Credit: Thanks to Edwin van Dam, Akihiro Munemasa, Misha Muzychuk and Jason Williford for their help with this data.
UPDATE: This data is outdated. Please refer instead to Williford's tables, which are up to date for d=3 (primitive) and d=4,5 (Q-bipartite).
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NOTE: In this table, we have not yet entered all schemes which are metric or duals of metric schemes. There is also an infinite family of Q-bipartite doubles of a certain family of Hermitian forms dual polar spaces. These have unbounded class number d and are described on p315 of the book by Bannai and Ito. So far, only one from this family appears in our table.
| d | |X| | Krein array | m1 | Cosines | Prop. | Description/Reference |
| 3 classes | ||||||
| 3 | wv | {v-1,(w-1)(s(v-2)+2k-v)/ws, 1; 1, (s(v-2)+2k-v)/ws, v-1} s2 = k - λ |
v-1 | (1, (v-k)/s(v-1), -1/(v-1), -k/s(v-1) ) | Q-antip. | linked systems of symmetric designs |
| 3 | 66 | { 10, 242/27, 11/5; 1, 55/27, 44/5 } | 10 | ( 1, 4/15, -1/10, -7/15 ) | prim. | block scheme of 4-(11,5,1) Witt design |
| 3 | 91 | { 12, 338/35, 39/25; 1, 312/175, 39/5 } | 12 | (1, 7/20, 2/15, -3/10) | prim. | (?) van Dam open case |
| 3 | 99 | { 14, 108/11, 15/4; 1, 24/11, 45/4 } | 14 | (1, 5/14, -1/14, -2/7 ) | prim. | (?) van Dam open case |
| 3 | 759 | { 23, 945/44, 1587/80; 1, 345/176, 207/20 } | 23 | (1, 1/4, -1/8, -1/2) | prim. | block scheme of 5-(24,8,1) Witt design |
| 3 | 2025 | {22,21,625/33; 1, 11/6, 30/11} | 22 | (1, 7/22, -1/44, -4/11) | prim. | Derived design from Leech lattice |
| 3 | 4324 | { 46, 77315/1782 , 24863/847; 1, 37835/19602 , 2162/231 } | 46 | (1, 19/66, 5/99, -37/198 ) | prim. | block scheme of a 4-(47,11,48) design (QR code) |
| 3 | 98280 | {299, 1800/7, 4563/20; 1, 156/35, 195/4} | 299 | (1, 5/23, 1/46, -1/23) | prim. | antipodal quotient of Leech lattice configuration |
| 4 classes | ||||||
| 4 | 2wd | { d, d-1, d(w-1)/w, 1; 1, d/w, d-1, d } | d | (1, d-1/2, 0, -d-1/2, -1) | Q-bip. Q-antip. |
mutually unbiased bases (MUBs) in Rd |
| 4 | 24 | {4,3,8/3,1; 1, 4/3, 3, 4} | 4 | (1, 1/2, 0, -1/2, -1) | Q-bip. Q-antip. |
24-cell , sample set of MUBs |
| 4 | 72 | {6, 5, 9/2, 3; 1, 3/2, 3, 6} | 6 | (1, 1/2, 0, -1/2, -1) | Q-bip. | shortest vectors of E6 lattice |
| 4 | 126 | {7, 6, 49/9, 35/8; 1, 14/9, 21/8, 7} | 7 | (1, 1/2, 0, -1/2, -1) | Q-bip. | shortest vectors of E7 lattice |
| 4 | 132 | { 11, 10, 242/27, 11/5; 1, 55/27, 44/5, 11 } | 11 | ( 1, 1/3, 0, -1/3, -1) | Q-bip. |
block scheme
of 5-(12,6,1) Witt design extended Q-bipartite double of Ex. d3v66m10 |
| 4 | 150 | { 21, 16, 8, 1; 1, 4, 16, 21 } | 21 | ( 1, 1/3, 1/21, -1/7, -3/7) | Q-antip. |
Higman's ULS-SRD
from U3(5) (3 linked copies of the Hoffman-Singleton graph) |
| 4 | 162 | { 20, 18, 3, 1; 1, 3, 18, 20 } | 20 | (1, 1/4, 1/10, -1/5, -7/20) | Q-antip. | dismantle dual scheme of coset graph of shortened ternary Golay code |
| 4 | 168 | { 20, 49/3, 56/9, 1; 1, 28/9, 49/3, 20 } | 20 | ( 1, 3/10, 1/15, -1/6, -2/5) | Q-antip. |
Higman's ULS-SRD
from L3(4) (3 linked copies of the Gewirtz graph) |
| 4 | 240 | {8,7,32/5,6; 1,8/5,2,8} | 8 | (1, 1/2, 0, -1/2, -1) | Q-bip. | shortest vectors of E8 lattice |
| 4 | 512 | { 16, 15, 128/9, 8; 1, 16/9, 8, 16 } | 16 | (1, 1/3, 0, -1/3, -1) | Q-bip. | shortest vectors of lattice OBW16 (Martinet) |
| 4 | 2576 | { 23, 22, 2645/126, 207/11; 1, 253/126, 46/11, 23 } | 23 | (1, 1/3, 0, -1/3, -1) | Q-bip. | block scheme of 5-(24,12,48) design (Golay code) |
| 4 | 2816 | {22, 21, 121/6, 55/3; 1, 11/6, 11/3, 22} | 22 | (1, 1/3, 0, -1/3, -1) | Q-bip. | 2nd derived design of Leech lattice |
| 4 | 4224 | {252, 605/3, 88/3, 1; 1, 44/3, 605/3, 252} | 252 | ( 1, 3/14, 13/189, -1/54, -1/21) | Q-antip. |
Higman's ULS-SRD
from U6(2) (3 orbits of points of type 222-2233 in Leech lattice) |
| 4 | 4600 | {23, 22, 529/25, 184/9; 1, 46/25, 23/9, 23 } | 23 | (1, 1/3, 0, -1/3, -1) | Q-bip. | Derived design of Leech lattice |
| 4 | 7128 | { 22, 21, 121/6, 2187/125; 1, 11/6, 363/125, 6} | 22 | ( 1, 2/5, 1/10, -1/5, -1/2) | prim. | 2nd derived design of Leech lattice example |
| 5 classes | ||||||
| 5 | 32 | { 5, 4, 3, 2, 1; 1, 2, 3, 4, 5 } | 5 | (1, 3/5, 1/5, -1/5, -3/5, -1) | Q-bip. Q-antip. |
Q-bip. double of Clebsch (d.r.g.) |
| 5 | 54 | { 6, 5, 9/2, 3/2, 1; 1, 3/2, 9/2, 5, 6 } | 6 | (1, 1/2, 1/4, -1/4, -1/2, -1) | Q-bip. Q-antip. |
Q-bip. double of Schläfli |
| 5 | 200 | { 22, 21, 16, 6, 1; 1, 6, 16, 21, 22 } | 22 | (1, 4/11, 1/11, -1/11, -4/11, -1) | Q-bip. Q-antip. |
Q-bip. double of Higman-Sims (d.r.g.) |
| 5 | 224 | { 21, 20, 49/3, 14/3, 1; 1, 14/3, 49/3, 20, 21 } | 21 | (1, 1/3, 1/9, -1/9, -1/3, -1) | Q-bip. Q-antip. |
Q-bip. double of McL1(x) (first subconstituent) |
| 5 | 324 | { 21, 20, 18, 3, 1; 1, 3, 18, 20, 21 } | 21 | (1, 2/7, 1/7, -1/7, -2/7, -1) | Q-bip. Q-antip. |
Q-bip. double of McL2(x) (second subconstituent) |
| 5 | 486 | { 22, 20, 27/2, 2, 1; 1, 2, 27/2, 20, 22 } | 22 | ( 1, 7/22, 2/11, -1/11, -5/22, -1/2 ) | Q-antip. | dismantle dual scheme of coset graph of shortened extended ternary Golay code |
| 5 | 550 | {22, 21, 121/6, 11/6, 1; 1, 11/6, 121/6, 21, 22} | 22 | (1, 1/4, 1/6, -1/6, -1/4, -1) | Q-bip. Q-antip. |
Q-bip. double of McL |
| 5 | 47104 | { 23, 22, 529/25, 184/9, 483/25; 1, 46/25, 23/9, 92/25, 23/3} | 23 | (1, 7/15, 1/5, -1/15, -1/3, -3/5) | prim. | Derived design of Leech lattice example |
| 6 classes | ||||||
| 6 | 1536 | { 21, 20, 16, 8, 2, 1; 1, 2, 4, 16, 20, 21 } | 21 | ( 1, 3/7, 5/21, 1/21, -1/7, -1/3, -11/21) | Q-antip. | dismantle dual code of triply truncated subcode of extended binary Golay code (see p365 in [BCN]) |
| 6 | 196560 | { 24, 23, 288/13, 150/7, 104/5, 81/4; 1, 24/13, 18/7, 16/5, 15/4, 24} | 24 | ( 1, 1/2, 1/4, 0, -1/4, -1/2, -1) | Q-bip. | shortest vectors in the Leech lattice |
| 7 classes | ||||||
| 7 | 1782 | {22,21,121/6,55/4,33/4,11/6,1; 1, 11/6, 33/4, 55/4, 121/6, 21, 22} | 22 | (1, 1/2, 1/4, 1/8, -1/8, -1/4, -1/2, -1) | Q-bip. Q-antip. |
Q-bipartite double of dual polar graph on 891 vertices |
| 8 classes | ||||||
| 9 classes | ||||||
| 10 classes | ||||||
| 11 classes | ||||||
| 11 | 94208 | { 23, 22, 529/25, 184/9, 483/25, 46/3, 23/3, 92/25, 23/9, 46/25, 1; 1, 46/25, 23/9, 92/25, 23/3, 46/3, 483/25, 184/9, 529/25, 22, 23 } | 23 | (1, 3/5, 7/15, 1/3, 1/5, 1/15, -1/15, -1/5, -1/3, -7/15, -3/5, -1) | Q-bip. Q-antip. |
Q-bipartite double of scheme on 47,104 above |