Sphere packings of dimension 8

Introduction

A page is devoted to eight-dimensional sphere packings since these are geometrically particularly appealing and have found usage in optical communication systems. The dimensions may in a communication scenario represent quadratures, time slots, frequency bands, or polarizations, as discussed on the page for other dimensions.

The best lattice in eight dimensions (8d) is the E8 lattice. It can be constructed from the 8d cubic lattice by shifting every second point along the main diagonal, halfway to the next point in the cubic lattice. This shifting operation increases the minimum distance by √2 without changing the density. Thus, the coding gain of E8 over the cubic lattice is a factor of 2, or 3.01 dB. The overall gain, which also includes a shaping gain, is 10log10((5π/6)(2/3)1/4) = 3.739 dB [Conway99, pp. 73, 121].

Subsets of the E8 lattice have been designed by numerical optimization for sizes up to 256 [Agrell14] and selected larger sizes [Agrell16]. These are included under the name l8_*.txt. We conjecture that these packings are optimal lattice subsets for all included sizes, however with no claim that they would be optimal among nonlattice packings. Only one case has been found when a nonlattice packing is better than the best E8-based packing, which occurs for M = 10.

The most power-efficient 8d packing known, in the sense of maximizing γ, is the biorthogonal packing with M = 16 and γ = 3.010 dB. As M increases, γ is irregular and displays two other prominent peaks with almost the same power efficiency, which are 3.007 dB at M = 58 and 2.981 dB at M = 241. As M continues to increase, the power efficiency drops rapidly [Agrell14]. At the same time, the gain G increases slowly towards 3.739 dB (see above), but it seems like very large sizes would be needed to get near this asymptotic value.

Database

FileNMdEEnLatβEbCFM [dB]γ [dB]G [dB]Comment
BPSK8_282210.25Y0.25112.04120-2.97094
triangle8_3834.2426460.333333Y0.3962413.7855810.79180.750803-1.99181
tetrahedron8_4845.65685120.375Y0.5610.28031.24939-1.32898
simplex8_5857.07107200.4Y0.5804828.61353101.61729-0.832402
simplex8_6868.48528300.416667Y0.64624111.60569.822711.90605-0.437666
simplex8_7879.89949420.428571Y0.70183914.96079.700372.14214-0.111403
ortho8_888480.5Y0.752.666679.03091.76091-0.414088An orthogonal packing [Viterbi61]. Becomes the 7-dimensional simplex if the mean is subtracted, i.e., if the first coordinate is deleted.
PPM8_8881.4142110.5Y0.750.3333339.03091.76091-0.414088Pulse-position modulation, where all coordinates except one are zero. Geometrically equivalent to ortho8_8.
simplex8_88811.3137560.4375Y0.7518.66679.610822.340830.165831
simplex8_9898.48528320.444444Y0.79248110.09499.542432.511720.406325The 8d analogy of the triangle and tetrahedron. One of the three regular convex polytopes in 8d [Coxeter73], [Agrell11a].
doublesimplex8_1081011.313759.20.4625N0.83048217.8219.369482.542180.499325Not a lattice subset, which makes it unique among best known packings in 8d.
l8_1181115.55631160.479339Y0.86485833.53159.214172.563020.576941
l8_1281216.97061400.486111Y0.89624139.0529.153242.656890.722821
l8_1381318.38481660.491124Y0.9251144.85959.108692.750020.863944
l8_1481419.7991940.494898Y0.95183950.9549.075442.840480.998964
l8_1581521.21322240.497778Y0.97672357.33469.050242.927361.12744
biortho8_168161.4142110.5Y10.259.03093.01031.24939The cross-polytope or hyperoctahedron, used in biorthogonal modulation [Viterbi61], [Eriksson14a]. One of the three regular convex polytopes in 8d [Coxeter73], [Agrell11a].
hamming8_16816480.5Y129.03093.01031.24939A subset of the hypercube, given by the extended (8,4) Hamming code [Hamming50, Table IV], [Viterbi61, Sec. III-B]. Also a Reed–Muller code. Equivalent to the cross-polytope.
l8_168161.4142110.5Y10.259.03093.01031.24939
l8_1781748.083312160.525952Y1.02187297.4958.811142.884481.16029
l8_1881825.45583540.546296Y1.0424884.89368.646322.80641.11691
l8_1981953.740116240.562327Y1.06198382.3048.520712.761281.10468
l8_2082028.28434600.575Y1.08048106.4348.423922.73951.11415
l8_2182159.39720640.585034Y1.09808469.9118.348792.734531.13897
l8_2282231.11275740.592975Y1.11486128.7168.290232.741831.17473
l8_2382365.053825360.599244Y1.13089560.628.244562.758171.2183
l8_2482411.313777.33330.604167Y1.1462416.86678.209032.781191.26743
l8_2582570.710730800.616Y1.16096663.2428.124792.752381.2637
l8_2682636.76968400.621302Y1.17511178.7078.087572.767761.30322
l8_2782776.367536480.625514Y1.18872767.2118.058232.788431.34714
l8_2882879.19639440.628827Y1.20184820.418.035292.813151.3943
l8_2982982.024442480.631391Y1.2145874.4378.017612.840971.4438
l8_308308.4852845.60.633333Y1.226729.293058.004282.871141.49494
l8_3183187.681249520.644121Y1.23855999.5577.930932.839461.48356
l8_3283245.254813380.65332Y1.25267.67.869342.817841.48162
l8_3383393.338157600.661157Y1.26111141.867.817552.804441.48733
l8_3483496.166561760.66782Y1.271871213.967.774012.797821.49925
l8_3583598.994966000.673469Y1.282321286.737.737422.796791.51625
l8_36836101.82370320.678241Y1.292481360.177.706762.80041.5374
rectsimplex8_368364.24264140.777778Y1.292482.707977.112042.205690.942684The rectified simplex, which is obtained by taking the midpoint of all edges in a simplex.
l8_37837104.65274560.680789Y1.302361431.247.690482.81721.57127
l8_3883853.740119840.686981Y1.31198378.0547.651162.809831.58054
l8_39839110.30984080.690993Y1.321351590.87.625862.815451.60237
l8_4084022.6274355.20.69375Y1.3304866.74277.608572.828061.6308
l8_41841115.96693920.698394Y1.339391753.047.57962.828061.64625
l8_4284259.39724780.702381Y1.34808459.5437.554872.831431.6647
l8_43843121.622104320.705246Y1.356571922.57.537192.8411.68901
l8_44844124.451109600.707645Y1.364862007.537.522452.852721.71515
l8_45845127.279114960.70963Y1.372962093.287.510282.866271.7428
l8_4684665.053830100.711248Y1.38089544.9387.500392.881381.77171
l8_47847132.936126320.714803Y1.388652274.167.478742.884061.7879
l8_4884867.882333080.717882Y1.39624592.3057.460072.889071.80615
l8_49849138.593138400.720533Y1.403682464.957.444062.896131.82619
l8_5085070.710736140.7228Y1.41096640.3427.430422.904981.84776
l8_5185148.08331674.670.724337Y1.41811295.2297.421192.917681.87294
l8_5285236.76969820.726331Y1.42511172.2677.409252.927141.89464
l8_53853149.907163600.728017Y1.431982856.187.399182.937951.91748
l8_5485476.367542540.729424Y1.43872739.1987.39082.949971.9413
l8_55855155.563176800.730579Y1.445343058.17.383932.963031.96595
l8_5685679.19645880.731505Y1.45184790.0337.378432.977011.99132
l8_57857161.22190320.732225Y1.458223262.887.374152.991792.0173
l8_588582.828435.862070.732759Y1.46451.00077.370993.007272.04379
l8_59859166.877206400.741166Y1.470663508.637.321442.975972.02331
l8_6086084.852853920.748889Y1.47672912.8327.276432.948822.00681
l8_61861172.534225040.75598Y1.482683794.477.23552.925391.99386
l8_6286287.681258620.762487Y1.48855984.5167.198282.905311.9841
l8_63863178.191244000.768456Y1.494324082.127.164412.888251.9772
l8_6486490.509763400.773926Y1.51056.677.133612.873921.97287
l8_65865183.848263280.778935Y1.505594371.77.105592.862061.97086
l8_6686693.338168320.784206Y1.51111130.37.07632.848631.96714
l8_67867189.505283200.788594Y1.516524668.587.052062.839951.96803
l8_68868192.333293200.792604Y1.521874816.467.030042.83321.97071
l8_69869195.161303600.797101Y1.527134970.17.005462.823631.97043
l8_70870197.99314080.801224Y1.532325124.256.983062.815951.97192
l8_71871200.818324640.804999Y1.537445278.926.962652.810021.97503
l8_72872203.647334960.807677Y1.542485428.926.948222.809821.98375
l8_73873206.475345200.80972Y1.547465576.896.937252.812831.99556
l8_74874209.304355520.811541Y1.552365725.466.927492.816832.00824
l8_75875212.132365920.813156Y1.55725874.636.918862.821722.0217
l8_76876107.4894100.814578Y1.561981506.16.911282.827442.03588
l8_77877217.789386960.815821Y1.56676174.786.904652.83392.0507
l8_78878220.617397600.816897Y1.571356325.776.898932.841062.06611
l8_79879223.446408320.817818Y1.575956477.386.894042.848852.08204
l8_8088014.14211640.82Y1.5804825.94156.882462.849762.091
l8_81881229.103432640.824265Y1.584966824.146.859932.839522.08872
l8_82882231.931445520.828227Y1.589397007.736.83912.83082.08786
l8_83883234.759458480.831906Y1.593767191.86.819862.823492.08831
l8_84884118.794117780.834609Y1.598081842.526.805772.821152.09365
l8_85885240.416484480.838201Y1.602357558.916.787122.814092.09417
l8_86886243.245497840.841401Y1.606577746.966.770572.808962.09655
l8_87887246.073511200.844233Y1.610747934.266.755982.805622.10063
l8_88888124.451131120.846591Y1.614862029.96.743862.804612.10695
l8_89889251.73537840.848756Y1.618938305.476.732772.804462.11406
l8_90890127.279137820.850741Y1.622962122.976.722632.805112.1219
l8_91891257.387564800.852554Y1.626958678.826.713382.806522.1304
l8_92892260.215578560.854442Y1.630898868.776.703772.807422.13833
l8_93893263.044592000.85559Y1.634799053.156.697942.811962.14982
l8_94894265.872606800.85842Y1.638659257.646.68362.807862.1526
l8_95895268.701621680.861053Y1.642469462.616.67032.804662.15621
l8_96896271.529636640.863498Y1.646249668.096.657992.802322.16061
l8_97897274.357651680.865767Y1.649989874.076.646592.800772.16574
l8_9889819.7993400.867347Y1.6536851.40066.638672.802582.17416
l8_99899280.014681520.869197Y1.6573410280.36.629422.802932.18105
l8_1008100282.843697280.8716Y1.6609610495.16.617432.800432.18503
l8_1018101285.671713120.873836Y1.6645510710.46.60632.798682.18969
l8_1028102288.5729040.875913Y1.6681110926.26.595992.797632.195
l8_1038103291.328745040.87784Y1.6716311142.56.586452.797242.20091
l8_1048104294.156760960.879438Y1.6751111356.96.578552.798382.20829
l8_1058105296.985776800.880726Y1.6785611569.46.572192.800972.21705
l8_1068106299.813793520.882787Y1.6819811794.46.562042.799652.22185
l8_1078107302.642810160.884531Y1.6853712017.66.553472.799812.22809
l8_1088108152.735206800.886488Y1.688723061.496.543872.798852.23314
l8_1098109308.299844160.888141Y1.6920512472.56.535782.79932.23955
l8_1108110311.127861520.89Y1.6953412704.26.52672.798672.24482
l8_1118111313.955878800.891567Y1.698612934.26.519062.799382.25139
l8_1128112158.392224120.893335Y1.701843292.326.510452.799042.25685
l8_1138113319.612914080.894823Y1.7050413402.66.503232.799982.26355
l8_1148114161.22233020.896507Y1.708223410.276.495062.799912.26918
l8_1158115325.269950160.898072Y1.7113713880.16.487492.800332.27526
l8_1168116328.098968480.899673Y1.714514121.96.479752.800522.28105
l8_1178117330.926986880.901162Y1.7175914364.36.472572.801172.28727
l8_1188118333.7541005520.902686Y1.7206614609.56.465232.801592.2932
l8_1198119336.5831024320.904173Y1.723714856.46.458082.802112.2992
l8_1208120169.706260820.905625Y1.726723776.236.451122.802742.30525
l8_1218121342.241062400.907042Y1.7297215355.16.444332.803472.31137
l8_1228122172.534270460.90856Y1.732683902.336.437072.803662.3169
l8_1238123347.8971101440.91004Y1.7356315865.16.429992.803962.3225
l8_1248124350.7251121200.911485Y1.7385516122.66.42312.804372.32817
l8_1258125353.5531141120.912896Y1.7414516381.86.416392.804892.3339
l8_1268126356.3821161280.914336Y1.7443216643.76.409542.80522.33939
l8_1278127359.211181520.91568Y1.7471716906.26.403162.805922.34524
parity8_12881282.8284381Y1.751.142866.02062.430381.9748Every second vertex of the hypercube, selected using a single-parity check [Hamming50], [Puttnam14]. Also a Reed–Muller code.
l8_1288128181.019300500.917053Y1.754292.866.396652.806442.35086
l8_1298129364.8671222560.918334Y1.7528117437.26.390592.807332.35681
l8_1308130183.848310880.919763Y1.755594427.6.383842.807472.36198
l8_1318131370.5241264400.920984Y1.7583617977.6.378082.808552.36803
l8_1328132373.3521285680.922348Y1.761118251.16.371652.808892.37332
l8_1338133376.1811306880.923512Y1.7638218523.46.366182.810122.37946
l8_1348134189.505332120.924816Y1.766524700.26.360052.810642.38486
l8_135813542.42641666.670.925926Y1.7692235.5116.354842.812022.39108
l8_1368136384.6661371920.927173Y1.7718719357.6.348992.81272.39656
l8_1378137387.4951393760.928233Y1.7745119635.96.344032.814212.40285
l8_1388138390.3231416480.929742Y1.7771319926.56.336982.813572.40695
l8_1398139393.1511439120.931059Y1.7797420215.46.330832.813782.41186
l8_1408140395.981461680.932194Y1.7823220502.56.325542.81482.41755
l8_1418141398.8081484320.933253Y1.7848920790.16.320612.816122.42351
l8_1428142200.818376760.934239Y1.787445269.566.316022.817732.42973
l8_143814331.1127905.2310.935156Y1.78997126.4316.311762.819612.43619
l8_1448144101.82397080.936343Y1.792481353.996.306252.82022.44133
l8_1458145410.1221577120.937646Y1.7949821965.76.300212.82022.44585
l8_1468146412.951600640.938638Y1.7974622262.66.295622.82162.45173
l8_1478147415.7791624240.939562Y1.7999222559.96.291342.823272.45786
l8_1488148418.6071647920.940422Y1.8023622857.86.287372.82522.46421
l8_1498149421.4361671680.941219Y1.8047923156.16.283692.827372.47078
l8_1508150424.2641696800.942667Y1.807223472.76.277022.826492.47427
l8_1518151427.0921721600.943818Y1.809623784.26.271722.826942.47907
l8_1528152429.9211746640.944988Y1.8119824098.56.266342.827282.48371
l8_1538153432.7491771760.946089Y1.8143524413.26.261282.827882.48861
l8_1548154217.789449240.947124Y1.81676182.16.256532.828752.49374
l8_1558155438.4061822240.948096Y1.8190325044.16.252082.829882.49909
l8_1568156220.617461880.948964Y1.821356339.86.24812.831442.50486
l8_1578157444.0631872640.949653Y1.8236625671.56.244952.833782.51138
l8_1588158223.446474460.950288Y1.825956496.096.242052.836322.51808
l8_1598159449.721924320.951466Y1.8282226314.16.236672.836362.52224
l8_1608160452.5481950080.952187Y1.8304826633.46.233382.838432.52842
l8_1618161455.3771975920.952857Y1.8327326953.26.230322.840712.53478
l8_1628162458.2052001840.953475Y1.8349627273.66.227512.843182.5413
l8_1638163461.0342027840.954044Y1.8371827594.46.224922.845842.54799
l8_1648164231.931513920.955384Y1.839396984.936.218822.844962.55112
l8_1658165466.692083600.956657Y1.8415828285.56.213042.844342.55449
l8_1668166469.5192111600.957868Y1.8437628631.76.207552.843992.5581
l8_1678167472.3472139680.959016Y1.8459328978.46.202342.843882.56193
l8_1688168475.1762167840.960105Y1.8480829325.66.197412.844022.56598
l8_1698169478.0042196080.961136Y1.8502229673.26.192752.844392.57024
l8_170817096.16658897.60.962111Y1.852351200.856.188352.844972.57469
l8_1718171483.6612252640.962963Y1.8544630367.86.18452.846092.57965
l8_1728172486.4892280640.963629Y1.8565730710.56.18152.848012.5854
l8_1738173489.3182308720.964249Y1.8586631053.66.178712.85012.5913
l8_1748174492.1462336880.964824Y1.8607431397.36.176122.852372.59735
l8_1758175494.9752365440.965486Y1.862831745.76.173142.854212.60295
l8_1768176497.8032394080.966103Y1.8648632094.76.170372.856222.60871
l8_1778177500.6322422640.966612Y1.866932442.6.168082.858692.6149
l8_1788178251.73612880.967176Y1.868938198.266.165552.860882.62079
l8_1798179506.2882480480.967698Y1.8709533144.66.16322.863232.62682
l8_1808180254.558627380.968179Y1.872968374.166.161042.865742.63298
l8_1818181511.9452538640.968621Y1.8749633849.26.159062.868382.63927
l8_1828182257.387641960.969025Y1.876958550.586.157252.871172.64568
l8_1838183517.6022597120.969393Y1.8789234555.96.15562.87412.65221
l8_1848184260.215656620.969725Y1.880898727.526.154122.877152.65885
l8_1858185523.2592658160.97084Y1.8828535294.56.149122.876672.66193
l8_1868186263.044672480.971904Y1.884798919.836.144372.876392.6652
l8_1878187528.9162721760.972919Y1.8867236064.66.139832.876322.66865
l8_1888188265.872688420.973885Y1.888659112.616.135522.876432.67227
l8_1898189534.5732785680.974805Y1.8905636836.76.131422.876732.67607
l8_1908190268.701704440.975679Y1.892469305.866.127532.877212.68002
l8_1918191540.232849920.976508Y1.8943637610.66.123842.877862.68412
l8_1928192543.0582882320.977349Y1.8962438000.56.12012.878442.68814
l8_1938193545.8862914720.97812Y1.8981138389.76.116682.87932.69243
l8_1948194274.357736800.97885Y1.899989694.856.113442.880322.69685
l8_1958195551.5432979760.97954Y1.9018339169.66.110382.88152.70142
l8_1968196554.3723012400.980191Y1.9036839560.36.107492.882832.70611
l8_1978197557.23045520.980932Y1.9055139956.76.104212.883732.71037
l8_1988198280.014769680.981635Y1.9073410088.46.10112.884782.71476
l8_1998199562.8573112000.982298Y1.9091640751.6.098172.885982.71928
l8_2008200565.6853145360.982925Y1.9109641148.96.09542.887322.72392
l8_2018201568.5143178720.983491Y1.9127641546.26.09292.888912.7288
l8_2028202571.3423212160.984021Y1.9145541944.6.090562.890632.73379
l8_2038203574.1713246000.984615Y1.9163342346.56.087942.892052.73847
l8_2048204288.5819980.985174Y1.9181110687.46.085472.89362.74326
l8_2058205115.96613254.40.985604Y1.919871725.956.083582.895692.74858
l8_2068206582.6563347920.986167Y1.9216343555.86.081092.897182.75328
l8_2078207585.4843382160.986651Y1.9233743961.36.078972.8992.75829
l8_2088208294.156854120.987102Y1.9251111091.86.076982.900932.76341
l8_2098209591.1413450880.987523Y1.9268444773.86.075132.902982.76863
l8_2108210593.973485360.987914Y1.9285645180.86.073412.905142.77394
l8_2118211596.7983519920.988275Y1.9302745588.36.071822.907412.77935
l8_2128212149.907222160.988608Y1.931982874.776.070362.909792.78484
l8_2138213602.4553590000.989112Y1.9336846414.26.068152.911392.78955
l8_2148214605.2833625520.989584Y1.9353746832.56.066072.913112.79437
l8_2158215608.1123661120.990027Y1.9370547251.36.064132.914932.79927
l8_2168216610.943696800.990441Y1.9387247670.66.062322.916872.80428
l8_2178217613.7693732560.990826Y1.9403948090.46.060632.918912.80937
l8_2188218616.5973768400.991183Y1.9420548510.76.059062.921062.81456
l8_2198219619.4263804480.991556Y1.943748933.66.057432.923112.81964
l8_2208220124.45115361.60.991839Y1.945341974.156.056192.925542.82508
l8_2218221625.0823876720.992179Y1.9469849778.76.05472.927712.83024
l8_2228222627.9113912960.992452Y1.948650202.16.053512.930142.83567
l8_2238223630.7393949600.992781Y1.9502250630.16.052072.932312.84081
l8_2248224633.5683986160.993044Y1.9518451056.56.050912.934752.84621
l8_2258225636.3964022800.993284Y1.9534551483.46.049872.937282.85168
l8_2268226319.6121014880.9935Y1.9550412977.76.048922.939892.85723
l8_2278227642.0534096320.993693Y1.9566452338.86.048082.942582.86284
l8_2288228322.4411033420.993979Y1.9582213193.36.046832.944852.86802
l8_2298229647.714171120.994241Y1.959853208.56.045682.94722.87327
l8_2308230325.2691052160.99448Y1.9613713411.6.044642.949642.8786
l8_2318231653.3674246240.994697Y1.9629454080.26.043692.952162.88399
l8_2328232656.1954283920.994891Y1.964554516.86.042852.954762.88945
l8_2338233659.0244321600.995045Y1.9660554952.96.042172.957512.89505
l8_2348234330.9261089880.995215Y1.9675913847.96.041432.960182.90056
l8_2358235664.684397520.995364Y1.9691355830.86.040782.962922.90612
l8_2368236667.5094435600.995493Y1.9706656270.56.040222.965732.91175
l8_2378237670.3374473760.995603Y1.9721956710.76.039742.968622.91744
l8_2388238673.1664512000.995692Y1.973757151.46.039352.971572.92318
l8_2398239675.9944550320.995763Y1.9752257592.76.039042.974592.92898
l8_2408240678.8234588720.995816Y1.9767258034.46.038812.977672.93483
l8_24182412.828437.96680.995851Y1.978221.006816.038662.980812.94073The 8-dimensinoal “kissing” packing, i.e., a central sphere surrounded by the maximum number of touching spheres.
l8_2428242342.241171240.999966Y1.9797214790.56.020752.966182.92884
l8_2438243687.3084742801.004Y1.981259847.56.003282.951972.91737
l8_2448244690.1364800721.00794Y1.9826860533.15.986232.938172.90629
l8_2458245692.9654858721.01181Y1.9841661218.95.96962.924772.89561
l8_2468246695.7934916801.0156Y1.9856361904.85.953382.911762.8853
l8_2478247698.6214974961.01931Y1.9870962591.5.937542.899122.87536
l8_2488248350.7251258301.02294Y1.9885515819.35.922092.886852.86577
l8_2498249704.2785091521.0265Y1.9963963.85.907012.874942.85653
l8_2508250707.1075150241.03005Y1.9914564654.55.892032.863112.84736
l8_2518251709.9355208801.03348Y1.9928965342.45.87762.851822.83872
l8_2528252356.3821316941.0369Y1.9943216508.65.863252.84062.83014
l8_2538253715.5925326801.04024Y1.9957566726.85.849252.82972.82187
l8_2548254359.211346481.04352Y1.9971716854.85.835582.819132.81392
l8_2558255721.2495445121.04674Y1.9985968112.15.822232.808862.80626
cube8_2568256282Y213.010300The hypercube, one of the three regular convex polytopes in 8d [Coxeter73], [Agrell11a].
iMDPM-QPSK8_2568256261.5Y20.754.259691.249391.24939A generalized PPM packing called 4iMDPM-QPSK [Eriksson14b].
APSK8_2568256261.5N20.754.259691.249391.24939[Zhang15]. Even though the listed performance metrics are the same as for 4iMDPM-QPSK, two packings are geometrically different. This is not a lattice packing and it has fewer nearest neighbors than 4iMDPM-QPSK.
l8_2568256362.0391376101.04988Y217201.35.80922.79892.7989[Agrell14], [Agrell16]
6P-QPSK8_5128512282Y2.250.8888893.01030.5115250.977001A constant-radius packing obtained from a pair of 4-dimensional 24-point packings [Agrell09, Fig. 3]. An analogous construction method was used in [Forney84, Sec. IV-C], [Forney89a] to construct cross-shaped packings.
l8_5128512724.0776980921.3315Y2.2577565.84.777172.27842.74387[Agrell16]
l8_1024810241448.1533156181.58101Y2.5331562.4.031251.990052.93067[Agrell16]
l8_2048820482896.31156398841.86442Y2.751421808.3.315161.687893.11305[Agrell16]
l8_4096840962896.31188471712.24676Y31570598.2.505041.255653.17451[Agrell16]

References

[Agrell14]
E. Agrell, unpublished, 2014.
[Agrell16]
E. Agrell, unpublished, 2016.