812510 /usr/rhh/grass/best.4.1.59 4 1 60 0.345491502812526 /usr/rhh/grass/best.4.1.60 4 1 61 0.310486069601458 /usr/rhh/grass/best.4.1.61 4 1 62 0.307837167823019 /usr/rhh/grass/best.4.1.62 4 1 63 0.305698279237185 /usr/rhh/grass/best.4.1.63 4 1 64 0.301104358765766 /usr/rhh/grass/best.4.1.64 4 1 65 0.298077392569535 /usr/rhh/grass/best.4.1.65 4 1 66 0.295222689346728 /usr/rhh/grass/best.4.1.66 4 1 67 0.292386986941781 /usr/rhh/grass/best.4.1.67 4 1 68 0.289663102341894 /usr/rhh/grass/best.4.1.68 4 1 69 0.286702482160932 /usr/rhh/grass/best.4.1.69 4 1 70 0.284301677163667 /usr/rhh/grass/best.4.1.70 4 1 71 0.280817814338253 /usr/rhh/grass/best.4.1.71 4 1 72 0.278012151534500 /usr/rhh/grass/best.4.1.72 4 1 73 0.275716805913258 /usr/rhh/grass/best.4.1.73 4 1 74 0.273142565598311 /usr/rhh/grass/best.4.1.74 4 1 75 0.270902912861171 /usr/rhh/grass/best.4.1.75 4 1 76 0.270577229158447 /usr/rhh/grass/best.4.1.76 4 1 77 0.266957652162514 /usr/rhh/grass/best.4.1.77 4 1 78 0.265289820501121 /usr/rhh/grass/best.4.1.78 4 1 79 0.263398269016847 /usr/rhh/grass/best.4.1.79 4 1 80 0.261902935253495 /usr/rhh/grass/best.4.1.80 4 1 81 0.260205788648508 /usr/rhh/grass/best.4.1.81 4 1 82 0.258870178618668 /usr/rhh/grass/best.4.1.82 4 1 83 0.258002287537352 /usr/rhh/grass/best.4.1.83 4 1 84 0.256932482813683 /usr/rhh/grass/best.4.1.84 4 1 85 0.255919909656052 /usr/rhh/grass/best.4.1.85 4 1 86 0.253959913298905 /usr/rhh/grass/best.4.1.86 4 1 87 0.253068543994900 /usr/rhh/grass/best.4.1.87 4 1 88 0.251751387335585 /usr/rhh/grass/best.4.1.88 4 1 89 0.249957992256154 /usr/rhh/grass/best.4.1.89 4 1 90 0.248933105457685 /usr/rhh/grass/best.4.1.90 4 1 91 0.247869481951401 /usr/rhh/grass/best.4.1.91 4 1 92 0.246166099232569 /usr/rhh/grass/best.4.1.92 4 1 93 0.243028531124567 /usr/rhh/grass/best.4.1.93 4 1 94 0.241495621156023 /usr/rhh/grass/best.4.1.94 4 1 95 0.241055021911130 /usr/rhh/grass/best.4.1.95 4 1 96 0.240156903720021 /usr/rhh/grass/best.4.1.96 4 1 97 0.239626075169748 /usr/rhh/grass/best.4.1.97 4 1 98 0.236078016657295 /usr/rhh/grass/best.4.1.98 4 1 99 0.232923321810800 /usr/rhh/grass/best.4.1.99 4 1 100 0.231360221151740 /usr/rhh/grass/best.4.1.100 4 2 2 2.000000000000000 /usr/rhh/grass/OFFICIAL/best.4.2.2 4 2 3 1.500000000000000 /usr/rhh/grass/OFFICIAL/best.4.2.3 4 2 4 1.333333333333333 /usr/rhh/grass/OFFICIAL/best.4.2.4 4 2 5 1.250000000000000 /usr/rhh/grass/OFFICIAL/best.4.2.5 4 2 6 1.200000000000000 /usr/rhh/grass/OFFICIAL/best.4.2.6 4 2 7 1.166666666666666 /usr/rhh/grass/OFFICIAL/best.4.2.7 4 2 8 1.142857142857142 /usr/rhh/grass/OFFICIAL/best.4.2.8 4 2 9 1.123062444911653 /usr/rhh/grass/best.4.2.9 4 2 10 1.111111111111111 /usr/rhh/grass/OFFICIAL/best.4.2.10 4 2 11 1.000000000000000 /usr/rhh/grass/best.4.2.11 4 2 12 1.000000000000000 /usr/rhh/grass/best.4.2.12 4 2 13 1.000000000000000 /usr/rhh/grass/best.4.2.13 4 2 14 1.000000000000000 /usr/rhh/grass/best.4.2.14 4 2 15 1.000000000000000 /usr/rhh/grass/best.4.2.15 4 2 16 0.999999999999999 /usr/rhh/grass/best.4.2.16 4 2 17 1.000000000000000 /usr/rhh/grass/best.4.2.17 4 2 18 1.000000000000000 /usr/rhh/grass/OFFICIAL/best.4.2.18 4 2 19 0.909090909090875 /usr/rhh/grass/best.4.2.19 4 2 20 0.909090909090878 /usr/rhh/grass/best.4.2.20 4 2 21 0.868365800153399 /usr/rhh/grass/best.4.2.21 4 2 22 0.862866615040500 /usr/rhh/grass/best.4.2.22 4 2 23 0.845064594531599 /usr/rhh/grass/best.4.2.23 4 2 24 0.837159106228446 /usr/rhh/grass/best.4.2.24 4 2 25 0.827532966874440 /usr/rhh/grass/best.4.2.25 4 2 26 0.814360510071649 /usr/rhh/grass/best.4.2.26 4 2 27 0.805602675769936 /usr/rhh/grass/best.4.2.27 4 2 28 0.800482081149339 /usr/rhh/grass/best.4.2.28 4 2 29 0.788884362266385 /usr/rhh/grass/best.4.2.29 4 2 30 0.780907917888556 /usr/rhh/grass/best.4.2.30 4 2 31 0.776021385855360 /usr/rhh/grass/best.4.2.31 4 2 32 0.769058853908720 /usr/rhh/grass/best.4.2.32 4 2 33 0.759170796821908 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/usr/rhh/grass/best.16.2.41 16 2 42 1.792682926829256 /usr/rhh/grass/best.16.2.42 16 2 43 1.791666666666567 /usr/rhh/grass/best.16.2.43 16 2 44 1.790697674414849 /usr/rhh/grass/best.16.2.44 16 2 45 1.789772727272719 /usr/rhh/grass/best.16.2.45 16 2 46 1.788888888888871 /usr/rhh/grass/best.16.2.46 16 2 47 1.788043478260841 /usr/rhh/grass/best.16.2.47 16 2 48 1.787234042553164 /usr/rhh/grass/best.16.2.48 16 2 49 1.786457816557789 /usr/rhh/grass/best.16.2.49 16 2 50 1.785711349102418 /usr/rhh/grass/best.16.2.50 16 2 64 1.772476323179199 /usr/rhh/grass/best.16.2.64 16 3 3 3.000000000000000 /usr/rhh/grass/best.16.3.3 16 3 4 3.000000000000000 /usr/rhh/grass/best.16.3.4 16 3 5 3.000000000000000 /usr/rhh/grass/best.16.3.5 16 3 6 2.920000000000000 /usr/rhh/grass/best.16.3.6 16 3 7 2.843749998056267 /usr/rhh/grass/best.16.3.7 16 3 8 2.785714285714285 /usr/rhh/grass/best.16.3.8 16 3 9 2.742187499999999 /usr/rhh/grass/best.16.3.9 16 3 10 2.708333333333332 /usr/rhh/grass/best.16.3.10 16 3 11 2.681249999999999 /usr/rhh/grass/best.16.3.11 16 3 12 2.659090909090899 /usr/rhh/grass/best.16.3.12 16 3 13 2.640624999999996 /usr/rhh/grass/best.16.3.13 16 3 14 2.624999999999993 /usr/rhh/grass/best.16.3.14 16 3 15 2.611607142857141 /usr/rhh/grass/best.16.3.15 16 3 16 2.600000000000000 /usr/rhh/grass/best.16.3.16 16 3 17 2.589843749999999 /usr/rhh/grass/best.16.3.17 16 3 18 2.580882352941174 /usr/rhh/grass/best.16.3.18 16 3 19 2.572916666666666 /usr/rhh/grass/best.16.3.19 16 3 20 2.565789473684208 /usr/rhh/grass/best.16.3.20 16 3 21 2.559374999999999 /usr/rhh/grass/best.16.3.21 16 3 22 2.553571428571428 /usr/rhh/grass/best.16.3.22 16 3 23 2.548295454545454 /usr/rhh/grass/best.16.3.23 16 3 24 2.543478260869564 /usr/rhh/grass/best.16.3.24 16 3 25 2.539062499999999 /usr/rhh/grass/best.16.3.25 16 3 26 2.534999999999999 /usr/rhh/grass/best.16.3.26 16 3 27 2.531250000000000 /usr/rhh/grass/best.16.3.27 16 3 28 2.527777777777776 /usr/rhh/grass/best.16.3.28 16 3 29 2.524553571428570 /usr/rhh/grass/best.16.3.29 16 3 30 2.521551724137929 /usr/rhh/grass/best.16.3.30 16 3 31 2.518749999999999 /usr/rhh/grass/best.16.3.31 16 3 32 2.516129032258063 /usr/rhh/grass/best.16.3.32 16 3 33 2.513671874999997 /usr/rhh/grass/best.16.3.33 16 3 34 2.511363636363634 /usr/rhh/grass/best.16.3.34 16 3 35 2.509191176470587 /usr/rhh/grass/best.16.3.35 16 3 36 2.507142857142846 /usr/rhh/grass/best.16.3.36 16 3 37 2.505208333333325 /usr/rhh/grass/best.16.3.37 16 3 38 2.503378378378378 /usr/rhh/grass/best.16.3.38 16 3 39 2.501644736842103 /usr/rhh/grass/best.16.3.39 16 3 40 2.499999999999976 /usr/rhh/grass/best.16.3.40 16 3 41 2.498437499999985 /usr/rhh/grass/best.16.3.41 16 3 42 2.496951219512177 /usr/rhh/grass/best.16.3.42 16 3 43 2.495535714285711 /usr/rhh/grass/best.16.3.43 16 3 44 2.494186046511615 /usr/rhh/grass/best.16.3.44 16 3 45 2.492897727272704 /usr/rhh/grass/best.16.3.45 16 3 46 2.491666666666656 /usr/rhh/grass/best.16.3.46 16 3 47 2.490489130434779 /usr/rhh/grass/best.16.3.47 16 3 48 2.489361702127657 /usr/rhh/grass/best.16.3.48 16 3 49 2.488281249999995 /usr/rhh/grass/best.16.3.49 16 3 50 2.487244897959170 /usr/rhh/grass/best.16.3.50 16 3 64 2.476190476190359 /usr/rhh/grass/best.16.3.64 16 3 80 2.466734782418892 /usr/rhh/grass/best.16.3.80 4 1 4 1. [CHS96] Hemicube 4 1 12 3/4 [grass3] (2a) 4 1 24 0.518284221171048 [grass3] \cite{grass1} 4 1 24 1/2 [grass3] (1) 4 2 4 1.33333333333 [CHS96] Skew Conf. Mx. 4 2 6 1 [grass3] (2d) 4 2 6 1.200000000000000 [grass3] \cite{grass1} 4 2 18 1 [grass3] (1) 5 1 10 .8888888889 [CHS96] Had 6 1 16 .888888888889 [CHS96] Hemicube 7 1 14 .9230769231 [CHS96] Had 8 1 64 .750000000000 [CHS96] Hemicube 8 1 240 .5 [CCKS96] [SS96] 2.BW 8 1 240 1/2 [grass3] (1) 8 2 20 1.578947368421039 [grass3] \cite{grass1} 8 2 20 3/2 [grass3] (2d) 8 2 40 3/2 [grass3] (2c) 8 2 44 3/2 [grass3] \cite{grass1} 8 2 420 1 [SS96] Cliff 8 2 420 1 [grass3] (1) 8 3 448 1.207222814956890 [grass3] QN+ 8 3 1680 .5 [grass3] Cliff 8 4 70 2 [CHS96],[SS96] Cliff 8 4 70 2 [grass3] (1) 9 1 18 .9411764706 [CHS96] Had 10 1 256 .640000000000 [CHS96] Hemicube 12 1 1024 .555555555556 [CHS96] Hemicube 12 2 12 1.81818181818 [CHS96] Skew Conf. Mx. 13 1 26 .9600000000 [CHS96] Had 14 1 4096 .489795918367 [CHS96] Hemicube 15 1 30 .9655172414 [CHS96] Had 16 1 144 .9375 [CCKS96], [Isometric embedd.] NR + coord frame 16 1 144 15/16 [grass3] (2a) 16 1 4320 0.5 [CCKS96], [SS96] 2.BW 16 1 4320 1/2 [grass3] (1) 16 1 16384 .437500000000 [CHS96] Hemicube 16 2 72 7/4 [grass3] (2d) 16 2 136 7/4 [grass3] (2c) 16 2 144 1.702398406826456 best.16.2.144 16 2 1040 3/2 [grass3] (2c) 16 2 16200 1 [SS96] Cliff 16 2 16200 1 [grass3] (1) 16 3 151200 .5 [grass3] Cliff 16 4 72 15/4 [grass3] (2e) 16 4 180 3 [grass3] (2b) 16 4 6300 2 [SS96] Cliff 16 4 6300 2 [grass3] (1) 16 5 453600 .5 [grass3] Cliff 16 6 113400 1 [grass3] Cliff 16 7 64800 .5 [grass3] Cliff 16 8 270 4 [SS96] Cliff 16 8 270 4 [grass3] (1) 18 1 65536 .395061728395 [CHS96] Hemicube 19 1 38 .9729729730 [CHS96] Had 20 1 262144 .360000000000 [CHS96] Hemicube 20 2 20 1.89473684211 [CHS96] Skew Conf. Mx. 21 1 42 .9756097561 [CHS96] Had 22 1 1048576 .330578512397 [CHS96] Hemicube 23 11 276 5.76000000000 [grass3] QN 24 1 4194304 .305555555556 [CHS96] Hemicube 25 1 50 .9795918367 [CHS96] Had 26 1 16777216 .284023668639 [CHS96] Hemicube 27 1 54 .9811320755 [CHS96] Had 28 1 67108864 .265306122449 [CHS96] Hemicube 28 2 28 1.92592592593 [CHS96] Skew Conf. Mx. 30 1 268435456 .248888888889 [CHS96] Hemicube 31 1 62 .9836065574 [CHS96] Had 31 15 496 7.75757575758 [grass3] QN 32 1 146880 .5 [CCKS96], [SS96] 2.BW 32 1 146880 1/2 [grass3] (1) 32 1 1073741824 .234375000000 [CHS96] Hemicube 32 2 272 15/8 [grass3] (2d) 32 2 1138320 1 [grass3] (1) 32 4 948600 2 [grass3] (1) 32 8 94860 4 [SS96] Cliff 32 8 94860 4 [grass3] (1) 32 16 1054 8 [SS96] Cliff 32 16 1054 8 [grass3] (1) 34 1 4294967296 .221453287197 [CHS96] Hemicube 36 1 17179869184 .209876543210 [CHS96] Hemicube 36 2 36 1.94285714286 [CHS96] Skew Conf. Mx. 37 1 74 .9863013699 [CHS96] Had 38 1 68719476736 .199445983380 [CHS96] Hemicube 40 1 274877906944 .190000000000 [CHS96] Hemicube 41 1 82 .9876543210 [CHS96] Had 42 1 1099511627776 .181405895692 [CHS96] Hemicube 44 1 4398046511104 .173553719008 [CHS96] Hemicube 44 2 44 1.95348837209 [CHS96] Skew Conf. Mx. 45 1 90 .9887640449 [CHS96] Had 46 1 17592186044416 .166351606805 [CHS96] Hemicube 47 23 1128 11.7551020408 [grass3] QN 48 1 70368744177664 .159722222222 [CHS96] Hemicube 49 1 98 .9896907216 [CHS96] Had 50 1 281474976710656 .153600000000 [CHS96] Hemicube 51 1 102 .9900990099 [CHS96] Had 52 1 1125899906842624 .147928994083 [CHS96] Hemicube 52 2 52 1.96078431373 [CHS96] Skew Conf. Mx. 54 1 4503599627370496 .142661179698 [CHS96] Hemicube 55 1 110 .9908256881 [CHS96] Had 56 1 18014398509481984 .137755102041 [CHS96] Hemicube 57 1 114 .9911504425 [CHS96] Had 58 1 72057594037927936 .133174791914 [CHS96] Hemicube 60 1 288230376151711744 .128888888889 [CHS96] Hemicube 60 2 60 1.96610169492 [CHS96] Skew Conf. Mx. 61 1 122 .9917355372 [CHS96] Had 62 1 1152921504606846976 .124869927159 [CHS96] Hemicube 63 1 126 .9920000000 [CHS96] Had 64 1 2112 .984375000000 [CCKS96] Spread 64 1 2112 63/64 [grass3] (2a) 64 1 9694080 .5 [CCKS96], [SS96] 2.BW 64 1 9694080 1/2 [grass3] (1) 64 1 4611686018427387904 .121093750000 [CHS96] Hemicube 64 2 1056 31/16 [grass3] (2d) 64 2 152681760 1 [grass3] (1) 64 4 1056 63/16 [grass3] (2e) 64 4 262951920 2 [grass3] (1) 64 8 2376 7 [grass3] (2b) 64 8 56346840 4 [grass3] (1) 64 16 2772 12 [grass3] (2b) 64 16 1460844 8 [grass3] (1) 64 32 4158 16 [SS96] Cliff 64 32 4158 16 [grass3] (1) 68 2 68 1.97014925373 [CHS96] Skew Conf. Mx. 69 1 138 .9927007299 [CHS96] Had 71 35 2556 17.7534246575 [grass3] QN 75 1 150 .9932885906 [CHS96] Had 76 2 76 1.97333333333 [CHS96] Skew Conf. Mx. 79 1 158 .9936305732 [CHS96] Had 79 39 3160 19.7530864198 [grass3] QN 84 2 84 1.97590361446 [CHS96] Skew Conf. Mx. 85 1 170 .9940828402 [CHS96] Had 87 1 174 .9942196532 [CHS96] Had 91 1 182 .9944751381 [CHS96] Had 92 2 92 1.97802197802 [CHS96] Skew Conf. Mx. 97 1 194 .9948186528 [CHS96] Had 99 1 198 .9949238579 [CHS96] Had 100 2 100 1.97979797980 [CHS96] Skew Conf. Mx. 103 51 5356 25.7523809524 [grass3] QN 108 2 108 1.98130841121 [CHS96] Skew Conf. Mx. 115 1 230 .9956331878 [CHS96] Had 116 2 116 1.98260869565 [CHS96] Skew Conf. Mx. 117 1 234 .9957081545 [CHS96] Had 121 1 242 .9958506224 [CHS96] Had 124 2 124 1.98373983740 [CHS96] Skew Conf. Mx. 127 63 8128 31.7519379845 [grass3] QN 128 1 1260230400 .5 [CCKS96], [SS96] 2.BW 128 1 1260230400 1/2 [grass3] (1) 128 2 4160 63/32 [grass3] (2d) 128 2 40012315200 1 [grass3] (1) 128 4 140043103200 2 [grass3] (1) 128 8 62019088560 4 [grass3] (1) 128 16 3445504920 8 [grass3] (1) 128 32 22882860 16 [grass3] (1) 128 64 16510 32 [SS96] Cliff 128 64 16510 32 [grass3] (1) 129 1 258 .9961089494 [CHS96] Had 132 2 132 1.98473282443 [CHS96] Skew Conf. Mx. 135 1 270 .9962825279 [CHS96] Had 139 1 278 .9963898917 [CHS96] Had 140 2 140 1.98561151079 [CHS96] Skew Conf. Mx. 141 1 282 .9964412811 [CHS96] Had 145 1 290 .9965397924 [CHS96] Had 147 1 294 .9965870307 [CHS96] Had 148 2 148 1.98639455782 [CHS96] Skew Conf. Mx. 151 75 11476 37.7516339869 [grass3] QN 156 2 156 1.98709677419 [CHS96] Skew Conf. Mx. 157 1 314 .9968051118 [CHS96] Had 159 1 318 .9968454259 [CHS96] Had 164 2 164 1.98773006135 [CHS96] Skew Conf. Mx. 167 83 14028 41.7514792899 [grass3] QN 169 1 338 .9970326409 [CHS96] Had 172 2 172 1.98830409357 [CHS96] Skew Conf. Mx. 175 1 350 .9971346705 [CHS96] Had 177 1 354 .9971671388 [CHS96] Had 180 2 180 1.98882681564 [CHS96] Skew Conf. Mx. 181 1 362 .9972299169 [CHS96] Had 187 1 374 .9973190349 [CHS96] Had 191 95 18336 47.7512953368 [grass3] QN 195 1 390 .9974293059 [CHS96] Had 199 1 398 .9974811083 [CHS96] Had 199 99 19900 49.7512437811 [grass3] QN 201 1 402 .9975062344 [CHS96] Had 205 1 410 .9975550122 [CHS96] Had 211 1 422 .9976247031 [CHS96] Had 217 1 434 .9976905312 [CHS96] Had 223 111 24976 55.7511111111 [grass3] QN 225 1 450 .9977728285 [CHS96] Had 229 1 458 .9978118162 [CHS96] Had 231 1 462 .9978308026 [CHS96] Had 239 119 28680 59.7510373444 [grass3] QN 255 1 510 .9980353635 [CHS96] Had 256 1 33024 .996093750000 [CCKS96] Spread 256 1 33024 255/256 [grass3] (2a) 256 1 325139443200 .5 [CCKS96], [SS96] 2.BW 256 1 325139443200 1/2 [grass3] (1) 256 2 16512 127/64 [grass3] (2d) 256 2 20727639504000 1 [grass3] (1) 256 4 16512 255/64 [grass3] (2e) 256 4 146245012056000 2 [grass3] (1) 256 8 131620510850400 4 [grass3] (1) 256 16 35088 15 [grass3] (2b) 256 16 15111984579120 8 [grass3] (1) 256 32 215066194200 16 [grass3] (1) 256 64 43860 48 [grass3] (2b) 256 64 362064300 32 [grass3] (1) 256 128 65790 64 [SS96] Cliff 256 128 65790 64 [grass3] (1) 261 1 522 .9980806142 [CHS96] Had 263 131 34716 65.7509433962 [grass3] QN 265 1 530 .9981096408 [CHS96] Had 271 1 542 .9981515712 [CHS96] Had 271 135 36856 67.7509157509 [grass3] QN 279 1 558 .9982046679 [CHS96] Had 285 1 570 .9982425308 [CHS96] Had 289 1 578 .9982668977 [CHS96] Had 297 1 594 .9983136594 [CHS96] Had 301 1 602 .9983361065 [CHS96] Had 307 1 614 .9983686786 [CHS96] Had 309 1 618 .9983792545 [CHS96] Had 311 155 48516 77.7507987220 [grass3] QN 313 1 626 .9984000000 [CHS96] Had 321 1 642 .9984399376 [CHS96] Had 327 1 654 .9984686064 [CHS96] Had 331 1 662 .9984871407 [CHS96] Had 337 1 674 .9985141159 [CHS96] Had 339 1 678 .9985228951 [CHS96] Had 351 1 702 .9985734665 [CHS96] Had 355 1 710 .9985895628 [CHS96] Had 359 179 64620 89.7506925208 [grass3] QN 365 1 730 .9986282579 [CHS96] Had 367 1 734 .9986357435 [CHS96] Had 367 183 67528 91.7506775068 [grass3] QN 379 1 758 .9986789960 [CHS96] Had 381 1 762 .9986859396 [CHS96] Had 383 191 73536 95.7506493506 [grass3] QN 385 1 770 .9986996099 [CHS96] Had 387 1 774 .9987063389 [CHS96] Had 399 1 798 .9987452949 [CHS96] Had 405 1 810 .9987639061 [CHS96] Had 411 1 822 .9987819732 [CHS96] Had 415 1 830 .9987937274 [CHS96] Had 421 1 842 .9988109394 [CHS96] Had 427 1 854 .9988276671 [CHS96] Had 429 1 858 .9988331389 [CHS96] Had 431 215 93096 107.750577367 [grass3] QN 439 1 878 .9988597491 [CHS96] Had 439 219 96580 109.750566893 [grass3] QN 441 1 882 .9988649262 [CHS96] Had 463 231 107416 115.750537634 [grass3] QN 465 1 930 .9989235737 [CHS96] Had 469 1 938 .9989327641 [CHS96] Had 471 1 942 .9989373007 [CHS96] Had 477 1 954 .9989506821 [CHS96] Had 479 239 114960 119.750519751 [grass3] QN 481 1 962 .9989594173 [CHS96] Had 487 243 118828 121.750511247 [grass3] QN 489 1 978 .9989764585 [CHS96] Had 499 1 998 .9989969910 [CHS96] Had 503 251 126756 125.750495050 [grass3] QN 505 1 1010 .9990089197 [CHS96] Had 507 1 1014 .9990128332 [CHS96] Had 511 1 1022 .9990205681 [CHS96] Had 512 1 167121673804800 .5 [CCKS96], [SS96] 2.BW 512 1 167121673804800 1/2 [grass3] (1) 512 2 65792 255/128 [grass3] (2d) 512 2 21349793828563200 1 [grass3] (1) 512 4 302455412571312000 2 [grass3] (1) 512 8 548740534236523200 4 [grass3] (1) 512 16 128039457988522080 8 [grass3] (1) 512 32 3765866411427120 16 [grass3] (1) 512 64 13585376664600 32 [grass3] (1) 512 128 5760002220 64 [grass3] (1) 512 256 262654 128 [SS96] Cliff 512 256 262654 128 [grass3] (1) 599 299 179700 149.750415973 [grass3] QN 607 303 184528 151.750410509 [grass3] QN 631 315 199396 157.750394945 [grass3] QN 647 323 209628 161.750385208 [grass3] QN 719 359 258840 179.750346741 [grass3] QN 727 363 264628 181.750342936 [grass3] QN 743 371 276396 185.750335570 [grass3] QN 751 375 282376 187.750332005 [grass3] QN 823 411 339076 205.750303030 [grass3] QN 839 419 352380 209.750297265 [grass3] QN 863 431 372816 215.750289017 [grass3] QN 887 443 393828 221.750281215 [grass3] QN 911 455 415416 227.750273823 [grass3] QN 919 459 422740 229.750271444 [grass3] QN 967 483 468028 241.750257998 [grass3] QN 983 491 483636 245.750253807 [grass3] QN 991 495 491536 247.750251762 [grass3] QN 1024 1 525312 .999023437500 [CCKS96] Spread 1024 1 525312 1023/1024 [grass3] (2a) 1024 1 171466837323724800 .5 [CCKS96], [SS96] 2.BW 1024 1 171466837323724800 1/2 [grass3] (1) 1024 2 262656 511/256 [grass3] (2d) 1024 2 43852643645542617600 1 [grass3] (1) 1024 4 262656 1023/256 [grass3] (2e) 1024 4 1244927827937348755200 2 [grass3] (1) 1024 8 4535094230343199036800 4 [grass3] (1) 1024 16 2133173952791060287680 8 [grass3] (1) 1024 32 541728 31 [grass3] (2b) 1024 32 127504704958099428960 16 [grass3] (1) 1024 64 950612278427388720 32 [grass3] (1) 1024 128 863669544301080 64 [grass3] (1) 1024 256 699732 192 [grass3] (2b) 1024 256 91893704364 128 [grass3] (1) 1024 512 1049598 256 [SS96] Cliff 1024 512 1049598 256 [grass3] (1) 2048 1 351507016513635840000 1/2 [grass3] (1) 2048 2 1049600 1023/512 [grass3] (2d) 2048 2 179883715700853141120000 1 [grass3] (1) 2048 4 10223391175665153520320000 2 [grass3] (1) 2048 8 74630755582355620698336000 4 [grass3] (1) 2048 16 70484602494446975103984000 8 [grass3] (1) 2048 32 8492926486522548233592000 16 [grass3] (1) 2048 64 128680704341250730812000 32 [grass3] (1) 2048 128 241617312815189137200 64 [grass3] (1) 2048 256 55088306615410200 128 [grass3] (1) 2048 512 1468160189100 256 [grass3] (1) 2048 1024 4196350 512 [grass3] (1) 4096 1 8392704 4095/4096 [grass3] (2a) 4096 1 1440475753672879672320000 1/2 [grass3] (1) 4096 2 4196352 2047/1024 [grass3] (2d) 4096 2 1474687052822610564537600000 1 [grass3] (1) 4096 4 4196352 4095/1024 [grass3] (2e) 4096 4 167704688729326879200470400000 2 [grass3] (1) 4096 8 2450884236715734248886874560000 4 [grass3] (1) 4096 16 4638525351710148893263677408000 8 [grass3] (1) 4096 32 1122223875413745699983147760000 16 [grass3] (1) 4096 64 8523840 63 [grass3] (2b) 4096 64 34276679215378957166392440000 32 [grass3] (1) 4096 128 130795323474795536128572000 64 [grass3] (1) 4096 256 8950032 240 [grass3] (2b) 4096 256 61630263379216622890800 128 [grass3] (1) 4096 512 9589320 448 [grass3] (2b) 4096 512 3519664466135103000 256 [grass3] (1) 4096 1024 11187540 768 [grass3] (2b) 4096 1024 23473416739500 512 [grass3] (1) 4096 2048 16781310 1024 [grass3] (1)

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