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Spherical codes

This web page deals only with small codes. For a more extensive table, see https://spherical-codes.org.

The spherical code problem asks how to arrange N points on the surface of the unit sphere Sn-1 in n dimensions so as to maximize the distance between the nearest pair of points. This problem is also known as the Tammes problem, due to its origins in botany [28]. The following table shows the best constructions known with up to to 32 points; it omits configurations with at most 2n points in n dimensions, for which the exact answer is known [22], as well as the trivial case of two dimensions. Please let me know of any improvements to these records.

The table lists the cosine of the minimal angular distance (i.e., the inner product of the closest points) and the number of rattlers (i.e., points with no neighbors at the minimal distance). Following the example of [3], it also includes the minimal polynomial of this inner product in many cases; the minimal polynomials I have computed that are too large to fit in this table are available here. The cosines are rounded up, so that codes achieving this bound provably exist. When a minimal polynomial is listed, I have checked that there exists a code achieving this exact value. Entries that are known to be optimal are marked with ∗.

All of this data is available from https://hdl.handle.net/1721.1/142661, including coordinates for these codes. If you cite this table, please refer to the data set, not this web page.

Plots of the data are available in terms of minimal angle and density.

Table of spherical codes (N points in n dimensions)

n N Cosine of minimal angle Rattlers Minimal polynomial References
∗ 3 7 0.21013831273060308486576530161854 3x3 – 9x2 – 3x + 1 [25]
∗ 3 8 0.26120387496374144251476820691706 7x2 + 2x – 1 [25]
∗ 3 9 0.33333333333333333333333333333334 3x – 1 [25]
∗ 3 10 0.40439432521625075684622475072579 7x3 – 4x2 – 2x + 1 [25, 7]
∗ 3 11 0.44721359549995793928183473374626 5x2 – 1 [25, 7]
∗ 3 12 0.44721359549995793928183473374626 5x2 – 1 [9]
∗ 3 13 0.54263648682963846368144007679523 degree 8 [25, 20]
∗ 3 14 0.56395030036050516749088520575670 4x4 – 2x3 + 3x2 – 1 [25, 21]
3 15 0.59260590292507377809642492233276 13x5 – x4 + 6x3 + 2x2 – 3x – 1 [25, 13]
3 16 0.61229461648269661600605156530452 23x6 + 6x5 + 5x4 + 4x3 – 3x2 – 2x – 1 [25]
3 17 0.62809441507002164642659266364200 degree 10 [7]
3 18 0.64869583222311652907905517143837 degree 11 [29]
3 19 0.67311688756005104893098993946251 1 degree 75 [14]
3 20 0.67647713812965145206767190546124 2 21x3 – 9x2 – 5x + 1 [31]
3 21 0.69949843106637709573776394272263 [13]
3 22 0.71030625857969039075297000237262 degree 18 [13]
3 23 0.72284698486839966689606899351518 1 [30]
∗ 3 24 0.72307846833350853703234480939452 7x3 + x2 – 3x – 1 [23]
3 25 0.74739862857799274075800073210958 degree 89 [13]
3 26 0.75427817712004151970111096057049 2 degree 101 [13]
3 27 0.75838921077657748391083424965931 degree 24 [29]
3 28 0.77323026233796340734575992835035 1 [13]
3 29 0.78028141594871705893938480666397 1 [13]
3 30 0.78155187509498732710358610409650 degree 29 [5]
3 31 0.79111861329867495834332727354269 degree 66 [27]
3 32 0.79361661487126244036481707479245 degree 22 [7]
4 9 0.16201519961163454918243428113270 16x3 – 16x2 – 4x + 1 [26]
∗ 4 10 0.16666666666666666666666666666667 6x – 1 [17, 2]
4 11 0.23040556359455544173706204865074 8x3 – 12x2 – 2x + 1 [26]
4 12 0.25000000000000000000000000000000 4x – 1 [17]
4 13 0.30729565398102882232528869633144 degree 9 [26]
4 14 0.31951859421260363549590568166773 degree 7 [26]
4 15 0.35099217594534630329905559676410 36x4 – 18x3 + 10x2 – 1 [26]
4 16 0.38762817712253427775854691441514 degree 10 [17]
4 17 0.41225936269326378906697367932150 [26]
4 18 0.42281941407305934402640028185634 degree 16 [26]
4 19 0.43425854591066488218653687791175 3x2 + x – 1 [26]
4 20 0.43425854591066488218653687791175 3x2 + x – 1 [26]
4 21 0.47138085850731791681783507846628 degree 8 [26]
4 22 0.49788413084355235628616910040615 [26]
4 23 0.50000000000000000000000000000000 2x – 1 [26]
4 24 0.50000000000000000000000000000000 2x – 1 [24]
4 25 0.53731605665507787659607001344589 [26]
4 26 0.54078961769753707672755075220843 degree 6 [26]
4 27 0.55759135118017018253232385918274 3 794x5 + 393x4 – 344x3 – 82x2 + 6x + 1 [26]
4 28 0.56733880407434859617396405079452 [26]
4 29 0.57314853044836189190193776787666 degree 58 [26]
4 30 0.58423281393058512894226706121300 [26]
4 31 0.59076590398070368826368709295480 3 [26]
4 32 0.59572014923551345643989842209366 2 [26]
5 11 0.13285354259858991808946447681952 45x3 – 25x2 – 5x + 1 [26]
5 12 0.15393160503302123094881763125084 25x4 + 30x3 + 24x2 + 2x – 1 [26]
5 13 0.18725985188285358701782399517981 17x3 – 5x2 – 5x + 1 [26]
5 14 0.20000000000000000000000000000000 5x – 1 [26]
5 15 0.20000000000000000000000000000000 5x – 1 [26]
∗ 5 16 0.20000000000000000000000000000000 5x – 1 [12, 15]
5 17 0.27047583526857362208626102246801 9x4 – 16x3 – 10x2 + 1 [26]
5 18 0.27550174165981923838704223579799 484x5 – 488x4 + 97x3 + 17x2 – x – 1 [26]
5 19 0.29182239902449014614857168904980 degree 6 [26]
5 20 0.29938569289912478230200302792897 1 5x3 + 13x2 – x – 1 [26]
5 21 0.31491695717530346284922041852029 degree 14 [26]
5 22 0.35499503416625620682992409117504 [26]
5 23 0.36977269694307633377246233586792 2 [26]
5 24 0.37423298246516725172655168161941 degree 10 [26]
5 25 0.37962102539378266282456421792923 [26]
5 26 0.39024065950484684004435526211173 degree 23 [26]
5 27 0.40165926427641808725922803276444 degree 15
5 28 0.40816969909292876817531302066075 [26]
5 29 0.41103443509195154800015221999063 degree 31 [26]
5 30 0.41302977612208581019088251209695 degree 7 [26]
5 31 0.43391186395954602878684987397560 degree 20 [26]
5 32 0.44183074392731126949046338768223 degree 10
6 13 0.11307975214744721384507044810795 96x3 – 36x2 – 6x + 1 [1]
6 14 0.13249092032347031437017906291052 degree 13 [32]
6 15 0.14494897427831780981972840747059 20x2 + 4x – 1 [32]
6 16 0.17114764942939365334044778754758 degree 8 [1]
6 17 0.18327433702314481857632435406165 400x4 + 240x3 + 16x2 – 8x – 1 [1]
6 18 0.19781218860197545240607808204678 degree 8 [32]
6 19 0.20022602589120548304270384498373 [32]
6 20 0.21428571428571428571428571428572 14x – 3 [32]
6 21 0.24285284801369170250759542313692 [32]
6 22 0.24886569945945498664968496352672 2
6 23 0.25000000000000000000000000000000 4x – 1 [32]
6 24 0.25000000000000000000000000000000 4x – 1 [32]
6 25 0.25000000000000000000000000000000 4x – 1 [32]
6 26 0.25000000000000000000000000000000 4x – 1 [32]
∗ 6 27 0.25000000000000000000000000000000 4x – 1 [12, 15]
6 28 0.30000000000000000000000000000000 7 10x – 3 [8]
6 29 0.32784483088118687202127443197487 2
6 30 0.33320223547405022827954928731378
6 31 0.33333333333333333333333333333334 3x – 1 [32]
6 32 0.33333333333333333333333333333334 3x – 1 [16]
7 15 0.09870177627236447932802936506891 175x3 – 49x2 – 7x + 1 [1]
7 16 0.11332087960014474124552027924541 degree 12 [32]
7 17 0.12484158381525719834719018255480 degree 48
7 18 0.12613198362288317391722947587285 47x2 + 2x – 1 [1]
7 19 0.15659738541709551030242683082792 1280x6 – 352x4 – 48x3 + 37x2 + 3x – 1 [32]
7 20 0.16952084719853722593019861519182 23x2 + 2x – 1 [8]
7 21 0.18152396080041583540526502110446 [32]
7 22 0.18274399763155681014833407039277 19x2 + 2x – 1 [32]
7 23 0.18274399763155681014833407039277 19x2 + 2x – 1 [32]
7 24 0.18274399763155681014833407039277 19x2 + 2x – 1 [32]
7 25 0.21473723385459290952279854505932 1 31x2 – 2x – 1 [32]
7 26 0.23133143037135692995849173196346 [32]
7 27 0.24054938200924358136050178936531 1
7 28 0.24735665120702230944063342932200
7 29 0.24893314468092960074198540286771
7 30 0.24946828687500069590353003579697 2
7 31 0.25450276751476520071094915336323 1
7 32 0.26559569943123793674839332081547
8 17 0.08773346332333854567255804176022 288x3 – 64x2 – 8x + 1 [8]
8 18 0.09946957270878709385964502330429 [32]
8 19 0.11140997502603998543258143242674 degree 29 [32]
8 20 0.11949686668719356518012001310428 224x3 + 60x2 – 2x – 1 [32]
8 21 0.13060193748187072125738410345853 28x2 + 4x – 1 [32]
8 22 0.13060193748187072125738410345853 28x2 + 4x – 1 [8]
8 23 0.15716994198931666717945817456695 degree 68 [32]
8 24 0.15769214493936087799410602949295 4x4 – 4x3 – 27x2 – 2x + 1 [32]
8 25 0.16433417412503162111153267794066 degree 18
8 26 0.17265209503507043373086916730531 degree 40
8 27 0.17985645705828421512952134000393 1 degree 32
8 28 0.18115842002150127258095821368305 36x4 + 24x3 – 36x2 + 1
8 29 0.18792741558936484033926552129998 degree 16
8 30 0.20391627940388098449483075238404
8 31 0.21970657429991018286120539055796
8 32 0.22732545202384657669029760629103 95x2 – 4x – 4
9 19 0.07906715121746358947508550780826 441x3 – 81x2 – 9x + 1 [8]
9 20 0.08706795832124714750492273563902 63x2 + 6x – 1
9 21 0.08706795832124714750492273563902 63x2 + 6x – 1
9 22 0.10901537523956808103629964436986 662x4 + 23x3 – 67x2 – 3x + 1
9 23 0.11427236968530968639282662571064 degree 16
9 24 0.12713870233144916446935012832137
9 25 0.13545157071140596843926701196576
9 26 0.14666206355598451422693622189428
9 27 0.14997512190732167370329398866647
9 28 0.15022110482233484500666951280126 31x2 + 2x – 1
9 29 0.15022110482233484500666951280126 31x2 + 2x – 1
9 30 0.15022110482233484500666951280126 31x2 + 2x – 1
9 31 0.15022110482233484500666951280126 31x2 + 2x – 1
9 32 0.15022110482233484500666951280126 31x2 + 2x – 1 [8]
10 21 0.07203313984214689097267814366206 640x3 – 100x2 – 10x + 1 [8]
10 22 0.08395615471838847066948704055021
10 23 0.08685732654080788416875987795916
10 24 0.09871013349961574400220437104983 1 degree 8
10 25 0.10447637455518529630303057158420 2
10 26 0.10447765212347525953178684587436 2
10 27 0.12421241419098055856947587657933
10 28 0.12823868446883383035640365542200 92x2 – 4x – 1
10 29 0.13063429594350723450192868698071 degree 13
10 30 0.13843956089234491690957456895262
10 31 0.14411493980324056199859132544531 degree 64
10 32 0.14695924301669103564198244582096 1 279x4 – 420x3 + 102x2 – 1
11 23 0.06620127253123946651867717228392 891x3 – 121x2 – 11x + 1 [8]
11 24 0.07729688069488853395571677916567 degree 54
11 25 0.08361875782883598879744264452381
11 26 0.09095710678149019861518700080992
11 27 0.09657976733774336498968777561990
11 28 0.10206207261596575409155350311275 4 96x2 – 1
11 29 0.10857194419424991594861451754294 3
11 30 0.11111111111111111111111111111112 9x – 1
11 31 0.11688271662887820937338660035349 1 degree 4
11 32 0.12712017879076092950462557449550
12 25 0.06128191457415120417840411389235 1200x3 – 144x2 – 12x + 1 [8]
12 26 0.07142857142857142857142857142858 14x – 1 [8]
12 27 0.07476492618150171900046741798712
12 28 0.08259220626797780620169942560462
12 29 0.08434800930948543817609523562697 1 496x4 + 92x3 – 57x2 – 8x + 1
12 30 0.08914660143341378396187447349699 2 3856x4 + 1456x3 + 100x2 – 12x – 1
12 31 0.09870409124008622425851938202713 1 863x2 + 6x – 9
12 32 0.09870409124008622425851938202713 2 863x2 + 6x – 9
13 27 0.05707252378225744920008530386950 1573x3 – 169x2 – 13x + 1 [8]
13 28 0.06666666666666666666666666666667 15x – 1 [8]
13 29 0.06809860947842661920489059247239 2576x4 – 640x3 – 184x2 + 1
13 30 0.07171403472725743246612172136228 55x2 + 10x – 1
13 31 0.07171403472725743246612172136228 55x2 + 10x – 1
13 32 0.07171403472725743246612172136228 55x2 + 10x – 1 [26]
14 29 0.05342698734150995608695304765938 2016x3 – 196x2 – 14x + 1 [8]
14 30 0.06250000000000000000000000000000 16x – 1
14 31 0.06758677109098814555067432244432
14 32 0.07143090184092342423297534986236
15 31 0.05023712037295819182915747580845 2535x3 – 225x2 – 15x + 1 [8]
15 32 0.05890080848833338725817955269602

Aside from four cases (12 points in 3 dimensions, 20 or 24 points in 4 dimensions, and 32 points in 6 dimensions), the spherical codes listed above never have antipodal symmetry. The case of N pairs of antipodal points is equivalent to maximizing the minimal angle between N lines through the origin, i.e., optimizing a real projective code. For comparison, here is a table of records for N ≤ 16, which omits the trivial case of up to n orthogonal lines in n dimensions and has a plot available:

Table of real projective codes (N lines in n dimensions)

n N Cosine of minimal angle Rattlers Minimal polynomial References
∗ 3 4 0.33333333333333333333333333333334 3x – 1 [10, 33]
∗ 3 5 0.44721359549995793928183473374626 5x2 – 1 [10]
∗ 3 6 0.44721359549995793928183473374626 5x2 – 1 [10, 33]
∗ 3 7 0.57735026918962576450914878050196 3x2 – 1 [10]
∗ 3 8 0.64758897873417862734156471888316 degree 9 [6, 19]
3 9 0.66936231928109901015298754554435 13x3 + x2 – 5x – 1 [6]
3 10 0.68614066163450716496265286705474 1 2x2 + 3x – 3 [6]
3 11 0.71443449739467850485464566840562 degree 6 [6]
3 12 0.74452083820543412912980911157876 17x2 – 14x + 1 [6]
3 13 0.76813737631458482294245319830313 x5 + 13x4 + 42x3 + 42x2 – 75x + 9 [6]
3 14 0.78062219278523604521158858789220 degree 111 [6]
3 15 0.78655857113558130772653953634602 degree 12 [6]
3 16 0.79465447229176612295553092832760 45x4 – 30x2 + 1 [6]
∗ 4 5 0.25000000000000000000000000000000 4x – 1 [33]
∗ 4 6 0.33333333333333333333333333333334 3x – 1 [6, 11, 4]
4 7 0.39038820320220756872767623199676 4x2 + x – 1 [6]
4 8 0.41421356237309504880168872420970 x2 + 2x – 1 [6]
4 9 0.43425854591066488218653687791175 3x2 + x – 1 [6]
4 10 0.43425854591066488218653687791175 3x2 + x – 1 [6]
∗ 4 11 0.50000000000000000000000000000000 2x – 1 [6]
∗ 4 12 0.50000000000000000000000000000000 2x – 1 [24, 6]
4 13 0.56691527068179906330992487897558 1 11x2 – 8x + 1 [6]
4 14 0.59007651527101322282935291615497 degree 14 [6]
4 15 0.60873941168052296928961201448826 degree 10 [6]
4 16 0.61803398874989484820458683436564 1 x2 + x – 1 [6]
∗ 5 6 0.20000000000000000000000000000000 5x – 1 [33]
∗ 5 7 0.28620826421558111221120097995740 x3 – 9x2 – x + 1 [6, 18]
5 8 0.32880251120576348843092644566680 degree 7 [6]
5 9 0.33333333333333333333333333333334 3x – 1 [6]
∗ 5 10 0.33333333333333333333333333333334 3x – 1 [16, 33]
5 11 0.38664119885581345037571121632975 1 29x5 + 43x4 – 38x3 – 10x2 + 9x – 1 [6]
5 12 0.39038820320220756872767623199676 4x2 + x – 1 [6]
5 13 0.41100667568798964514107873257679 degree 7 [6]
5 14 0.41113055081625813204082106348155 49x4 – 26x3 – 16x2 + 10x – 1 [6]
5 15 0.41421356237309504880168872420970 x2 + 2x – 1 [6]
∗ 5 16 0.44721359549995793928183473374626 5x2 – 1 [6]
∗ 6 7 0.16666666666666666666666666666667 6x – 1 [33]
∗ 6 8 0.24094310926034164895054811006593 degree 6 [6, 18]
6 9 0.27825804947907715858960015942485 degree 7 [6]
6 10 0.28077640640441513745535246399352 2x2 + 3x – 1 [6]
6 11 0.31622776601683793319988935444328 10x2 – 1 [6]
6 12 0.31622776601683793319988935444328 10x2 – 1 [6]
6 13 0.33333333333333333333333333333334 1 3x – 1 [6]
6 14 0.33333333333333333333333333333334 3x – 1 [6]
6 15 0.33333333333333333333333333333334 3x – 1 [6]
∗ 6 16 0.33333333333333333333333333333334 3x – 1 [16, 33]
∗ 7 8 0.14285714285714285714285714285715 7x – 1 [33]
∗ 7 9 0.20000000000000000000000000000000 5x – 1 [6, 4]
7 10 0.23606797749978969640917366873128 x2 + 4x – 1 [6]
7 11 0.25851694486106515028856211348079 76x5 – 44x4 – 31x3 + 13x2 + 3x – 1 [6]
7 12 0.27272727272727272727272727272728 11x – 3 [6]
7 13 0.27735009811261456100917086672850 13x2 – 1 [6]
∗ 7 14 0.27735009811261456100917086672850 13x2 – 1 [16, 33]
7 15 0.31618306625810221110773896871360 degree 8 [6]
7 16 0.32579741728791719861570522781737 [6]
∗ 8 9 0.12500000000000000000000000000000 8x – 1 [33]
8 10 0.18274399763155681014833407039277 19x2 + 2x – 1 [6]
8 11 0.21013725661767148165690452663564 degree 19
8 12 0.23166247903553998491149327366707 10x2 + 2x – 1 [6]
8 13 0.23906104311673605013508516180858
8 14 0.25822993164268312301618267398557 degree 11 [6]
8 15 0.26996519072292358265868340141656
8 16 0.27395147170889821586916164817386 9x4 – 14x2 + 1 [6]
∗ 9 10 0.11111111111111111111111111111112 9x – 1 [33]
9 11 0.16297443277324822659060918082968 degree 6 [6]
∗ 9 12 0.18274399763155681014833407039277 19x2 + 2x – 1 [6, 4]
9 13 0.20887624097518068198468149347108
9 14 0.22089903147727419339287508022539 degree 8 [6]
9 15 0.23330570880288229977333738574545
9 16 0.24253562503633297351890646211613 17x2 – 1 [6]
∗ 10 11 0.10000000000000000000000000000000 10x – 1 [33]
∗ 10 12 0.14285714285714285714285714285715 7x – 1 [6, 4]
10 13 0.17273960369212825269887122191439 158x5 + 349x4 + 28x3 – 38x2 – 2x + 1 [6]
10 14 0.19054473957762242321193235020642 degree 84
10 15 0.19472111361678885284305008159890 6x3 – 7x2 – 4x + 1 [6]
∗ 10 16 0.20000000000000000000000000000000 5x – 1 [16, 33]
∗ 11 12 0.09090909090909090909090909090910 11x – 1 [33]
11 13 0.13498920127459953264187217399185 89x5 – 103x4 – 138x3 – 42x2 + x + 1 [6]
11 14 0.15777454145867061988851120512478 x3 + 21x2 + 3x – 1 [6]
11 15 0.17427391850565105193695154876787 degree 102
11 16 0.17839458616266547701324680749673 9x2 + 4x – 1 [6]
∗ 12 13 0.08333333333333333333333333333334 12x – 1 [33]
12 14 0.12295337815830641009836335476174 degree 6
12 15 0.14620609896519678926426427287249 degree 6
12 16 0.16133398878131368888992070727444 degree 8 [6]
∗ 13 14 0.07692307692307692307692307692308 13x – 1 [33]
∗ 13 15 0.11111111111111111111111111111112 9x – 1 [6, 4]
13 16 0.13499680245071983220881421251035 degree 8 [6]
∗ 14 15 0.07142857142857142857142857142858 14x – 1 [33]
14 16 0.10643341651544703618977147422985 degree 5 [6]
∗ 15 16 0.06666666666666666666666666666667 15x – 1 [33]

References

  1. P. G. Adams, A numerical approach to Tamme’s problem in Euclidean n-space, Master of Science thesis, Oregon State University, 1997, https://hdl.handle.net/1957/33911.
  2. C. Bachoc and F. Vallentin, Optimality and uniqueness of the (4,10,1/6) spherical code, J. Combin. Theory Ser. A 116 (2009), 195–204, doi:10.1016/j.jcta.2008.05.001.
  3. J. Buddenhagen and D. A. Kottwitz, Multiplicity and symmetry breaking in (conjectured) densest packings of congruent circles on a circle, undated preprint. [I obtained a copy in 2004 and have seen it cited with a date of 2001.]
  4. B. Bukh and C. Cox, Nearly orthogonal vectors and small antipodal spherical codes, Israel J. Math. 238 (2020), 359–388, doi:10.1007/s11856-020-2027-7.
  5. B. W. Clare and D. L. Kepert, The closest packing of equal circles on a sphere, Proc. Roy. Soc. London Ser. A 405 (1986), 329–344, doi:10.1098/rspa.1986.0056.
  6. J. H. Conway, R. H. Hardin, and N. J. A. Sloane, Packing lines, planes, etc.: packings in Grassmannian spaces, Experiment. Math. 5 (1996), 139–159, https://projecteuclid.org/euclid.em/1047565645.
  7. L. Danzer, Finite point-sets on S2 with minimum distance as large as possible, Discrete Math. 60 (1986), 3–66, doi:10.1016/0012-365X(86)90002-6.
  8. T. Ericson and V. Zinoviev, Codes on Euclidean spheres, North-Holland Mathematical Library 63, North-Holland Publishing Co., Amsterdam, 2001.
  9. L. Fejes [Tóth], Über eine Abschätzung des kürzesten Abstandes zweier Punkte eines auf einer Kugelfläche liegenden Punktsystems, Jber. Deutsch. Math.-Verein. 53 (1943), 66–68.
  10. L. Fejes Tóth, Distribution of points in the elliptic plane, Acta Math. Acad. Sci. Hungar. 16 (1965), 437–440, doi:10.1007/BF01904849.
  11. M. Fickus, J. Jasper, and D. G. Mixon, Packings in real projective spaces, SIAM J. Appl. Algebra Geom. 2 (2018), 377–409, doi:10.1137/17M1137528.
  12. T. Gosset, On the regular and semi-regular figures in space of n dimensions, Messenger Math. 29 (1900), 43–48.
  13. D. A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370.
  14. D. E. Lazić, V. Šenk, and I. Šeškar, Arrangements of points on a sphere which maximize the least distance, Bull. Appl. Math. (Technical University Budapest) 47 (1987), 7–21.
  15. V. I. Levenšteĭn, On bounds for packings in n-dimensional Euclidean space (Russian), Dokl. Akad. Nauk SSSR 245 (1979), 1299–1303; English translation in Soviet Math. Dokl. 20 (1979), 417–421.
  16. J. H. van Lint and J. J. Seidel, Equilateral point sets in elliptic geometry, Indag. Math. 28 (1966), 335–348, https://research.tue.nl/en/publications/equilateral-point-sets-in-elliptic-geometry.
  17. A. L. Mackay, The packing of three-dimensional spheres on the surface of a four-dimensional hypersphere, J. Phys. A 13 (1980), 3373–3379, doi:10.1088/0305-4470/13/11/011.
  18. D. G. Mixon and H. Parshall, Globally optimizing small codes in real projective spaces, SIAM J. Discrete Math. 35 (2021), 234–249, doi:10.1137/19M1307895.
  19. D. G. Mixon and H. Parshall, The optimal packing of eight points in the real projective plane, Experiment. Math., to appear, doi:10.1080/10586458.2019.1641767, arXiv:1902.10177.
  20. O. R. Musin and A. S. Tarasov, The strong thirteen spheres problem, Discrete Comput. Geom. 48 (2012), 128–141, doi:10.1007/s00454-011-9392-2.
  21. O. R. Musin and A. S. Tarasov, The Tammes problem for N=14, Exp. Math. 24 (2015), 460–468, doi:10.1080/10586458.2015.1022842.
  22. R. A. Rankin, The closest packing of spherical caps in n dimensions, Proc. Glasgow Math. Assoc. 2 (1955), 139–144, doi:10.1017/S2040618500033219.
  23. R. M. Robinson, Arrangements of 24 points on a sphere, Math. Ann. 144 (1961), 17–48, doi:10.1007/BF01396539.
  24. L. Schläfli, Theorie der vielfachen Kontinuität [1852], Denkschriften der Schweizerischen naturforschenden Gesellschaft, vol. 38, J. H. Graf, Bern, 1901.
  25. K. Schütte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96–124, doi:10.1007/BF02054944.
  26. N. J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994, http://neilsloane.com/packings.
  27. J. Strohmajer, Über die Verteilung von Punkten auf der Kugel, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 6 (1963), 49–53.
  28. P. M. L. Tammes, On the origin of number and arrangement of the places of exit on the surface of pollen-grains, Recueil des travaux botaniques néerlandais 27 (1930), no. 1, 1–84, https://natuurtijdschriften.nl/pub/552640.
  29. T. Tarnai and Zs. Gáspár, Improved packing of equal circles on a sphere and rigidity of its graph, Math. Proc. Cambridge Philos. Soc. 93 (1983), 191–218, doi:10.1017/S0305004100060485.
  30. T. Tarnai and Zs. Gáspár, Arrangement of 23 points on a sphere (on a conjecture of R. M. Robinson), Proc. Roy. Soc. London Ser. A 433 (1991), 257–267, doi:10.1098/rspa.1991.0046.
  31. B. L. van der Waerden, Punkte auf der Kugel. Drei Zusätze, Math. Ann. 125 (1952), 213–222, doi:10.1007/BF01343118.
  32. J. Wang, Finding and investigating exact spherical codes, Experiment. Math. 18 (2009), 249–256, https://projecteuclid.org/euclid.em/1259158434.
  33. L. R. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inform. Theory 20 (1974), 397–399, doi:10.1109/TIT.1974.1055219.