Sphere packings of dimension 4

Introduction

Four-dimensional (4d) sphere packings have a natural application in optical communications, where the electromagnetic field has four degrees of freedom: two orthogonal polarizations and two orthogonal quadratures in each polarization [Betti90], [Betti91], [Agrell09]. However, 4d sphere packings were studied even before their applications in optical communications arose. In the context of modulation, the early works constructed a 4d signal space by encoding data simultaneously over a pair of frequencies or a pair of time slots [Welti74], [Zetterberg77], [Saha89].

In a pure mathematical context, 4d geometry is special due to the existence of the so-called D4 lattice. If a single parity check is applied to the 4d cubic lattice, thus removing half the points, the D4 lattice is obtained. Applying another parity check to D4 gives the cubic lattice back, but rescaled and rotated compared with the original. This so-called set partitioning (SP) process is often applied recursively [Ungerboeck82], [Forney88].

Conversely, the D4 lattice be constructed by interlacing two 4d cubic lattices with each other. To see how this is possible, we first consider the two-dimensional pattern obtained by placing circles centered on a square grid. If the circles barely touch each other, there will be diamond-shaped “holes” between the circles. These size of these holes are large enough so that a circle with radius √2−1 can be fitted inside each of them, if the original circles have radius 1. The holes are also located on a square grid. If we repeat the same exercise in three dimensions, it can be shown that if unit-radius spheres are placed on a cubic grid, the holes between allows spheres of radius √3−1 to be fitted inside. And finally, generalizing the pattern to four dimensions, the holes between the unit-radius (hyper)spheres centered on a cubic lattice allows another set of spheres of radius √4−1 = 1, i.e., the same size as the original spheres. Therefore, two cubic lattices can coexist in 4d space, without reducing the minimum distance compared with a single cubic lattice. This construction yields the D4 lattice, which has twice the density, measured as the number of spheres per unit volume, of the cubic lattice.

If the number of points is large, the best 4d sphere packings are thus subsets of the D4 lattice. Good packings of selected sizes were designed in [Welti74] by selecting D4 lattice points inside suitably chosen hyperspheres. These packings are included, rotated and rescaled into integer coordinates, under the name w4_*.txt.

Optimal lattice subsets (which are not necessarily the optimal packings) of arbitrary sizes can be designed by enumerating and testing a finite number of possible centroids inside the fundamental simplex [Conway82] of the lattice. This algorithm [Agrell14] straightforwardly generalizes the algorithm in [Chow95] for optimizing 2-dimensional lattice subsets. Packings obtained by this technique are included under the name l4_*.txt.

Conjectured optimal sphere packings of sizes up to M ≤ 32 are collected online in Sloane's unpublished tables [Sloane-web]. These packings are included here, sometimes after rescaling and rotation, under the name c4_*.txt.

The most power-efficient 4d packing, in the sense of maximizing γ, is the biorthogonal packing with size M = 8, whose power efficiency is γ = 3/2 = 1.761 dB. Its coordinates are all 0 except one, which is ±1.

The 24 nearest neighbors of the D4 lattice form the 24-cell. Coxeter remarked that a “peculiarity of four-dimensional space is the occurrence of the 24-cell, having no analogue above or below” [Coxeter73, p. 289].

As M → ∞, the gain asymptotically tends to G → 10log10(π/2) = 1.961 dB, which equals the coding gain of the D4 lattice over the cubic lattice [Conway99, pp. 10, 73]. The largest 4d packing in this database comes rather close, with a gain of 1.903 dB.

Database

FileNMdEEnLatβEbCFM [dB]γ [dB]G [dB]Comment
BPSK4_242210.25Y0.519.03090-2.57837
triangle4_3434.2426460.333333Y0.7924813.785587.781510.750803-1.35459
ortho4_4442.8284340.5Y126.02060-1.76091An orthogonal packing [Viterbi61]. Becomes the 3-dimensional simplex if the mean is subtracted, i.e., if the first coordinate is deleted.
PPM4_4441.4142110.5Y10.56.02060-1.76091Pulse-position modulation, where all coordinates except one are zero. Geometrically equivalent to ortho4_4.
tetrahedron4_4445.65685120.375Y167.269991.24939-0.511525[Dochhan13]
l4_54510440.44Y1.1609618.94986.575771.20336-0.285316
simplex4_5453.908796.111460.4Y1.160962.632066.98971.617290.128611[Coxeter73], [Zetterberg77], [Betti91], [Biglieri92], [Karlsson10b], [Agrell11a]. The 4d analogy of the triangle and tetrahedron. One of the six regular convex polytopes in 4d.
l4_6466170.472222Y1.292486.57656.268841.362480.0994747
doublesimplex4_6465.6568514.66670.458333N1.292485.673846.398491.492130.229124Two simplices arranged so that they share a facet [Karlsson10b].
l4_74714960.489796Y1.4036834.19596.110151.562220.492278
c4_7471.756971.511970.489796Y1.403680.5385756.110151.562220.492278[Sloane-web], [Karlsson10b]
voronoi4_8488520.8125Y1.517.33333.91207-0.347621-1.24867This Voronoi code, given in [Forney89b, Table III], does not have zero mean and can be improved by translation.
biortho4_8481.4142110.5Y1.50.3333336.02061.760910.859864[Welti74], [Zetterberg77], [Saha89], [Betti91], [Biglieri92]. The cross-polytope or hyperoctahedron. One of the six regular convex polytopes in 4d [Coxeter73], [Agrell11a]. Used in biorthogonal modulation.
parity4_8482.8284340.5Y1.51.333336.02061.760910.859864A hypercube with every second point removed [Hamming50]. Equivalent to biortho4_8 by rotation.
l4_848220.5Y1.50.6666676.02061.760910.859864Known as polarization-switched QPSK (PS-QPSK) in optical communications [Karlsson09], [Karlsson10b].
c4_8481.958281.917440.5Y1.50.6391466.02061.760910.859864[Sloane-web]. Equivalent to biortho4_8 by rotation.
tetra4_9491.732052.666670.888889Y1.584960.841243.52183-0.498585-1.24939The ternary (4,2,3) tetracode, mapped onto a sequence of four 3-PAM symbols, i.e., a 3×3×3×3 hypercube [Conway99, p. 81].
l4_949181920.592593Y1.5849660.56935.282741.262330.511525
c4_9491.608841.481660.572431N1.584960.4674135.433071.412660.661857[Sloane-web]
w4_1041022.60.65Y1.660960.7826784.881171.064170.448769[Welti74]
l4_1041022.60.65Y1.660960.7826784.881171.064170.448769
c4_104102.512933.788880.6Y1.660961.140575.228791.411790.79639[Sloane-web]
rectsimplex4_104101.95442.29180.6Y1.660960.6898995.228791.411790.79639[Lachs63, Table IV], [Karlsson10b]. The rectified simplex, which is obtained by taking the midpoint of all edges in a simplex.
l4_11411223320.68595Y1.7297295.96954.647371.006520.514417
c4_114112.779614.882980.632001N1.729721.41155.003121.362270.870166[Sloane-web]
w4_124122.828435.666670.708333Y1.792481.580684.507921.021870.642997[Welti74]
dicyclic4_12412111Y1.792480.2789433.0103-0.475754-0.854626[Zetterberg77, Fig. 1 and Sec. 8]
l4_12412411.33330.708333Y1.792483.161354.507921.021870.642997
c4_124122.828435.258280.657285N1.792481.466764.832761.346710.967833[Sloane-web], [Karlsson10b]
l4_13413265080.751479Y1.85022137.2814.251130.9027630.628614
c4_134131.734752.088210.693908N1.850220.5643154.597281.248910.974763[Sloane-web]
l4_14414141530.780612Y1.9036840.18544.085950.861280.684566
c4_144144.4497814.21870.718099N1.903683.734544.448461.223791.04708[Sloane-web]
l4_15415628.80.8Y1.953457.371593.97940.8668130.781219
c4_154152.552784.843780.74329N1.953451.239814.298711.186131.10053[Sloane-web]
dicyclic4_164160.76536711.70711N20.250.687693-2.32261-2.32261[Zetterberg77, Sec. 8]
cube4_16416241Y213.010300[Welti74], [Zetterberg77], [Saha89], [Biglieri92]. The hypercube or tesseract, one of the six regular convex polytopes in 4d [Coxeter73], [Agrell11a]. Also known as PM-QPSK.
SO-PM-QPSK4_164162.828437.236070.904508N21.809023.446170.4358730.435873Constructed from the hypercube by changing the amplitude of half of the points [Sjodin13], [Alvarado15].
l4_164168530.828125Y213.253.829340.8190410.819041
c4_1641623.09520.7738N20.77384.124011.113711.11371[Sloane-web], [Karlsson10a], [Karlsson10b], [Alvarado15].
l4_17417349920.858131Y2.04373242.6933.674760.75840.839116
c4_174173.016747.374830.810358N2.043731.804263.923531.007171.08788[Sloane-web]
l4_18418182870.885802Y2.0849668.82623.536930.7073140.864407
c4_184182.842436.810550.842953N2.084961.633263.752270.922651.07974[Sloane-web]
l4_194193812960.897507Y2.12396305.093.479920.7307930.960379
c4_194192.952667.560890.867253N2.123961.77993.628840.8797151.1093[Sloane-web]
l4_20420203670.9175Y2.1609684.91583.384240.7101151.00869
c4_204204.03114.20590.874264N2.160963.286933.593870.9197491.21833[Sloane-web]
l4_214214216440.931973Y2.19616374.293.316270.7123051.07671
c4_214215.0044722.92440.91534N2.196165.219213.394470.7905111.15492[Sloane-web]
l4_22422224560.942149Y2.22972102.2553.269110.7311.15835
c4_224223.2729810.0840.941336N2.229722.261273.272850.7347481.16209[Sloane-web]
l4_234234620120.950851Y2.26178444.7823.229180.7530821.24073
c4_234232.710436.985340.950851Y2.261781.544213.229180.7530821.24073[Sloane-web], [Karlsson10b]
dicyclic4_244240.51763813.73205N2.292480.218104-2.70918-5.12672-4.58118[Zetterberg77, Sec. 8]
24cell4_24424241Y2.292480.8724173.01030.5927581.1383The vertices of the 24-cell, a regular polytope with 24 faces [Coxeter73], [Zetterberg77], [Biglieri92], [Agrell09], [Bulow09], [Karlsson10b], [Agrell11a].
l4_24424245510.956597Y2.29248120.1753.203010.7854671.33101
c4_244243.6310112.6120.956597Y2.292482.750733.203010.7854671.33101[Sloane-web], [Karlsson10b]
doubleprism4_254251.1755721.44721N2.321930.4306771.40497-0.957139-0.355939Independent 5-PSK packings in each pair of dimensions [Zetterberg77, Sec. 8].
l4_2542523.840.96Y2.321930.8268993.187590.8254751.42668Equivalent to c4_25 by rotation.
c4_2542523.840.96Y2.321930.8268993.187590.8254751.42668[Sloane-web] The 24-cell with a central point added [Gilbert52], [Welti74], [Karlsson10b]. The 4-dimensional “kissing” packing, i.e., a central sphere surrounded by the maximum number of touching spheres [Agrell09].
l4_26426266740.997041Y2.35022143.3913.023170.7136531.36846
c4_264261.546522.384650.997041Y2.350220.5073243.023170.7136531.36846[Sloane-web], [Karlsson10b]
l4_274275430001.02881Y2.37744630.932.886960.6274651.33396
c4_274271.445922.150921.02881Y2.377440.4523612.886960.6274651.33396[Sloane-web], [Karlsson10b]
l4_28428142071.05612Y2.4036843.0592.773160.5613191.31774
c4_284281.683562.992441.05577N2.403680.6224722.774610.5627671.31919[Sloane-web]
l4_294295836401.08205Y2.42899749.2822.667850.5015041.30619
c4_294292.99219.624571.07505N2.428991.981192.696010.5296691.33436[Sloane-web]
l4_30430309941.10444Y2.45345202.5722.578860.4560251.30744
c4_304302.08214.739051.09317N2.453450.9657942.623420.5005861.352[Sloane-web]
l4_314316243201.12383Y2.4771871.9882.50330.4221291.31882
c4_314312.224125.530181.11796N2.47711.116262.526050.4448861.34158[Sloane-web]
b4_3243211.751.75Y2.50.350.579919-1.46128-0.520667A subset of the cubic lattice [Biglieri92].
voronoi4_3243281071.67188Y2.521.40.778262-1.26294-0.322324A Voronoi code based on the cubic lattice [Forney89b, Table III].
SP-QAM4_324324201.25Y2.542.041200.940614[Coelho11], [Karlsson12]. Two interlaced hypercubes. It has the same γ as one hypercube but twice as many points. Also, the 3rd step when applying SP recursively to QAM4_256.
l4_324323211781.15039Y2.5235.62.401850.3606471.30126
c4_324322.241195.69181.13316N2.51.138362.467370.4261741.36679[Sloane-web]
l4_334336651121.17355Y2.52221013.42.315270.312461.29572
l4_344343413781.19204Y2.54373270.8622.247390.2814991.3062
l4_354357059521.21469Y2.564641160.42.165630.2352981.30031
l4_364363616021.23611Y2.58496309.8692.089720.1936671.29791
l4_374377468801.25639Y2.604731320.682.019050.1560721.29854
l4_384383818461.27839Y2.62396351.7581.943650.1126331.29235
l4_394397878721.29389Y2.64271489.391.891340.09122171.30728
w4_404401.414212.61.3Y2.660960.4885451.870870.1006571.35219[Welti74]
l4_4044025.21.3Y2.660960.9770891.870870.1006571.35219
l4_414418289281.32778Y2.678781666.431.779040.03779931.32397
l4_42442142651.35204Y2.6961649.1441.7004-0.01274341.30728
l4_4344386101561.37317Y2.713131871.641.63304-0.0528481.30028
l4_4444488107801.39205Y2.729721974.561.57377-0.08565981.29984
l4_4544590114081.4084Y2.745932077.261.52306-0.1106561.30654
l4_464464630101.4225Y2.76178544.9381.47979-0.1289161.31931
l4_4744794127201.43957Y2.777292290.1.42799-0.1563951.32223
48cell4_484481.5307341.70711N2.792480.7162090.687693-0.8730040.635411The union of the 24-cell and its dual [Zetterberg77], [Biglieri92].
w4_484482.8284311.66671.45833Y2.792482.088941.37173-0.1889651.31945[Welti74]
l4_484484833541.45573Y2.79248600.5411.37949-0.1812031.32721
doubleprism4_494490.86776722.65597N2.807350.356207-1.23193-2.76956-1.23193Independent 7-PSK packings in each pair of dimensions [Zetterberg77, Sec. 8].
w4_494491.414212.938781.46939Y2.807350.5234061.33894-0.1986911.33894[Welti74]
l4_4944925.877551.46939Y2.807351.046811.33894-0.1986911.33894
l4_504505037181.4872Y2.82193658.7691.28661-0.2285341.33774
l4_51451102156321.5025Y2.836212755.791.24216-0.2510541.34334
w4_524522.8284312.23081.52885Y2.850222.145581.16666-0.3051541.31684This packing, suggested in [Welti74], does not have zero mean and can be improved by translation.
l4_524525241021.51701Y2.85022719.5941.20041-0.2714061.35058
l4_53453106172201.53257Y2.863963006.331.15609-0.2948441.35425
l4_544545445091.5463Y2.87744783.5081.11737-0.3131591.36255
l4_55455110188561.55835Y2.890683261.521.08366-0.3269421.37492
l4_56456112197321.57302Y2.903683397.761.04295-0.3481671.37941
l4_57457114206121.58603Y2.916453533.751.0072-0.3648661.38799
l4_584585853811.59958Y2.92899918.5760.97023-0.383191.39452
l4_59459118224641.61333Y2.941323818.690.933069-0.4021061.40007
l4_60460120234361.6275Y2.953453967.570.89509-0.4222211.40402
l4_61461122244121.64015Y2.965374116.180.861463-0.438351.41158
l4_62462124254361.65427Y2.97714271.950.824245-0.4584231.41483
l4_63463126264641.66692Y2.988644427.430.791156-0.4747081.42152
b4_6446412.3752.375Y30.395833-0.746336-1.99572-0.0768683A subset of the cubic lattice [Biglieri92].
w4_6446426.751.6875Y31.1250.737862-0.5115251.40733[Welti74], [Biglieri92]
l4_644646468741.67822Y31145.670.761804-0.4875831.43127
l4_65465130286161.69325Y3.011184751.620.723078-0.5101491.431
l4_664666674331.70638Y3.02221229.730.689537-0.5278351.43529
l4_67467134308441.71775Y3.033045084.660.660689-0.5411231.44366
w4_684682.8284313.82351.72794Y3.043732.270820.63501-0.5515261.45461[Welti74]
l4_68468427.64711.72794Y3.043734.541640.63501-0.5515261.45461
l4_69469662.60871.73913Y3.0542610.24940.606978-0.5645581.46263
l4_704707085841.75184Y3.064641400.490.575364-0.5814391.46652
l4_71471142355841.76473Y3.074875786.250.543519-0.5988081.46964
l4_724726641.77778Y3.0849610.37290.511525-0.6165761.47209
l4_73473146382401.79396Y3.094916177.880.472178-0.6419381.46668
l4_744747498981.80752Y3.104731594.020.43946-0.6609061.46741
l4_75475150409961.82204Y3.114416581.670.40471-0.6821331.46563
l4_7647676106011.83535Y3.123961696.720.373103-0.7004371.46652
l4_77477154438281.84804Y3.133396993.70.343198-0.7172531.46867
l4_7847878113131.85947Y3.14271799.880.316414-0.7311551.4735
l4_79479158467321.87198Y3.151897413.330.287298-0.7475911.47556
l4_8048080120581.88406Y3.160961907.330.259347-0.7630571.47838
l4_81481162497121.89422Y3.169937841.190.235988-0.7741221.48538
l4_8248282128021.90393Y3.178782013.670.213799-0.7842021.49315
l4_83483166527521.91436Y3.187528274.770.190072-0.7959991.499
l4_84484168543001.92389Y3.196168494.570.168487-0.8058291.50662
l4_85485170558521.9326Y3.20478714.090.148891-0.8138411.51586
l4_8648686143531.94064Y3.213132233.490.130842-0.8204721.52628
l4_87487174589761.94795Y3.221479153.580.114532-0.8255251.5381
w4_884881.414213.909091.95455Y3.229720.6051760.0998422-0.8291151.5512[Welti74]
l4_8848827.818181.95455Y3.229721.210350.0998422-0.8291151.5512
l4_89489178624641.97147Y3.237879645.860.0624021-0.8556081.54122
l4_9049090160971.98728Y3.245932479.570.0277007-0.8795121.53364
l4_91491182663162.00205Y3.253910190.2-0.00445551-0.9010171.5283
w4_924922.8284316.21742.02717Y3.261782.48597-0.0586101-0.9446621.50065[Welti74]
l4_9249292170622.01583Y3.261782615.44-0.0342429-0.9202951.52502
l4_93493186702202.02971Y3.2695810738.4-0.0640494-0.939731.52141
l4_9449494180462.04233Y3.277292753.19-0.0909525-0.9563981.52041
l4_95495190741762.05474Y3.2849311290.4-0.117262-0.9726041.51971
l4_9649696190502.06706Y3.292482892.95-0.143225-0.9885921.51908
l4_97497194782522.07918Y3.2999611856.5-0.16862-1.004141.51874
l4_9849898200812.0909Y3.307353035.81-0.193032-1.018821.51911
l4_99499198824002.10183Y3.3146812429.6-0.215668-1.031851.52099
l4_1004100100211262.1126Y3.321933179.78-0.237873-1.044571.52304
l4_1014101202866562.12371Y3.3291113014.9-0.260659-1.057981.52425
l4_1024102342467.672.13466Y3.33621369.831-0.282987-1.071051.52568
l4_1034103206910402.14535Y3.3432513615.5-0.304678-1.083591.52749
l4_10441042614572.15533Y3.35022217.448-0.324829-1.09471.53061
l4_1054105210955362.16635Y3.3571214228.9-0.346985-1.107911.53149
l4_1064106106244582.17675Y3.363963635.3-0.367792-1.119881.53348
l4_10741072141001282.18639Y3.3707314852.6-0.38698-1.130341.53687
l4_10841085464042.19616Y3.37744948.054-0.406338-1.141061.53987
l4_10941092181048962.20722Y3.3840915498.4-0.428159-1.154341.54019
l4_1104110110268332.2176Y3.390683956.88-0.448539-1.166271.54174
l4_11141112221097922.22774Y3.3972116159.2-0.468348-1.177731.54364
l4_1124112112280702.23772Y3.403684123.48-0.487764-1.188881.54574
l4_11341132261147922.24747Y3.4100916831.2-0.506647-1.199591.54817
l4_1144114114293142.25562Y3.416454290.13-0.522354-1.207211.55358
l4_1154115464790.42.26389Y3.42275699.789-0.538261-1.215121.55859
l4_1164116116306112.2749Y3.428994463.56-0.559315-1.228251.55826
l4_11741172341251322.28527Y3.4351818213.3-0.579067-1.240171.55906
l4_11841182361278122.29481Y3.4413218570.2-0.597176-1.250521.5613
l4_11941192381304642.30323Y3.4474118922.-0.613068-1.258741.56559
600cell4_12041201.2360742.61803N3.453450.579132-1.16945-1.807531.02921The vertices of the 600-cell, a regular polytope with 600 faces [Coxeter73], [Zetterberg77], [Biglieri92], [Agrell11a].
l4_1204120683.22.31111Y3.4534512.0459-0.627908-1.265981.57075
doubleprism4_12141210.56346526.29935N3.459430.289065-4.98266-5.61321-2.76417Independent 11-PSK packings in each pair of dimensions [Zetterberg77, Sec. 8].
l4_12141212421359362.32115Y3.4594319647.2-0.646738-1.277291.57175
l4_12241222441389242.33345Y3.4653720044.6-0.669676-1.292781.56847
l4_12341238215757.32.34345Y3.471262269.69-0.68825-1.303981.56938
l4_12441246290422.35224Y3.47711300.22-0.704511-1.312941.57244
l4_12541252501475522.36083Y3.4828921182.4-0.720351-1.321551.57576
l4_1264126126376172.36943Y3.488645391.36-0.736131-1.330171.57898
l4_12741272541534082.37783Y3.4943421950.9-0.751506-1.338451.58244
SP-QAM4_12841282.82843202.5Y3.52.85714-0.9691-1.549021.38354QAM4_256 with a single-parity check constraint [Coelho11], [Karlsson12].
l4_1284128128390892.3858Y3.55584.14-0.766046-1.345971.58659
w4_14541451.414214.965522.48276Y3.589950.691585-0.939045-1.408761.70976[Welti74]
w4_15241521.414215.210532.60526Y3.623960.718899-1.14822-1.576981.61212Characterized in [Welti74], Table IV, where it was incorrectly labeled M=164.
w4_16941691.414215.396452.69822Y3.700440.729163-1.30048-1.638551.70982[Welti74]
w4_19341932.8284323.50782.93847Y3.796233.0962-1.67091-1.897991.65093This packing, suggested in [Welti74], does not have zero mean and can be improved by translation.
w4_2004200211.962.99Y3.821931.56466-1.74641-1.944191.65874[Welti74]
w4_21242122.8284324.73583.09198Y3.863963.20084-1.89207-2.042341.64909[Welti74]
QAM4_25642562205Y42.5-3.9794-3.97940[Alvarado15]. The Cartesian product of two 16-QAM or four 4-PAM packings. Also known as PM-16QAM.
voronoi4_256425645.254870423.43848Y4880.25-2.35336-2.353361.62604Conjectured in [Conway83], [Forney89b] to be the best Voronoi code.
w4_256425613.3753.375Y40.421875-2.27244-2.272441.70696As defined in [Welti74], not rotated or rescaled.
a4_25642561.414216.753.375Y40.84375-2.27244-2.272441.70696Integer coordinates with odd parity [Conway83], [Alvarado15], [Eriksson15]. Equivalent to w4_256 by rotation.
ab4_25642562.82843273.375Y43.375-2.27244-2.272441.70696Odd integer coordinates [Eriksson15]. Equivalent to w4_256 by rotation.
w4_26842682.8284327.71643.46455Y4.033043.43617-2.38617-2.350441.69925[Welti74]
w4_31343131.414217.514383.75719Y4.145010.906437-2.73833-2.583671.70519[Welti74]
w4_36043602.8284332.15564.01944Y4.245933.78664-3.03136-2.772241.73349[Welti74]
w4_40944091.414218.567244.28362Y4.337980.987469-3.30781-2.955531.74906[Welti74]
w4_4144414217.27054.31763Y4.346741.98661-3.34216-2.981121.74246[Welti74]
w4_43244322.8284335.29634.41204Y4.377444.03161-3.43609-3.044491.74567[Welti74]
w4_46444641.414219.137934.56897Y4.428991.0316-3.58788-3.145431.75675[Welti74]
w4_4944494218.84624.71154Y4.474182.1061-3.72133-3.234791.76586[Welti74]
w4_50445042.8284338.11114.76389Y4.488644.24528-3.76932-3.268771.76343[Welti74]
cross4_51245122287Y4.53.11111-5.44068-4.929160.127854A cross-shaped section of the cubic lattice [Forney84, Sec. IV-C], [Forney89a].
SP-QAM4_51245124845.25Y4.59.33333-4.19129-3.679771.37724[Karlsson12]. Two interlaced QAM4_256 packings. Also, the 3rd step when applying SP recursively to QAM4_4096.
SP-cross4_51245122.82843405Y4.54.44444-3.9794-3.467871.58913A single parity applied to the Cartesian product of two cross2_32 packings [Fischer14].
a4_5124512102450159564.78359Y4.5557328.-3.78724-3.275711.7813[Agrell14]
120cell4_60046000.763932813.7082N4.614410.86685-8.35951-7.73894-2.43128The vertices of the 120-cell, a regular polytope with 120 faces [Coxeter73], [Zetterberg77]. Similarly to the dodecahedron in 3 dimensions, it is very weak as a sphere packing [Agrell11a].
w4_60146011.4142110.30285.15141Y4.615611.11609-4.10896-3.487271.82303[Welti74]
w4_65646561.4142110.85375.42683Y4.678781.15988-4.33516-3.654441.79488[Welti74]
w4_66846682.8284343.86235.48278Y4.691854.6743-4.37971-3.686871.79129[Welti74]
w4_68846882.8284344.52335.56541Y4.713134.72332-4.44467-3.732171.79295This packing, suggested in [Welti74], does not have zero mean and can be improved by translation.
w4_7144714222.69195.67297Y4.739892.39371-4.5278-3.790721.79352[Welti74]
w4_73647362.82843465.75Y4.761784.83013-4.58638-3.829281.80339[Welti74]
w4_80048001.4142111.965.98Y4.821931.24017-4.75671-3.94511.8209[Welti74]
cross4_20484204825714.25Y5.55.18182-8.52785-7.144820.150246A cross-shaped section of the cubic lattice [Forney84, Sec. IV-C], [Forney89a].
a4_1024410241281111506.78406Y511115-5.3046-4.33551.82751[Agrell14]
SP-QAM4_2048420482.828438410.5Y5.57.63636-7.20159-5.818571.4765
a4_2048420482048402438189.59487Y5.53658529.-6.81009-5.427071.868[Agrell14]
QAM4_40964409628421Y67-10.2119-8.450980The Cartesian product of two 64-QAM or four 8-PAM packings [Alvarado15]. Also known as PM-64QAM.
a4_409644096409622772382013.5734Y618976985-8.31659-6.555671.89531[Agrell14], [Alvarado15]
w4_569845698264.070616.0176Y6.238125.13541-9.03568-7.105751.9035[Welti74]

References

[Agrell14]
E. Agrell, unpublished, 2014.
[Sloane-web]
http://neilsloane.com/cluster (previously http://www2.research.att.com/~njas/cluster/), accessed April 2010.