Sphere packings of dimension 2

Introduction

Sphere packings in two dimensions (2d), i.e., circle packings, are well studied, due to their usage in coherent modulation. The two dimensions represent the in-phase and quadrature (I/Q) components.

The best known 2d packings are subsets of the hexagonal lattice (honeycomb) [Gilbert52, p. 520], [Simon73], [Foschini74]. It was shown in [Foschini74] that a circular subset of the hexagonal lattice is optimal at asymptotically high M, and it was conjectured in [Graham90] that for M ≠ 4, every optimal 2d packing is a subset of the hexagonal lattice, possibly translated. All best known 2d packings comply with this conjecture. Even for M = 4, a translated subset of the hexagonal lattice is optimal, but in this case, there exist also other optimal packings.

Numerical algorithms to design sphere packings were given in [Foschini74], [Graham90]. The best found normalized second moments were tabulated in [Graham90] for all sizes M ≤ 100 and selected larger sizes. The packings were reproduced using an improved algorithm in [Chow95], where it was proven that the algorithm finds the optimal subset of the hexagonal lattice. Hence, if the conjecture in [Graham90] is true, these are optimal packings. Packings designed using the algorithm of [Chow95] are named c2_*.txt in this database.

The database also contains many suboptimal packings which have been studied in the past. Such packings include quadrature amplitude modulation (QAM) and phase shift keying (PSK) constellations, which form the most popular modulation formats in coherent digital communications, and various special packings from other sources.

The most power-efficient 2d packing, in the sense of maximizing γ, is the equilateral triangle (M = 3). It is the only 2d packing with a positive γ. The highest gain for any 2d packing in the database is G = 0.815 dB and occurs for M = 253, but it would not be hard to design a constellation with marginally higher gains by intersecting the hexagonal lattice with an even large circle. As M → ∞, the gain asymptotically tends to G → 10log10(2π/(3√3)) = 0.825 dB, which equals the coding gain of the hexagonal lattice over the cubic lattice [Conway99, pp. 7, 73].

The 2d packings in the database are, in most cases, rotated and scaled so that the first coordinate is an integer and the second coordinate is a rational number times √3.

Database

FileNMdEEnLatβEbCFM [dB]γ [dB]G [dB]Comment
BPSK2_222210.25Y116.02060-1.76091
PSK2_3231.7320510.333333Y1.584960.630934.771210.7508030An equilateral triangle, or a 2d simplex [Gilbert52], [Cahn59], [Pierce80], [Ekanayake82].
tri2_42410.750.75Y20.3751.24939-1.76091-1.76091An equilateral triangle with a central point [Thomas74].
QPSK2_4241.4142110.5Y20.53.010300A square. Quadrature phase shift keying (QPSK), possibly the most common coherent modulation format. First standardized 1968 in the V.26 2400 bit/s modem [Hanzo04, Sec. 8.1].
QAM2_424220.5Y213.010300Another square [Gilbert52], [Campopiano62], equivalent to QPSK2_4 by rotation.
doublesimplex2_424220.5Y213.010300Two equilateral triangles that share an edge. Has the same En as QPSK. Every rhombic packing with a vertex angle between 60° and 120° is equally good [Foschini74], [Forney84], [Graham90]. Described as a Voronoi constellation in [Forney89b].
PSK2_5251.1755710.723607N2.321930.4306771.40497-0.957139-0.355939A pentagon [Karlsson10b].
c2_52522.720.68Y2.321931.171441.67491-0.687201-0.0860017[Gilbert52], [Graham90], [Karlsson10b]
PSK2_626111Y2.584960.3868530-1.89606-0.791812A hexagon.
c2_6266290.805556Y2.5849611.21870.939045-0.9570120.147233[Gilbert52], [Karlsson10b]
PSK2_7270.86776711.32799N2.807350.356207-1.23193-2.76956-1.23193A heptagon.
c2_72723.428570.857143Y2.807351.221280.669468-0.8681590.669468A hexagon with a central point [Gilbert52], [Foschini74], [Graham90], [Karlsson10b]. The “kissing” packing, i.e., a central circle surrounded by the maximum number of touching circles [Agrell09].
PSK2_8280.76536711.70711N30.333333-2.32261-3.57199-1.65314An octagon. Standardized 1972 in the V.27 4800 bit/s modem [Hanzo04, Sec. 8.1].
doubleQPSK2_82826.828431.70711N32.27614-2.32261-3.57199-1.65314[Proakis01, Sec. 5.2.9]. The same performance as PSK.
square2_82811.51.5Y30.5-1.76091-3.0103-1.09144[Foschini74], [Thomas74], [Proakis01, Sec. 5.2.9]
rectangle2_828261.5Y32-1.76091-3.0103-1.09144[Proakis01, Sec. 5.2.9]
v29modem2_82825.51.375N31.83333-1.38303-2.63241-0.713559[Campopiano62]. In the 7200 bit/s mode of the V.29 modem standard [Hanzo04, Sec. 8.3]. Similar to star2_8.
DSQ2_8282.82843101.25Y33.33333-0.9691-2.21849-0.299632Called 8AMPM in [Ungerboeck82]. [Forney88], [Agrell11b] Described as a Voronoi constellation in [Forney89b].
star2_82824.732051.18301N31.57735-0.729894-1.97928-0.0604262[Foschini74], [Thomas74]. Incorrectly claimed in [Lucky67], [Proakis01, Sec. 5.2.9] to be optimal. Similar to v29modem2_8.
hepta2_8280.8677670.8751.16199N30.291667-0.652013-1.90140.0174548[Foschini74], [Thomas74]
modem2_82824.51.125Y31.5-0.511525-1.760910.157943[Gilbert52], [Foschini74], [Thomas74]. Used in a 4800 bit/s Hycom modem [Forney84].
c2_82824.31251.07813Y31.4375-0.326691-1.576080.342777[Foschini74], [Forney88]
QAM2_92911.333331.33333Y3.169930.42062-1.24939-2.25950[Gilbert52]
c2_9296441.22222Y3.1699313.8805-0.871502-1.881610.377886
c2_1021025.41.35Y3.321931.62556-1.30334-2.110040.457575[Gilbert52]
c2_11211227241.49587Y3.45943209.283-1.74893-2.379480.469555
c2_1221226.333331.58333Y3.584961.76664-1.99572-2.471480.636691
c2_132132611961.76923Y3.70044323.205-2.47784-2.815910.532455
c2_1421427.571431.89286Y3.807351.98863-2.77118-2.985540.586743
c2_152153018562.06222Y3.90689475.058-3.14335-3.245640.536413
PSK2_162160.39018116.56854N40.25-8.17469-8.17469-4.19529[Simon73]
multiPSK2_16216214.05473.51367N43.51367-5.45761-5.45761-1.47821[Salz71]
v29modem2_16216213.53.375N43.375-5.28274-5.28274-1.30334In the V.29 9600 bit/s modem standard [Hanzo04, Sec. 8.3]. It is more tolerant to phase errors than most of the other 16-point packings.
honeycomb2_162164523.25Y413-5.11883-5.11883-1.13943[Thomas74]. Contains holes where further circles would fit.
octa2_16216212.013.00249N43.00249-4.77482-4.77482-0.795418Two octagons forming a star shape [Simon73]. Incorrectly claimed in [Lucky67] to be optimal. Weaker versions of this packing are shown in [Foschini74], [Thomas74], where the ratio between the two radii is higher.
dodeca2_16216211.69622.92404N42.92404-4.65983-4.65983-0.68043Two polygons with the same center [Thomas74].
ennea2_16216210.3232.58075N42.58075-4.11746-4.11746-0.138062Two polygons with the same center. Adapted from a packing in [Thomas74] by optimizing the two radii.
penta2_1621612.573772.57377N40.643443-4.1057-4.1057-0.126299[Simon73], [Foschini74]. A suboptimal version, where the ratio between the two nonzero radii is higher, is shown in [Thomas74].
QAM2_162162102.5Y42.5-3.9794-3.97940[Campopiano62], [Simon73], [Foschini74], [Thomas74]. In the V.32 9600 bit/s modem [Hanzo04, Sec. 8.4] and many other standards.
cross2_162162102.5Y42.5-3.9794-3.97940[Forney84]. The same performance as QAM.
t2_162162102.5Y42.5-3.9794-3.97940[Thomas74]. The same performance as QAM.
tri2_1621629.083332.27083Y42.27083-3.56185-3.561850.417547This packing, suggested in [Simon73], does not have zero mean and can be improved by translation.
hex2_1621612.252.25Y40.5625-3.52183-3.521830.457575Incorrectly claimed in [Simon73] to be optimal.
compact2_1621628.828132.20703Y42.20703-3.43808-3.438080.541315[Conway83]. The same constellation as tri2_16, but recentered to zero mean.
c2_162164352.1875Y48.75-3.39948-3.399480.579919Conjectured optimal in [Conway83], where it was described as a Voronoi code. A similar constellation, optimized for a slightly different criterion, was shown in [Foschini74]. Also shown in [Forney88], [Forney89b], [Graham90].
c2_1721729.245672.31142Y4.087462.26196-3.63879-3.544850.620901
c2_18218187882.4321Y4.16993188.972-3.85981-3.679130.663165
c2_19219210.10532.52632Y4.247932.37887-4.02488-3.763710.746336[Simon73], [Foschini74], [Graham90], [Karlsson10b]
c2_202208174.882.7325Y4.3219340.4634-4.3656-4.029430.640422
c2_21221211.56462.89116Y4.392322.63292-4.61072-4.204380.618071
greedy2_222222214663.02893Y4.45943328.741-4.81289-4.340690.627794The smallest size for which the “greedy algorithm” does not yield the best known packing [Graham90].
c2_22222212.09093.02273Y4.459432.71131-4.80399-4.331790.636691[Graham90]
c2_232234666683.15123Y4.523561474.06-4.9848-4.450590.657915
c2_242242418953.28993Y4.58496413.308-5.17187-4.579110.663899
c2_25225213.67683.4192Y4.643862.94514-5.33925-4.691060.681355
c2_262262624033.55473Y4.70044511.229-5.50807-4.807290.689817
c2_27227214.66673.66667Y4.754893.08455-5.64271-4.891910.725507
c2_28228215.29083.8227Y4.807353.18071-5.82371-5.025240.708418
c2_29229215.91443.9786Y4.857983.27593-5.9973-5.153340.692768
c2_30230216.46674.11667Y4.906893.35583-6.14546-5.257990.69701
c2_31231217.03234.25806Y4.95423.43795-6.29212-5.362990.697578[Karlsson10b]
multiPSK2_32232236.50729.1268N57.30144-9.60318-8.63408-2.47108[Salz71]
honeycomb2_32232226.56.625Y55.3-8.21186-7.24276-1.07975[Thomas74]. Contains holes where further circles would fit.
tb2_32232221.77635.44407N54.35526-7.35924-6.39014-0.227132[Thomas74]
circb2_3223215.353485.35348N51.0707-7.28636-6.31726-0.154258Adapted from a packing in [Thomas74] by optimizing the radii.
DSQ2_322322.82843425.25Y58.4-7.20159-6.23249-0.0694886Called 32AMPM in [Ungerboeck82] and "square" in [Forney89a]. Described as a Voronoi constellation in [Forney89b].
circa2_3223215.235375.23537N51.04707-7.18947-6.22037-0.0573697Adapted from a packing in [Thomas74] by optimizing the radii.
ta2_32232220.00115.00028N54.00023-6.98994-6.020840.142159[Thomas74]
cross2_322322205Y54-6.9897-6.02060.142404[Campopiano62], [Thomas74], [Forney84]. In the V.32 9600 bit/s modem standard [Hanzo04, Sec. 8.4].
tri2_3223214.68754.6875Y50.9375-6.70941-5.740310.422692[Thomas74]
c2_322323245034.39746Y5900.6-6.43202-5.462920.700085[Forney84]
c2_33233218.08264.52066Y5.044393.5847-6.55202-5.544530.717968
c2_342343453704.64533Y5.087461055.54-6.67016-5.625750.733462
c2_35235219.11184.77796Y5.129283.72602-6.79242-5.712460.740852
c2_36236127074.90972Y5.16993136.752-6.91057-5.796330.748599
c2_37237220.10815.02703Y5.209453.85993-7.01311-5.865790.7684
c2_38238220.80895.20222Y5.247933.96516-7.16188-5.982610.738621
c2_392392636245.36095Y5.2854685.662-7.29241-6.082230.723909
c2_40240222.03255.50813Y5.321934.13995-7.41004-6.169950.719096
c2_4124182379525.64426Y5.357557083.83-7.51607-6.247010.723018
c2_42242223.04765.7619Y5.392324.27416-7.60566-6.308510.740665
c2_4324386436965.90806Y5.426268052.68-7.71445-6.390040.736533
c2_44244224.15916.03977Y5.459434.4252-7.81021-6.459330.742966
c2_4524590500246.1758Y5.491859108.77-7.90693-6.530340.74608
c2_46246225.25716.31427Y5.523564.57261-8.00323-6.601640.74738
c2_4724794570406.45541Y5.5545910269.-8.09924-6.673320.746828
c2_48248226.33336.58333Y5.584964.71504-8.18446-6.734860.755008
cb2_4824862376.58333Y5.5849642.4354-8.18446-6.734860.755008The best known packing with M = 48 is not unique. This packing has the same En as c2_48, although they are geometrically different [Graham90].
c2_4924998646726.73386Y5.6147111518.3-8.28264-6.809970.748258
c2_50250227.48326.8708Y5.643864.86958-8.37007-6.874910.750375
c2_51251102730407.02038Y5.6724312876.3-8.4636-6.946520.744583
c2_52252228.63467.15865Y5.700445.02323-8.54831-7.009830.745876
c2_53253106820607.30331Y5.7279214326.3-8.6352-7.075830.743323
c2_5425454216417.42147Y5.754893760.46-8.7049-7.125130.756348
c2_55255230.10917.52727Y5.781365.20796-8.76638-7.166680.776049[Karlsson10b]
c2_562561415047.67347Y5.80735258.982-8.84992-7.230730.772197
c2_57257231.23927.80979Y5.832895.35569-8.92639-7.288150.773976
c2_5825858267327.94649Y5.857984563.35-9.00175-7.344870.775481
c2_59259232.35168.08791Y5.882645.4995-9.07836-7.403240.774407
c2_602603074068.22889Y5.906891253.79-9.15341-7.460420.773596
c2_61261233.44268.36066Y5.930745.63886-9.2224-7.511920.777597
c2_6226262327798.52732Y5.95425505.19-9.30812-7.580490.763663
c2_63263234.71818.67952Y5.977285.80834-9.38496-7.640520.757449
honeycomb2_64264253.513.375Y68.91667-11.2629-9.50203-1.05104[Thomas74] Contains holes where further circles would fit.
circa2_64264245.635311.4088N67.60588-10.5724-8.8115-0.360516Adapted from a packing in [Thomas74] by optimizing the radii.
circb2_64264243.21610.804N67.20266-10.3358-8.57493-0.123951Adapted from a packing in [Thomas74] by optimizing the radii.
QAM2_6426424210.5Y67-10.2119-8.450980[Campopiano62], [Thomas74], [Forney84]
cross2_6426424110.25Y66.83333-10.1072-8.346330.104654[Forney84], [Forney88]
circular2_64264240.640510.1601N66.77342-10.069-8.308080.142898Called "circular" in [Forney84].
voronoi2_64264235.43758.85938Y65.90625-9.47403-7.713120.737862This packing was presented in [Forney89b] as a Voronoi constellation, which enables low-complexity implementation. Combined with translated replicas of itself, it tiles the plane.
tri2_64264236.6259.15625Y66.10417-9.61718-7.856260.594717[Thomas74]
modem2_64264235.43758.85938Y65.90625-9.47403-7.713120.737862Used in the Codex/ESE SP14.4 modem [Forney84].
c2_64264235.258.8125Y65.875-9.45099-7.690080.760902[Forney84]
c2_65265266052.488.95337Y6.022371005.-9.51987-7.742790.760421
c2_66266236.34999.08747Y6.044396.01381-9.58443-7.79150.763193
c2_67267236.85729.2143Y6.066096.07594-9.64462-7.836140.769303
c2_6826834108029.34429Y6.087461774.47-9.70546-7.88170.773772
c2_69269237.85519.46377Y6.108526.19709-9.76064-7.921880.782936
c2_70270287538.089.6149Y6.129281229.85-9.82945-7.975950.777532
c2_71271239.0289.75699Y6.149756.34627-9.89316-8.025190.776308
c2_722728633.6679.90104Y6.16993102.702-9.95681-8.074610.774262
c2_7327314621402010.0403Y6.1898234576.1-10.0175-8.12130.774326
c2_74274745574710.1802Y6.209458977.76-10.0776-8.167650.774135
c2_752755025782.710.3131Y6.228824139.25-10.1339-8.210420.776926
c2_76276241.789510.4474Y6.247936.68853-10.1901-8.253310.779031
c2_7727715425146810.6033Y6.2667940127.1-10.2544-8.304560.772211
c2_78278267266.3310.749Y6.28541156.06-10.3137-8.350960.769709
c2_7927915827172810.8848Y6.3037843105.6-10.3682-8.392790.771231
c2_80280204406.8111.017Y6.32193697.068-10.4206-8.432750.774113
c2_81281244.592611.1481Y6.339857.0337-10.472-8.471840.77736
c2_82282827585811.2817Y6.3575511932.-10.5237-8.511440.779601
c2_8328316631441211.4099Y6.3750449319.2-10.5728-8.54860.783797
c2_84284289041.6711.5327Y6.392321414.46-10.6193-8.583340.789944
c2_85285246.588211.6471Y6.409397.26875-10.6622-8.614590.799118
c2_86286868726511.7989Y6.4262613579.4-10.7184-8.659450.794245
c2_87287247.776511.9441Y6.442947.41532-10.7715-8.70130.791933
c2_88288442340512.0894Y6.459433623.38-10.824-8.742690.789647
c2_89289248.950912.2377Y6.475737.55913-10.877-8.784720.786308
c2_9029018401312.3858Y6.49185618.159-10.9292-8.826150.783146
c2_91291250.109912.5275Y6.507797.69998-10.9786-8.86490.782278
c2_922929210725912.6724Y6.5235616441.8-11.0286-8.904330.780321
c2_932936249234.712.8082Y6.539167529.2-11.0749-8.940260.78149
c2_942949411434512.9408Y6.5545917445.-11.1196-8.974760.783702
c2_952953818893.313.084Y6.569862875.75-11.1674-9.012440.782364
c2_96296252.87513.2188Y6.584968.02966-11.2119-9.046970.78382
c2_9729719450276013.3585Y6.5999176176.8-11.2576-9.082790.783627
c2_98298142644.7313.4935Y6.61471399.826-11.3013-9.116750.784944
c2_99299226594.9613.626Y6.62936994.812-11.3437-9.149560.787078
c2_1002100255.014813.7537Y6.643868.28055-11.3842-9.180590.790644
c2_101210120256657213.8852Y6.6582185093.7-11.4255-9.212550.792964
c2_1022102256.039214.0098Y6.672438.39863-11.4643-9.242080.797381
c2_103210320660112414.1654Y6.686589901.1-11.5123-9.280910.792193
c2_1042104257.250414.3126Y6.700448.54427-11.5572-9.316750.789677
c2_1052105142832.8514.4533Y6.71425421.917-11.5997-9.350310.789141
c2_1062106258.357814.5894Y6.727928.67397-11.6404-9.382180.789993
c2_107210721467472414.7333Y6.74147100086.-11.683-9.416040.78856
c2_1082108259.481514.8704Y6.754898.8057-11.7232-9.447640.789107
c2_109210921871328415.0089Y6.76818105388.-11.7635-9.479370.78923
c2_1102110260.615.15Y6.781368.93626-11.8041-9.511560.788626
c2_111211122275372815.2936Y6.79442110933.-11.8451-9.544170.787327
c2_1122112163950.4515.4314Y6.80735580.321-11.8841-9.574880.787652
c2_1132113262.25815.5645Y6.820189.12849-11.9213-9.603990.789318
c2_114211411420397515.6952Y6.8328929851.9-11.9577-9.632230.7916
c2_115211523083710415.8243Y6.84549122285.-11.9932-9.659790.794299
c2_116211681021.4215.9596Y6.85798148.938-12.0302-9.688870.795233
c2_1172117264.372616.0931Y6.870369.3696-12.0664-9.717210.796659
c2_118211811822597716.2293Y6.8826432832.9-12.103-9.746050.797344
c2_119211923892676416.3612Y6.89482134415.-12.1382-9.773520.799155
c2_12021206059362.316.4895Y6.906898594.64-12.1721-9.799850.801879
c2_1212121266.446316.6116Y6.918869.60364-12.2041-9.824360.806193
c2_122212212224957616.7681Y6.9307436010.-12.2448-9.857640.80151
c2_1232123267.678816.9197Y6.942519.74845-12.2839-9.889360.798162
c2_12421246265621.317.0711Y6.95429436.21-12.3226-9.920740.794927
c2_1252125268.862217.2156Y6.965789.88578-12.3592-9.950110.793495
c2_126212612627561817.3607Y6.9772839502.2-12.3957-9.97940.791923
c2_1272127254112913217.5016Y6.98868161566.-12.4308-10.00740.79142
fivecircle2_12821282115.53928.8846N716.5055-14.6067-12.1763-1.35014Adapted from a packing in [Thomas74] by optimizing the radii.
sixcircle2_1282128285.721721.4304N712.246-13.3103-10.8799-0.0537841Adapted from a packing in [Thomas74] by optimizing the radii.
DSQ2_12821282.8284317021.25Y724.2857-13.2736-10.8432-0.0170646128-ary double square (DSQ) packing is used in the 10-gigabit Ethernet standard 10GBASE-T.
cross2_128212828220.5Y711.7143-13.1175-10.68720.138986[Forney84]. In the V.33 14400 bit/s modem standard [Hanzo04, Sec. 8.5].
b2_128212828220.5Y711.7143-13.1175-10.68720.138986[Campopiano62], [Thomas74]. Has, as pointed out in [Forney84], the same performance as cross2_128, although the constellations are not equivalent.
tri2_1282128271.96217.9905N710.2803-12.5504-10.12010.706091[Thomas74]
c2_1282128270.544217.636Y710.0777-12.464-10.03360.792512
c2_1292129258118292017.7712Y7.01123168718.-12.4972-10.05980.793428
c2_1302130271.615417.9038Y7.0223710.1982-12.5295-10.08520.794921
c2_1312131262123872018.0456Y7.03342176119.-12.5637-10.11260.794215
c2_1322132228799.6718.1811Y7.044391249.17-12.5962-10.13840.794992
c2_1332133266129616018.3187Y7.05528183715.-12.629-10.16440.795273
c2_134213413433135618.4538Y7.0660946893.8-12.6609-10.18970.796151
c2_1352135274.34818.587Y7.0768210.5059-12.6921-10.21430.797443
c2_13621366886588.318.7258Y7.0874612217.1-12.7244-10.24010.797414
c2_1372137274141619618.8635Y7.09803199520.-12.7562-10.26540.797654
c2_138213813836182918.9996Y7.1085250900.7-12.7875-10.29030.798241
c2_1392139276.528819.1322Y7.1189410.75-12.8176-10.31410.799632
c2_1402140709440919.2671Y7.1292813242.4-12.8482-10.33830.800462
c2_1412141277.588719.3972Y7.1395510.8674-12.8774-10.36130.802386
c2_1422142278.170419.5426Y7.1497510.9333-12.9098-10.38750.800855
c2_1432143278.75219.688Y7.1598710.9991-12.942-10.41360.799354
c2_1442144279.318919.8297Y7.1699311.0627-12.9732-10.43860.798683
c2_1452145290167973619.9731Y7.17991233949.-13.0045-10.46390.797662
c2_1462146280.451920.113Y7.1898211.1897-13.0348-10.48820.797406
c2_1472147294175100020.2578Y7.19967243206.-13.0659-10.51340.796101
c2_148214874111691.20.3965Y7.2094515492.3-13.0956-10.53710.796104
c2_1492149282.121220.5303Y7.2191711.3754-13.124-10.55970.797154
c2_15021505051645.720.6583Y7.228827144.41-13.1509-10.58090.799411
c2_1512151283.125820.7815Y7.238411.484-13.1768-10.60090.80264
c2_15221523830216.820.9258Y7.247934169.03-13.2068-10.62530.801442
c2_1532153306197241221.0647Y7.25739271780.-13.2355-10.64830.801376
c2_154215415450280221.201Y7.2667969191.8-13.2636-10.67070.801846
c2_1552155285.334921.3337Y7.2761211.7281-13.2907-10.69230.803026
c2_15621562412366.421.4694Y7.28541697.42-13.3182-10.71430.803608
c2_1572157314212998421.6031Y7.29462291994.-13.3452-10.73570.804563
c2_158215815854277321.7422Y7.3037874314.-13.373-10.75820.804443
c2_1592159287.517321.8793Y7.3128811.9675-13.4003-10.78010.804721
c2_16021604035230.322.0189Y7.321934811.61-13.428-10.80230.804501
c2_1612161322229711622.155Y7.33092313346.-13.4547-10.82370.804975
c2_162216216258493922.2885Y7.3398579693.6-13.4808-10.84450.805941
c2_1632163289.668722.4172Y7.3487312.2019-13.5058-10.86430.807828
c2_1642164290.285522.5714Y7.3575512.2711-13.5356-10.88880.804781
c2_165216511027493622.722Y7.3663237323.4-13.5645-10.91260.802464
c2_166216616663027322.8724Y7.3750485460.3-13.5931-10.93610.800202
c2_1672167334256783623.0184Y7.3837347771.-13.6207-10.95860.798825
c2_1682168292.618623.1547Y7.3923212.529-13.6464-10.97920.799269
c2_1692169338266075623.2901Y7.40088359519.-13.6717-10.99950.799864
c2_1702170293.685423.4213Y7.4093912.6441-13.6961-11.01890.801235
c2_1712171342275516023.5556Y7.41785371423.-13.7209-11.03880.802031
c2_1722172294.751423.6878Y7.4262612.759-13.7453-11.05810.803194
c2_1732173346285200823.8231Y7.43463383611.-13.77-11.0780.803787
c2_1742174295.816123.954Y7.4429412.8734-13.7938-11.09690.805164
c2_17521755060243.624.0974Y7.451218085.07-13.8197-11.1180.804271
c2_1762176296.948924.2372Y7.4594312.9968-13.8448-11.13840.804041
c2_1772177354305494424.3779Y7.46761409093.-13.87-11.15880.803647
c2_178217817877676524.516Y7.47573103905.-13.8945-11.17860.803724
c2_1792179358315955624.6524Y7.48382422185.-13.9186-11.1980.804087
c2_1802180299.161124.7903Y7.4918513.2359-13.9428-11.21750.804204
c2_1812181362326716424.9318Y7.49985435631.-13.9675-11.23760.803675
c2_18221822616945.925.068Y7.507792257.11-13.9912-11.25670.804084
c2_1832183366337656825.2065Y7.5157449269.-14.0151-11.2760.804068
c2_18421842101.36925.3422Y7.5235613.4735-14.0384-11.29480.804563
c2_1852185370348834425.481Y7.53138463175.-14.0622-11.3140.804507
c2_18621862102.46225.6156Y7.5391613.5907-14.085-11.33240.805161
c2_1872187374360309225.7592Y7.54689477427.-14.1093-11.35220.804295
c2_18821882103.625.9001Y7.5545913.7136-14.133-11.37150.80389
c2_1892189378372111226.0429Y7.56224492065.-14.1569-11.3910.803175
c2_19021902104.72126.1802Y7.5698613.8339-14.1797-11.40950.803377
c2_1912191382384092826.3214Y7.57743506891.-14.2031-11.42850.802929
c2_19221922105.83326.4583Y7.5849613.953-14.2256-11.44670.803196
c2_1932193386396185626.5904Y7.59246521815.-14.2472-11.4640.804259
c2_1942194194100574826.723Y7.59991132337.-14.2689-11.48140.805203
c2_19521952107.42526.8563Y7.6073314.1213-14.2905-11.49870.806038
c2_1962196145290.0226.9899Y7.61471694.711-14.312-11.51610.806821
c2_1972197394421028827.1219Y7.62205552382.-14.3332-11.53310.807855
c2_1982198198106835627.2512Y7.62936140032.-14.3539-11.54960.809294
c2_19921992109.50827.3769Y7.6366214.3398-14.3738-11.56540.811299
c2_200220080176151.27.5236Y7.6438623044.8-14.397-11.58450.80997
c2_201220113449677227.6661Y7.6510564928.6-14.4195-11.60290.809312
c2_2022202202113462127.8066Y7.65821148157.-14.4415-11.62080.808967
c2_20322032111.77727.9442Y7.6653414.5821-14.4629-11.63820.809083
c2_204220468129862.28.0843Y7.6724316925.8-14.4846-11.65590.808816
c2_20522058218976828.2225Y7.6794824711.-14.506-11.67320.808836
c2_2062206206120370528.3652Y7.6865156600.-14.5279-11.69120.80817
c2_20722072114.02428.506Y7.6934914.8208-14.5494-11.70870.807799
c2_20822083229336.628.649Y7.700443809.73-14.5711-11.72650.807094
c2_2092209418503014828.7891Y7.70736652642.-14.5923-11.74380.806839
c2_21022102822679.28.9273Y7.714252939.88-14.6131-11.76070.806875
c2_21122112116.24629.0616Y7.721115.0557-14.6332-11.7770.807484
c2_2122212212131285529.2109Y7.72792169885.-14.6555-11.79540.805862
c2_2132213142591853.29.352Y7.7347176519.1-14.6764-11.81250.805472
c2_2142214214135043629.4881Y7.74147174442.-14.6965-11.82880.805819
c2_2152215430547814429.6276Y7.74819707022.-14.717-11.84560.80566
c2_216221610834720729.7674Y7.7548944772.7-14.7374-11.86230.805462
c2_217221762114953.29.9044Y7.7615514810.5-14.7574-11.87850.805669
c2_2182218218142750330.0375Y7.76818183763.-14.7766-11.8940.806444
c2_21922192120.67630.1689Y7.7747915.5214-14.7956-11.90930.807451
c2_220222088234680.30.3048Y7.7813630159.3-14.8151-11.92520.807812
c2_22122212121.74930.4373Y7.787915.6331-14.8341-11.94050.808651
c2_222222274167412.30.572Y7.7944221478.5-14.8532-11.9560.80917
c2_22322232122.82230.7055Y7.800915.7446-14.8722-11.97130.809858
c2_2242224112386870.30.8411Y7.8073549552.-14.8913-11.98690.810243
c2_22522252123.89330.9733Y7.8137815.8557-14.9099-12.00190.811088
c2_2262226226158946931.1197Y7.82018203252.-14.9304-12.01880.809961
c2_22722272125.05331.2632Y7.8265515.978-14.9503-12.03520.809236
c2_22822282125.62831.4069Y7.8328916.0385-14.9703-12.05160.808493
c2_2292229458661762431.5479Y7.8392844170.-14.9897-12.06760.808125
c2_23022302126.72231.6804Y7.8454916.1522-15.0079-12.08230.808931
c2_2312231462679229631.8224Y7.85175865068.-15.0273-12.09830.808439
c2_23222325810751631.9608Y7.8579813682.4-15.0462-12.11370.808436
c2_2332233466697052032.0991Y7.86419886363.-15.0649-12.1290.808434
c2_2342234234176514832.2366Y7.87036224278.-15.0835-12.14410.808553
c2_2352235470715209632.3771Y7.87652908028.-15.1024-12.15960.80827
c2_236223611845276132.5166Y7.8826457437.7-15.121-12.17490.808116
c2_2372237474733702432.656Y7.88874930062.-15.1396-12.19020.807975
c2_23822382131.16832.792Y7.8948216.6145-15.1577-12.20490.80829
c2_2392239478752560032.9371Y7.90087952503.-15.1769-12.22070.807404
c2_2402240240190525133.0773Y7.90689240961.-15.1953-12.23580.807169
c2_24122412132.84633.2116Y7.9128916.7886-15.2129-12.25010.8077
c2_24222422216140.833.3488Y7.918862038.27-15.2308-12.26480.807858
c2_24322432133.9433.4851Y7.9248116.9014-15.2485-12.27920.808127
c2_2442244122500380.33.6187Y7.9307463093.8-15.2658-12.29330.808745
c2_24522452135.00233.7505Y7.9366417.01-15.2828-12.3070.809587
c2_2462246246205046333.883Y7.94251258163.-15.2998-12.32080.810331
c2_2472247494830103234.0156Y7.948371044369.-15.3168-12.33460.811057
c2_248224862131273.34.1502Y7.954216503.7-15.3339-12.34860.811526
c2_24922492137.13934.2848Y7.9617.2285-15.351-12.36250.811988
c2_2502250250215125134.42Y7.96578270061.-15.3681-12.37640.81237
c2_2512251502870767234.5537Y7.971541092345.-15.3849-12.39010.812941
c2_2522252126550666.34.6855Y7.9772869029.3-15.4015-12.40350.813751
c2_25322532139.25734.8142Y7.9829917.4442-15.4176-12.41650.814925
c2_2542254254225570134.9634Y7.98868282362.-15.4361-12.4320.813552
c2_25522553031598.435.1093Y7.994353952.58-15.4542-12.4470.812605
c2_256225612857759535.2536Y872199.4-15.472-12.46170.811854
QAM2_2562256217042.5Y821.25-16.2839-13.27360[Campopiano62], [Forney84]. In the 12000 bit/s mode of the V.33 modem standard [Hanzo04, Sec. 8.5].
DSQ2_51225122.8284368285.25Y975.7778-19.3069-15.7851-0.00424738
cross2_5122512233082.5Y936.6667-19.1645-15.64270.138157
QAM2_1024210242682170.5Y1068.2-22.3172-18.33780