The circuit
The 93\times256 matrix. Rows are wires and columns are rotations. A 1 means the wire takes part in the rotation, and the number beside each wire is its row weight. Scroll sideways to see every column.
| wire | 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 |
|---|---|
| out 0 (64) | 1111111111111111111111111111111111111111111111111111111111111111································································································································································································ |
| out 1 (64) | 11111111111111111111111111111111································11111111111111111111111111111111································································································································································ |
| out 2 (64) | 1111111111111111················1111111111111111················1111111111111111················1111111111111111················································································································································ |
| out 3 (64) | 11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········································································································································ |
| out 4 (64) | 1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111···································································································································· |
| out 5 (64) | 11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11·································································································································· |
| out 6 (64) | 1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1································································································································· |
| out 7 (64) | 11111111111111111111111111111111·······························································································11111111111111111111111111111111································································································· |
| out 8 (64) | 1111111111111111················1111111111111111···············································································1111111111111111················1111111111111111················································································· |
| out 9 (64) | 11111111········11111111········11111111········11111111·······································································11111111········11111111········11111111········11111111········································································· |
| out 10 (64) | 1111····1111····1111····1111····1111····1111····1111····1111···································································1111····1111····1111····1111····1111····1111····1111····1111····································································· |
| out 11 (64) | 11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11·································································11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··································································· |
| out 12 (64) | 1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1································································1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·································································· |
| out 13 (64) | 1111111111111111················································1111111111111111···············································1111111111111111···············································1111111111111111·················································· |
| out 14 (64) | 11111111········11111111········································11111111········11111111·······································11111111········11111111·······································11111111········11111111·········································· |
| out 15 (64) | 1111····1111····1111····1111····································1111····1111····1111····1111···································1111····1111····1111····1111···································1111····1111····1111····1111······································ |
| out 16 (64) | 11··11··11··11··11··11··11··11··································11··11··11··11··11··11··11··11·································11··11··11··11··11··11··11··11·································11··11··11··11··11··11··11··11···································· |
| out 17 (64) | 1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·································1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1································1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1································1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1··································· |
| out 18 (64) | 11111111························11111111························11111111························11111111·······················11111111························11111111·······················11111111·······················11111111··························· |
| out 19 (64) | 1111····1111····················1111····1111····················1111····1111····················1111····1111···················1111····1111····················1111····1111···················1111····1111···················1111····1111······················· |
| out 20 (64) | 11··11··11··11··················11··11··11··11··················11··11··11··11··················11··11··11··11·················11··11··11··11··················11··11··11··11·················11··11··11··11·················11··11··11··11····················· |
| out 21 (64) | 1·1·1·1·1·1·1·1·················1·1·1·1·1·1·1·1·················1·1·1·1·1·1·1·1·················1·1·1·1·1·1·1·1················1·1·1·1·1·1·1·1·················1·1·1·1·1·1·1·1················1·1·1·1·1·1·1·1················1·1·1·1·1·1·1·1···················· |
| out 22 (64) | 1111············1111············1111············1111············1111············1111············1111············1111···········1111············1111············1111············1111···········1111············1111···········1111···········1111················ |
| out 23 (64) | 11··11··········11··11··········11··11··········11··11··········11··11··········11··11··········11··11··········11··11·········11··11··········11··11··········11··11··········11··11·········11··11··········11··11·········11··11·········11··11·············· |
| out 24 (64) | 1·1·1·1·········1·1·1·1·········1·1·1·1·········1·1·1·1·········1·1·1·1·········1·1·1·1·········1·1·1·1·········1·1·1·1········1·1·1·1·········1·1·1·1·········1·1·1·1·········1·1·1·1········1·1·1·1·········1·1·1·1········1·1·1·1········1·1·1·1············· |
| out 25 (64) | 11······11······11······11······11······11······11······11······11······11······11······11······11······11······11······11·····11······11······11······11······11······11······11······11·····11······11······11······11·····11······11·····11·····11··········· |
| out 26 (64) | 1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1····1·1·····1·1·····1·1·····1·1····1·1·····1·1····1·1····1·1·········· |
| out 27 (64) | 1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1··1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1··1···1···1···1···1···1···1···1··1···1···1···1··1···1··1··1········· |
| out 28 (32) | 11111111111111111111111111111111································································································································································································································ |
| out 29 (32) | 1111111111111111················1111111111111111················································································································································································································ |
| out 30 (32) | 11111111········11111111········11111111········11111111········································································································································································································ |
| out 31 (32) | 1111····1111····1111····1111····1111····1111····1111····1111···································································································································································································· |
| out 32 (32) | 11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11·································································································································································································· |
| out 33 (32) | 1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1································································································································································································· |
| out 34 (32) | 1111111111111111················································1111111111111111················································································································································································ |
| out 35 (32) | 11111111········11111111········································11111111········11111111········································································································································································ |
| out 36 (32) | 1111····1111····1111····1111····································1111····1111····1111····1111···································································································································································· |
| out 37 (32) | 11··11··11··11··11··11··11··11··································11··11··11··11··11··11··11··11·································································································································································· |
| out 38 (32) | 1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·································1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1································································································································································· |
| out 39 (32) | 11111111························11111111························11111111························11111111························································································································································ |
| out 40 (32) | 1111····1111····················1111····1111····················1111····1111····················1111····1111···················································································································································· |
| out 41 (32) | 11··11··11··11··················11··11··11··11··················11··11··11··11··················11··11··11··11·················································································································································· |
| out 42 (32) | 1·1·1·1·1·1·1·1·················1·1·1·1·1·1·1·1·················1·1·1·1·1·1·1·1·················1·1·1·1·1·1·1·1················································································································································· |
| out 43 (32) | 1111············1111············1111············1111············1111············1111············1111············1111············································································································································ |
| out 44 (32) | 11··11··········11··11··········11··11··········11··11··········11··11··········11··11··········11··11··········11··11·········································································································································· |
| out 45 (32) | 1·1·1·1·········1·1·1·1·········1·1·1·1·········1·1·1·1·········1·1·1·1·········1·1·1·1·········1·1·1·1·········1·1·1·1········································································································································· |
| out 46 (32) | 11······11······11······11······11······11······11······11······11······11······11······11······11······11······11······11······································································································································ |
| out 47 (32) | 1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1····································································································································· |
| out 48 (32) | 1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1··································································································································· |
| out 49 (32) | 1111111111111111···············································································································1111111111111111················································································································· |
| out 50 (32) | 11111111········11111111·······································································································11111111········11111111········································································································· |
| out 51 (32) | 1111····1111····1111····1111···································································································1111····1111····1111····1111····································································································· |
| out 52 (32) | 11··11··11··11··11··11··11··11·································································································11··11··11··11··11··11··11··11··································································································· |
| out 53 (32) | 1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1································································································1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·································································································· |
| out 54 (32) | 11111111························11111111·······················································································11111111························11111111························································································· |
| out 55 (32) | 1111····1111····················1111····1111···················································································1111····1111····················1111····1111····················································································· |
| out 56 (32) | 11··11··11··11··················11··11··11··11·················································································11··11··11··11··················11··11··11··11··················································································· |
| out 57 (32) | 1·1·1·1·1·1·1·1·················1·1·1·1·1·1·1·1················································································1·1·1·1·1·1·1·1·················1·1·1·1·1·1·1·1·················································································· |
| out 58 (32) | 1111············1111············1111············1111···········································································1111············1111············1111············1111············································································· |
| out 59 (32) | 11··11··········11··11··········11··11··········11··11·········································································11··11··········11··11··········11··11··········11··11··········································································· |
| out 60 (32) | 1·1·1·1·········1·1·1·1·········1·1·1·1·········1·1·1·1········································································1·1·1·1·········1·1·1·1·········1·1·1·1·········1·1·1·1·········································································· |
| out 61 (32) | 11······11······11······11······11······11······11······11·····································································11······11······11······11······11······11······11······11······································································· |
| out 62 (32) | 1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1····································································1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1······································································ |
| out 63 (32) | 1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1··································································1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···································································· |
| out 64 (32) | 11111111························································11111111·······················································11111111·······················································11111111·························································· |
| out 65 (32) | 1111····1111····················································1111····1111···················································1111····1111···················································1111····1111······················································ |
| out 66 (32) | 11··11··11··11··················································11··11··11··11·················································11··11··11··11·················································11··11··11··11···················································· |
| out 67 (32) | 1·1·1·1·1·1·1·1·················································1·1·1·1·1·1·1·1················································1·1·1·1·1·1·1·1················································1·1·1·1·1·1·1·1··················································· |
| out 68 (32) | 1111············1111············································1111············1111···········································1111············1111···········································1111············1111·············································· |
| out 69 (32) | 11··11··········11··11··········································11··11··········11··11·········································11··11··········11··11·········································11··11··········11··11············································ |
| out 70 (32) | 1·1·1·1·········1·1·1·1·········································1·1·1·1·········1·1·1·1········································1·1·1·1·········1·1·1·1········································1·1·1·1·········1·1·1·1··········································· |
| out 71 (32) | 11······11······11······11······································11······11······11······11·····································11······11······11······11·····································11······11······11······11········································ |
| out 72 (32) | 1·1·····1·1·····1·1·····1·1·····································1·1·····1·1·····1·1·····1·1····································1·1·····1·1·····1·1·····1·1····································1·1·····1·1·····1·1·····1·1······································· |
| out 73 (32) | 1···1···1···1···1···1···1···1···································1···1···1···1···1···1···1···1··································1···1···1···1···1···1···1···1··································1···1···1···1···1···1···1···1····································· |
| out 74 (32) | 1111····························1111····························1111····························1111···························1111····························1111···························1111···························1111······························· |
| out 75 (32) | 11··11··························11··11··························11··11··························11··11·························11··11··························11··11·························11··11·························11··11····························· |
| out 76 (32) | 1·1·1·1·························1·1·1·1·························1·1·1·1·························1·1·1·1························1·1·1·1·························1·1·1·1························1·1·1·1························1·1·1·1···························· |
| out 77 (32) | 11······11······················11······11······················11······11······················11······11·····················11······11······················11······11·····················11······11·····················11······11························· |
| out 78 (32) | 1·1·····1·1·····················1·1·····1·1·····················1·1·····1·1·····················1·1·····1·1····················1·1·····1·1·····················1·1·····1·1····················1·1·····1·1····················1·1·····1·1························ |
| out 79 (32) | 1···1···1···1···················1···1···1···1···················1···1···1···1···················1···1···1···1··················1···1···1···1···················1···1···1···1··················1···1···1···1··················1···1···1···1······················ |
| out 80 (32) | 11··············11··············11··············11··············11··············11··············11··············11·············11··············11··············11··············11·············11··············11·············11·············11·················· |
| out 81 (32) | 1·1·············1·1·············1·1·············1·1·············1·1·············1·1·············1·1·············1·1············1·1·············1·1·············1·1·············1·1············1·1·············1·1············1·1············1·1················· |
| out 82 (32) | 1···1···········1···1···········1···1···········1···1···········1···1···········1···1···········1···1···········1···1··········1···1···········1···1···········1···1···········1···1··········1···1···········1···1··········1···1··········1···1··············· |
| out 83 (32) | 1·······1·······1·······1·······1·······1·······1·······1·······1·······1·······1·······1·······1·······1·······1·······1······1·······1·······1·······1·······1·······1·······1·······1······1·······1·······1·······1······1·······1······1······1············ |
| check 84 (256) | 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 |
| check 85 (128) | 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·························································································································1······· |
| check 86 (128) | 1111111111111111111111111111111111111111111111111111111111111111·······························································111111111111111111111111111111111111111111111111111111111111111···························································1······ |
| check 87 (128) | 11111111111111111111111111111111································11111111111111111111111111111111·······························11111111111111111111111111111111·······························1111111111111111111111111111111·····························1····· |
| check 88 (128) | 1111111111111111················1111111111111111················1111111111111111················1111111111111111···············1111111111111111················1111111111111111···············1111111111111111···············111111111111111···············1···· |
| check 89 (128) | 11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111·······11111111········11111111········11111111········11111111·······11111111········11111111·······11111111·······1111111·········1··· |
| check 90 (128) | 1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111···1111····1111····1111····1111····1111····1111····1111····1111···1111····1111····1111····1111···1111····1111···1111···111·······1·· |
| check 91 (128) | 11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11·11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11·11··11··11··11··11··11··11··11·11··11··11··11·11··11·11·1·······1· |
| check 92 (128) | 1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·11·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·11·1·1·1·1·1·1·1·1·1·1·1·1·1·1·11·1·1·1·1·1·1·11·1·1·11·11········1 |
The gate is read off the output rows. An output row of odd weight gives a T, a pair of output rows with odd overlap gives a CS, and an odd triple overlap gives a CCZ. The circuit is a valid factory because every such parity that touches a check row is even.
Figures of merit
inputs n256Noisy T states per run.
outputs k84
distance d4Exact, proved and witnessed here.
error coefficient A_{4}690,880Faults of weight 4 in the circuit shown that pass every check and damage the output. The output error is A_{4}\,p^{4} to leading order.
wires N93
checks r9
V_{\mathrm{ex}}not definedUndefined, because the gate's terms overlap.
\gamma_\rho = \log(n/V_{\mathrm{ex}})/\log dnot definedOverhead per extractable T state. Lower is better.
\gamma = \log(n/k)/\log d0.8038Overhead per output qubit.
\gamma_T = \log(n/T)/\log dnot computedOverhead per unit of T-count.
T-countnot computedMinimal T-count of the output gate. Not computable at this width.
phase-polynomial degreenot computedMinimised over changes of output basis.
effective width84Rank of the output rows modulo the check rows.
Output gate
0 T0 CS1120 CCZon 84 output wires
As gates: CCZ0,64,83·CCZ0,65,82·CCZ0,66,81·CCZ0,67,80·CCZ0,68,79·CCZ0,69,78·CCZ0,70,77·CCZ0,71,76·CCZ0,72,75·CCZ0,73,74·CCZ1,54,83·CCZ1,55,82·CCZ1,56,81·CCZ1,57,80·CCZ1,58,79·CCZ1,59,78·CCZ1,60,77·CCZ1,61,76·CCZ1,62,75·CCZ1,63,74·CCZ2,50,83·CCZ2,51,82·CCZ2,52,81·CCZ2,53,80·CCZ2,58,73·CCZ2,59,72·CCZ2,60,71·CCZ2,61,70·CCZ2,62,69·CCZ2,63,68·CCZ3,49,83·CCZ3,51,79·CCZ3,52,78·CCZ3,53,77·CCZ3,55,73·CCZ3,56,72·CCZ3,57,71·CCZ3,61,67·CCZ3,62,66·CCZ3,63,65·CCZ4,49,82·CCZ4,50,79·CCZ4,52,76·CCZ4,53,75·CCZ4,54,73·CCZ4,56,70·CCZ4,57,69·CCZ4,59,67·CCZ4,60,66·CCZ4,63,64·CCZ5,49,81·CCZ5,50,78·CCZ5,51,76·CCZ5,53,74·CCZ5,54,72·CCZ5,55,70·CCZ5,57,68·CCZ5,58,67·CCZ5,60,65·CCZ5,62,64·CCZ6,49,80·CCZ6,50,77·CCZ6,51,75·CCZ6,52,74·CCZ6,54,71·CCZ6,55,69·CCZ6,56,68·CCZ6,58,66·CCZ6,59,65·CCZ6,61,64·CCZ7,39,83·CCZ7,40,82·CCZ7,41,81·CCZ7,42,80·CCZ7,43,79·CCZ7,44,78·CCZ7,45,77·CCZ7,46,76·CCZ7,47,75·CCZ7,48,74·CCZ8,35,83·CCZ8,36,82·CCZ8,37,81·CCZ8,38,80·CCZ8,43,73·CCZ8,44,72·CCZ8,45,71·CCZ8,46,70·CCZ8,47,69·CCZ8,48,68·CCZ9,34,83·CCZ9,36,79·CCZ9,37,78·CCZ9,38,77·CCZ9,40,73·CCZ9,41,72·CCZ9,42,71·CCZ9,46,67·CCZ9,47,66·CCZ9,48,65·CCZ10,34,82·CCZ10,35,79·CCZ10,37,76·CCZ10,38,75·CCZ10,39,73·CCZ10,41,70·CCZ10,42,69·CCZ10,44,67·CCZ10,45,66·CCZ10,48,64·CCZ11,34,81·CCZ11,35,78·CCZ11,36,76·CCZ11,38,74·CCZ11,39,72·CCZ11,40,70·CCZ11,42,68·CCZ11,43,67·CCZ11,45,65·CCZ11,47,64·CCZ12,34,80·CCZ12,35,77·CCZ12,36,75·CCZ12,37,74·CCZ12,39,71·CCZ12,40,69·CCZ12,41,68·CCZ12,43,66·CCZ12,44,65·CCZ12,46,64·CCZ13,30,83·CCZ13,31,82·CCZ13,32,81·CCZ13,33,80·CCZ13,43,63·CCZ13,44,62·CCZ13,45,61·CCZ13,46,60·CCZ13,47,59·CCZ13,48,58·CCZ14,29,83·CCZ14,31,79·CCZ14,32,78·CCZ14,33,77·CCZ14,40,63·CCZ14,41,62·CCZ14,42,61·CCZ14,46,57·CCZ14,47,56·CCZ14,48,55·CCZ15,29,82·CCZ15,30,79·CCZ15,32,76·CCZ15,33,75·CCZ15,39,63·CCZ15,41,60·CCZ15,42,59·CCZ15,44,57·CCZ15,45,56·CCZ15,48,54·CCZ16,29,81·CCZ16,30,78·CCZ16,31,76·CCZ16,33,74·CCZ16,39,62·CCZ16,40,60·CCZ16,42,58·CCZ16,43,57·CCZ16,45,55·CCZ16,47,54·CCZ17,29,80·CCZ17,30,77·CCZ17,31,75·CCZ17,32,74·CCZ17,39,61·CCZ17,40,59·CCZ17,41,58·CCZ17,43,56·CCZ17,44,55·CCZ17,46,54·CCZ18,28,83·CCZ18,31,73·CCZ18,32,72·CCZ18,33,71·CCZ18,36,63·CCZ18,37,62·CCZ18,38,61·CCZ18,46,53·CCZ18,47,52·CCZ18,48,51·CCZ19,28,82·CCZ19,30,73·CCZ19,32,70·CCZ19,33,69·CCZ19,35,63·CCZ19,37,60·CCZ19,38,59·CCZ19,44,53·CCZ19,45,52·CCZ19,48,50·CCZ20,28,81·CCZ20,30,72·CCZ20,31,70·CCZ20,33,68·CCZ20,35,62·CCZ20,36,60·CCZ20,38,58·CCZ20,43,53·CCZ20,45,51·CCZ20,47,50·CCZ21,28,80·CCZ21,30,71·CCZ21,31,69·CCZ21,32,68·CCZ21,35,61·CCZ21,36,59·CCZ21,37,58·CCZ21,43,52·CCZ21,44,51·CCZ21,46,50·CCZ22,28,79·CCZ22,29,73·CCZ22,32,67·CCZ22,33,66·CCZ22,34,63·CCZ22,37,57·CCZ22,38,56·CCZ22,41,53·CCZ22,42,52·CCZ22,48,49·CCZ23,28,78·CCZ23,29,72·CCZ23,31,67·CCZ23,33,65·CCZ23,34,62·CCZ23,36,57·CCZ23,38,55·CCZ23,40,53·CCZ23,42,51·CCZ23,47,49·CCZ24,28,77·CCZ24,29,71·CCZ24,31,66·CCZ24,32,65·CCZ24,34,61·CCZ24,36,56·CCZ24,37,55·CCZ24,40,52·CCZ24,41,51·CCZ24,46,49·CCZ25,28,76·CCZ25,29,70·CCZ25,30,67·CCZ25,33,64·CCZ25,34,60·CCZ25,35,57·CCZ25,38,54·CCZ25,39,53·CCZ25,42,50·CCZ25,45,49·CCZ26,28,75·CCZ26,29,69·CCZ26,30,66·CCZ26,32,64·CCZ26,34,59·CCZ26,35,56·CCZ26,37,54·CCZ26,39,52·CCZ26,41,50·CCZ26,44,49·CCZ27,28,74·CCZ27,29,68·CCZ27,30,65·CCZ27,31,64·CCZ27,34,58·CCZ27,35,55·CCZ27,36,54·CCZ27,39,51·CCZ27,40,50·CCZ27,43,49·CCZ28,39,83·CCZ28,40,82·CCZ28,41,81·CCZ28,42,80·CCZ28,43,79·CCZ28,44,78·CCZ28,45,77·CCZ28,46,76·CCZ28,47,75·CCZ28,48,74·CCZ28,54,83·CCZ28,55,82·CCZ28,56,81·CCZ28,57,80·CCZ28,58,79·CCZ28,59,78·CCZ28,60,77·CCZ28,61,76·CCZ28,62,75·CCZ28,63,74·CCZ28,64,83·CCZ28,65,82·CCZ28,66,81·CCZ28,67,80·CCZ28,68,79·CCZ28,69,78·CCZ28,70,77·CCZ28,71,76·CCZ28,72,75·CCZ28,73,74·CCZ28,74,79·CCZ28,74,82·CCZ28,74,83·CCZ28,75,78·CCZ28,75,81·CCZ28,75,83·CCZ28,76,77·CCZ28,76,80·CCZ28,76,83·CCZ28,77,81·CCZ28,77,82·CCZ28,78,80·CCZ28,78,82·CCZ28,79,80·CCZ28,79,81·CCZ29,35,83·CCZ29,36,82·CCZ29,37,81·CCZ29,38,80·CCZ29,43,73·CCZ29,44,72·CCZ29,45,71·CCZ29,46,70·CCZ29,47,69·CCZ29,48,68·CCZ29,50,83·CCZ29,51,82·CCZ29,52,81·CCZ29,53,80·CCZ29,58,73·CCZ29,59,72·CCZ29,60,71·CCZ29,61,70·CCZ29,62,69·CCZ29,63,68·CCZ29,64,83·CCZ29,65,82·CCZ29,66,81·CCZ29,67,80·CCZ29,68,73·CCZ29,68,79·CCZ29,68,82·CCZ29,68,83·CCZ29,69,72·CCZ29,69,78·CCZ29,69,81·CCZ29,69,83·CCZ29,70,71·CCZ29,70,77·CCZ29,70,80·CCZ29,70,83·CCZ29,71,76·CCZ29,71,81·CCZ29,71,82·CCZ29,72,75·CCZ29,72,80·CCZ29,72,82·CCZ29,73,74·CCZ29,73,80·CCZ29,73,81·CCZ30,34,83·CCZ30,36,79·CCZ30,37,78·CCZ30,38,77·CCZ30,40,73·CCZ30,41,72·CCZ30,42,71·CCZ30,46,67·CCZ30,47,66·CCZ30,48,65·CCZ30,49,83·CCZ30,51,79·CCZ30,52,78·CCZ30,53,77·CCZ30,55,73·CCZ30,56,72·CCZ30,57,71·CCZ30,61,67·CCZ30,62,66·CCZ30,63,65·CCZ30,64,83·CCZ30,65,73·CCZ30,65,79·CCZ30,65,82·CCZ30,65,83·CCZ30,66,72·CCZ30,66,78·CCZ30,66,81·CCZ30,66,83·CCZ30,67,71·CCZ30,67,77·CCZ30,67,80·CCZ30,67,83·CCZ30,68,79·CCZ30,69,78·CCZ30,70,77·CCZ30,71,76·CCZ30,71,78·CCZ30,71,79·CCZ30,72,75·CCZ30,72,77·CCZ30,72,79·CCZ30,73,74·CCZ30,73,77·CCZ30,73,78·CCZ31,34,82·CCZ31,35,79·CCZ31,37,76·CCZ31,38,75·CCZ31,39,73·CCZ31,41,70·CCZ31,42,69·CCZ31,44,67·CCZ31,45,66·CCZ31,48,64·CCZ31,49,82·CCZ31,50,79·CCZ31,52,76·CCZ31,53,75·CCZ31,54,73·CCZ31,56,70·CCZ31,57,69·CCZ31,59,67·CCZ31,60,66·CCZ31,63,64·CCZ31,64,73·CCZ31,64,79·CCZ31,64,82·CCZ31,64,83·CCZ31,65,82·CCZ31,66,70·CCZ31,66,76·CCZ31,66,81·CCZ31,66,82·CCZ31,67,69·CCZ31,67,75·CCZ31,67,80·CCZ31,67,82·CCZ31,68,79·CCZ31,69,76·CCZ31,69,78·CCZ31,69,79·CCZ31,70,75·CCZ31,70,77·CCZ31,70,79·CCZ31,71,76·CCZ31,72,75·CCZ31,73,74·CCZ31,73,75·CCZ31,73,76·CCZ32,34,81·CCZ32,35,78·CCZ32,36,76·CCZ32,38,74·CCZ32,39,72·CCZ32,40,70·CCZ32,42,68·CCZ32,43,67·CCZ32,45,65·CCZ32,47,64·CCZ32,49,81·CCZ32,50,78·CCZ32,51,76·CCZ32,53,74·CCZ32,54,72·CCZ32,55,70·CCZ32,57,68·CCZ32,58,67·CCZ32,60,65·CCZ32,62,64·CCZ32,64,72·CCZ32,64,78·CCZ32,64,81·CCZ32,64,83·CCZ32,65,70·CCZ32,65,76·CCZ32,65,81·CCZ32,65,82·CCZ32,66,81·CCZ32,67,68·CCZ32,67,74·CCZ32,67,80·CCZ32,67,81·CCZ32,68,76·CCZ32,68,78·CCZ32,68,79·CCZ32,69,78·CCZ32,70,74·CCZ32,70,77·CCZ32,70,78·CCZ32,71,76·CCZ32,72,74·CCZ32,72,75·CCZ32,72,76·CCZ32,73,74·CCZ33,34,80·CCZ33,35,77·CCZ33,36,75·CCZ33,37,74·CCZ33,39,71·CCZ33,40,69·CCZ33,41,68·CCZ33,43,66·CCZ33,44,65·CCZ33,46,64·CCZ33,49,80·CCZ33,50,77·CCZ33,51,75·CCZ33,52,74·CCZ33,54,71·CCZ33,55,69·CCZ33,56,68·CCZ33,58,66·CCZ33,59,65·CCZ33,61,64·CCZ33,64,71·CCZ33,64,77·CCZ33,64,80·CCZ33,64,83·CCZ33,65,69·CCZ33,65,75·CCZ33,65,80·CCZ33,65,82·CCZ33,66,68·CCZ33,66,74·CCZ33,66,80·CCZ33,66,81·CCZ33,67,80·CCZ33,68,75·CCZ33,68,77·CCZ33,68,79·CCZ33,69,74·CCZ33,69,77·CCZ33,69,78·CCZ33,70,77·CCZ33,71,74·CCZ33,71,75·CCZ33,71,76·CCZ33,72,75·CCZ33,73,74·CCZ34,43,63·CCZ34,44,62·CCZ34,45,61·CCZ34,46,60·CCZ34,47,59·CCZ34,48,58·CCZ34,50,83·CCZ34,51,82·CCZ34,52,81·CCZ34,53,80·CCZ34,54,83·CCZ34,55,82·CCZ34,56,81·CCZ34,57,80·CCZ34,58,63·CCZ34,58,73·CCZ34,58,79·CCZ34,58,82·CCZ34,58,83·CCZ34,59,62·CCZ34,59,72·CCZ34,59,78·CCZ34,59,81·CCZ34,59,83·CCZ34,60,61·CCZ34,60,71·CCZ34,60,77·CCZ34,60,80·CCZ34,60,83·CCZ34,61,70·CCZ34,61,76·CCZ34,61,81·CCZ34,61,82·CCZ34,62,69·CCZ34,62,75·CCZ34,62,80·CCZ34,62,82·CCZ34,63,68·CCZ34,63,74·CCZ34,63,80·CCZ34,63,81·CCZ35,40,63·CCZ35,41,62·CCZ35,42,61·CCZ35,46,57·CCZ35,47,56·CCZ35,48,55·CCZ35,49,83·CCZ35,51,79·CCZ35,52,78·CCZ35,53,77·CCZ35,54,83·CCZ35,55,63·CCZ35,55,73·CCZ35,55,79·CCZ35,55,82·CCZ35,55,83·CCZ35,56,62·CCZ35,56,72·CCZ35,56,78·CCZ35,56,81·CCZ35,56,83·CCZ35,57,61·CCZ35,57,71·CCZ35,57,77·CCZ35,57,80·CCZ35,57,83·CCZ35,58,79·CCZ35,59,78·CCZ35,60,77·CCZ35,61,67·CCZ35,61,76·CCZ35,61,78·CCZ35,61,79·CCZ35,62,66·CCZ35,62,75·CCZ35,62,77·CCZ35,62,79·CCZ35,63,65·CCZ35,63,74·CCZ35,63,77·CCZ35,63,78·CCZ36,39,63·CCZ36,41,60·CCZ36,42,59·CCZ36,44,57·CCZ36,45,56·CCZ36,48,54·CCZ36,49,82·CCZ36,50,79·CCZ36,52,76·CCZ36,53,75·CCZ36,54,63·CCZ36,54,73·CCZ36,54,79·CCZ36,54,82·CCZ36,54,83·CCZ36,55,82·CCZ36,56,60·CCZ36,56,70·CCZ36,56,76·CCZ36,56,81·CCZ36,56,82·CCZ36,57,59·CCZ36,57,69·CCZ36,57,75·CCZ36,57,80·CCZ36,57,82·CCZ36,58,79·CCZ36,59,67·CCZ36,59,76·CCZ36,59,78·CCZ36,59,79·CCZ36,60,66·CCZ36,60,75·CCZ36,60,77·CCZ36,60,79·CCZ36,61,76·CCZ36,62,75·CCZ36,63,64·CCZ36,63,74·CCZ36,63,75·CCZ36,63,76·CCZ37,39,62·CCZ37,40,60·CCZ37,42,58·CCZ37,43,57·CCZ37,45,55·CCZ37,47,54·CCZ37,49,81·CCZ37,50,78·CCZ37,51,76·CCZ37,53,74·CCZ37,54,62·CCZ37,54,72·CCZ37,54,78·CCZ37,54,81·CCZ37,54,83·CCZ37,55,60·CCZ37,55,70·CCZ37,55,76·CCZ37,55,81·CCZ37,55,82·CCZ37,56,81·CCZ37,57,58·CCZ37,57,68·CCZ37,57,74·CCZ37,57,80·CCZ37,57,81·CCZ37,58,67·CCZ37,58,76·CCZ37,58,78·CCZ37,58,79·CCZ37,59,78·CCZ37,60,65·CCZ37,60,74·CCZ37,60,77·CCZ37,60,78·CCZ37,61,76·CCZ37,62,64·CCZ37,62,74·CCZ37,62,75·CCZ37,62,76·CCZ37,63,74·CCZ38,39,61·CCZ38,40,59·CCZ38,41,58·CCZ38,43,56·CCZ38,44,55·CCZ38,46,54·CCZ38,49,80·CCZ38,50,77·CCZ38,51,75·CCZ38,52,74·CCZ38,54,61·CCZ38,54,71·CCZ38,54,77·CCZ38,54,80·CCZ38,54,83·CCZ38,55,59·CCZ38,55,69·CCZ38,55,75·CCZ38,55,80·CCZ38,55,82·CCZ38,56,58·CCZ38,56,68·CCZ38,56,74·CCZ38,56,80·CCZ38,56,81·CCZ38,57,80·CCZ38,58,66·CCZ38,58,75·CCZ38,58,77·CCZ38,58,79·CCZ38,59,65·CCZ38,59,74·CCZ38,59,77·CCZ38,59,78·CCZ38,60,77·CCZ38,61,64·CCZ38,61,74·CCZ38,61,75·CCZ38,61,76·CCZ38,62,75·CCZ38,63,74·CCZ39,46,53·CCZ39,47,52·CCZ39,48,51·CCZ39,49,83·CCZ39,50,83·CCZ39,51,63·CCZ39,51,73·CCZ39,51,79·CCZ39,51,82·CCZ39,51,83·CCZ39,52,62·CCZ39,52,72·CCZ39,52,78·CCZ39,52,81·CCZ39,52,83·CCZ39,53,61·CCZ39,53,71·CCZ39,53,77·CCZ39,53,80·CCZ39,53,83·CCZ39,55,73·CCZ39,56,72·CCZ39,57,71·CCZ39,58,73·CCZ39,59,72·CCZ39,60,71·CCZ39,61,67·CCZ39,61,70·CCZ39,61,72·CCZ39,61,73·CCZ39,62,66·CCZ39,62,69·CCZ39,62,71·CCZ39,62,73·CCZ39,63,65·CCZ39,63,68·CCZ39,63,71·CCZ39,63,72·CCZ40,44,53·CCZ40,45,52·CCZ40,48,50·CCZ40,49,82·CCZ40,50,63·CCZ40,50,73·CCZ40,50,79·CCZ40,50,82·CCZ40,50,83·CCZ40,51,82·CCZ40,52,60·CCZ40,52,70·CCZ40,52,76·CCZ40,52,81·CCZ40,52,82·CCZ40,53,59·CCZ40,53,69·CCZ40,53,75·CCZ40,53,80·CCZ40,53,82·CCZ40,54,73·CCZ40,56,70·CCZ40,57,69·CCZ40,58,73·CCZ40,59,67·CCZ40,59,70·CCZ40,59,72·CCZ40,59,73·CCZ40,60,66·CCZ40,60,69·CCZ40,60,71·CCZ40,60,73·CCZ40,61,70·CCZ40,62,69·CCZ40,63,64·CCZ40,63,68·CCZ40,63,69·CCZ40,63,70·CCZ41,43,53·CCZ41,45,51·CCZ41,47,50·CCZ41,49,81·CCZ41,50,62·CCZ41,50,72·CCZ41,50,78·CCZ41,50,81·CCZ41,50,83·CCZ41,51,60·CCZ41,51,70·CCZ41,51,76·CCZ41,51,81·CCZ41,51,82·CCZ41,52,81·CCZ41,53,58·CCZ41,53,68·CCZ41,53,74·CCZ41,53,80·CCZ41,53,81·CCZ41,54,72·CCZ41,55,70·CCZ41,57,68·CCZ41,58,67·CCZ41,58,70·CCZ41,58,72·CCZ41,58,73·CCZ41,59,72·CCZ41,60,65·CCZ41,60,68·CCZ41,60,71·CCZ41,60,72·CCZ41,61,70·CCZ41,62,64·CCZ41,62,68·CCZ41,62,69·CCZ41,62,70·CCZ41,63,68·CCZ42,43,52·CCZ42,44,51·CCZ42,46,50·CCZ42,49,80·CCZ42,50,61·CCZ42,50,71·CCZ42,50,77·CCZ42,50,80·CCZ42,50,83·CCZ42,51,59·CCZ42,51,69·CCZ42,51,75·CCZ42,51,80·CCZ42,51,82·CCZ42,52,58·CCZ42,52,68·CCZ42,52,74·CCZ42,52,80·CCZ42,52,81·CCZ42,53,80·CCZ42,54,71·CCZ42,55,69·CCZ42,56,68·CCZ42,58,66·CCZ42,58,69·CCZ42,58,71·CCZ42,58,73·CCZ42,59,65·CCZ42,59,68·CCZ42,59,71·CCZ42,59,72·CCZ42,60,71·CCZ42,61,64·CCZ42,61,68·CCZ42,61,69·CCZ42,61,70·CCZ42,62,69·CCZ42,63,68·CCZ43,48,49·CCZ43,49,63·CCZ43,49,73·CCZ43,49,79·CCZ43,49,82·CCZ43,49,83·CCZ43,50,79·CCZ43,51,79·CCZ43,52,57·CCZ43,52,67·CCZ43,52,76·CCZ43,52,78·CCZ43,52,79·CCZ43,53,56·CCZ43,53,66·CCZ43,53,75·CCZ43,53,77·CCZ43,53,79·CCZ43,54,73·CCZ43,55,73·CCZ43,56,67·CCZ43,56,70·CCZ43,56,72·CCZ43,56,73·CCZ43,57,66·CCZ43,57,69·CCZ43,57,71·CCZ43,57,73·CCZ43,59,67·CCZ43,60,66·CCZ43,61,67·CCZ43,62,66·CCZ43,63,64·CCZ43,63,65·CCZ43,63,66·CCZ43,63,67·CCZ44,47,49·CCZ44,49,62·CCZ44,49,72·CCZ44,49,78·CCZ44,49,81·CCZ44,49,83·CCZ44,50,78·CCZ44,51,57·CCZ44,51,67·CCZ44,51,76·CCZ44,51,78·CCZ44,51,79·CCZ44,52,78·CCZ44,53,55·CCZ44,53,65·CCZ44,53,74·CCZ44,53,77·CCZ44,53,78·CCZ44,54,72·CCZ44,55,67·CCZ44,55,70·CCZ44,55,72·CCZ44,55,73·CCZ44,56,72·CCZ44,57,65·CCZ44,57,68·CCZ44,57,71·CCZ44,57,72·CCZ44,58,67·CCZ44,60,65·CCZ44,61,67·CCZ44,62,64·CCZ44,62,65·CCZ44,62,66·CCZ44,62,67·CCZ44,63,65·CCZ45,46,49·CCZ45,49,61·CCZ45,49,71·CCZ45,49,77·CCZ45,49,80·CCZ45,49,83·CCZ45,50,77·CCZ45,51,56·CCZ45,51,66·CCZ45,51,75·CCZ45,51,77·CCZ45,51,79·CCZ45,52,55·CCZ45,52,65·CCZ45,52,74·CCZ45,52,77·CCZ45,52,78·CCZ45,53,77·CCZ45,54,71·CCZ45,55,66·CCZ45,55,69·CCZ45,55,71·CCZ45,55,73·CCZ45,56,65·CCZ45,56,68·CCZ45,56,71·CCZ45,56,72·CCZ45,57,71·CCZ45,58,66·CCZ45,59,65·CCZ45,61,64·CCZ45,61,65·CCZ45,61,66·CCZ45,61,67·CCZ45,62,66·CCZ45,63,65·CCZ46,49,60·CCZ46,49,70·CCZ46,49,76·CCZ46,49,81·CCZ46,49,82·CCZ46,50,57·CCZ46,50,67·CCZ46,50,76·CCZ46,50,78·CCZ46,50,79·CCZ46,51,76·CCZ46,52,76·CCZ46,53,54·CCZ46,53,64·CCZ46,53,74·CCZ46,53,75·CCZ46,53,76·CCZ46,54,67·CCZ46,54,70·CCZ46,54,72·CCZ46,54,73·CCZ46,55,70·CCZ46,56,70·CCZ46,57,64·CCZ46,57,68·CCZ46,57,69·CCZ46,57,70·CCZ46,58,67·CCZ46,59,67·CCZ46,60,64·CCZ46,60,65·CCZ46,60,66·CCZ46,60,67·CCZ46,62,64·CCZ46,63,64·CCZ47,49,59·CCZ47,49,69·CCZ47,49,75·CCZ47,49,80·CCZ47,49,82·CCZ47,50,56·CCZ47,50,66·CCZ47,50,75·CCZ47,50,77·CCZ47,50,79·CCZ47,51,75·CCZ47,52,54·CCZ47,52,64·CCZ47,52,74·CCZ47,52,75·CCZ47,52,76·CCZ47,53,75·CCZ47,54,66·CCZ47,54,69·CCZ47,54,71·CCZ47,54,73·CCZ47,55,69·CCZ47,56,64·CCZ47,56,68·CCZ47,56,69·CCZ47,56,70·CCZ47,57,69·CCZ47,58,66·CCZ47,59,64·CCZ47,59,65·CCZ47,59,66·CCZ47,59,67·CCZ47,60,66·CCZ47,61,64·CCZ47,63,64·CCZ48,49,58·CCZ48,49,68·CCZ48,49,74·CCZ48,49,80·CCZ48,49,81·CCZ48,50,55·CCZ48,50,65·CCZ48,50,74·CCZ48,50,77·CCZ48,50,78·CCZ48,51,54·CCZ48,51,64·CCZ48,51,74·CCZ48,51,75·CCZ48,51,76·CCZ48,52,74·CCZ48,53,74·CCZ48,54,65·CCZ48,54,68·CCZ48,54,71·CCZ48,54,72·CCZ48,55,64·CCZ48,55,68·CCZ48,55,69·CCZ48,55,70·CCZ48,56,68·CCZ48,57,68·CCZ48,58,64·CCZ48,58,65·CCZ48,58,66·CCZ48,58,67·CCZ48,59,65·CCZ48,60,65·CCZ48,61,64·CCZ48,62,64
As monomials: 0,64,83+0,65,82+0,66,81+0,67,80+0,68,79+0,69,78+0,70,77+0,71,76+0,72,75+0,73,74+1,54,83+1,55,82+1,56,81+1,57,80+1,58,79+1,59,78+1,60,77+1,61,76+1,62,75+1,63,74+2,50,83+2,51,82+2,52,81+2,53,80+2,58,73+2,59,72+2,60,71+2,61,70+2,62,69+2,63,68+3,49,83+3,51,79+3,52,78+3,53,77+3,55,73+3,56,72+3,57,71+3,61,67+3,62,66+3,63,65+4,49,82+4,50,79+4,52,76+4,53,75+4,54,73+4,56,70+4,57,69+4,59,67+4,60,66+4,63,64+5,49,81+5,50,78+5,51,76+5,53,74+5,54,72+5,55,70+5,57,68+5,58,67+5,60,65+5,62,64+6,49,80+6,50,77+6,51,75+6,52,74+6,54,71+6,55,69+6,56,68+6,58,66+6,59,65+6,61,64+7,39,83+7,40,82+7,41,81+7,42,80+7,43,79+7,44,78+7,45,77+7,46,76+7,47,75+7,48,74+8,35,83+8,36,82+8,37,81+8,38,80+8,43,73+8,44,72+8,45,71+8,46,70+8,47,69+8,48,68+9,34,83+9,36,79+9,37,78+9,38,77+9,40,73+9,41,72+9,42,71+9,46,67+9,47,66+9,48,65+10,34,82+10,35,79+10,37,76+10,38,75+10,39,73+10,41,70+10,42,69+10,44,67+10,45,66+10,48,64+11,34,81+11,35,78+11,36,76+11,38,74+11,39,72+11,40,70+11,42,68+11,43,67+11,45,65+11,47,64+12,34,80+12,35,77+12,36,75+12,37,74+12,39,71+12,40,69+12,41,68+12,43,66+12,44,65+12,46,64+13,30,83+13,31,82+13,32,81+13,33,80+13,43,63+13,44,62+13,45,61+13,46,60+13,47,59+13,48,58+14,29,83+14,31,79+14,32,78+14,33,77+14,40,63+14,41,62+14,42,61+14,46,57+14,47,56+14,48,55+15,29,82+15,30,79+15,32,76+15,33,75+15,39,63+15,41,60+15,42,59+15,44,57+15,45,56+15,48,54+16,29,81+16,30,78+16,31,76+16,33,74+16,39,62+16,40,60+16,42,58+16,43,57+16,45,55+16,47,54+17,29,80+17,30,77+17,31,75+17,32,74+17,39,61+17,40,59+17,41,58+17,43,56+17,44,55+17,46,54+18,28,83+18,31,73+18,32,72+18,33,71+18,36,63+18,37,62+18,38,61+18,46,53+18,47,52+18,48,51+19,28,82+19,30,73+19,32,70+19,33,69+19,35,63+19,37,60+19,38,59+19,44,53+19,45,52+19,48,50+20,28,81+20,30,72+20,31,70+20,33,68+20,35,62+20,36,60+20,38,58+20,43,53+20,45,51+20,47,50+21,28,80+21,30,71+21,31,69+21,32,68+21,35,61+21,36,59+21,37,58+21,43,52+21,44,51+21,46,50+22,28,79+22,29,73+22,32,67+22,33,66+22,34,63+22,37,57+22,38,56+22,41,53+22,42,52+22,48,49+23,28,78+23,29,72+23,31,67+23,33,65+23,34,62+23,36,57+23,38,55+23,40,53+23,42,51+23,47,49+24,28,77+24,29,71+24,31,66+24,32,65+24,34,61+24,36,56+24,37,55+24,40,52+24,41,51+24,46,49+25,28,76+25,29,70+25,30,67+25,33,64+25,34,60+25,35,57+25,38,54+25,39,53+25,42,50+25,45,49+26,28,75+26,29,69+26,30,66+26,32,64+26,34,59+26,35,56+26,37,54+26,39,52+26,41,50+26,44,49+27,28,74+27,29,68+27,30,65+27,31,64+27,34,58+27,35,55+27,36,54+27,39,51+27,40,50+27,43,49+28,39,83+28,40,82+28,41,81+28,42,80+28,43,79+28,44,78+28,45,77+28,46,76+28,47,75+28,48,74+28,54,83+28,55,82+28,56,81+28,57,80+28,58,79+28,59,78+28,60,77+28,61,76+28,62,75+28,63,74+28,64,83+28,65,82+28,66,81+28,67,80+28,68,79+28,69,78+28,70,77+28,71,76+28,72,75+28,73,74+28,74,79+28,74,82+28,74,83+28,75,78+28,75,81+28,75,83+28,76,77+28,76,80+28,76,83+28,77,81+28,77,82+28,78,80+28,78,82+28,79,80+28,79,81+29,35,83+29,36,82+29,37,81+29,38,80+29,43,73+29,44,72+29,45,71+29,46,70+29,47,69+29,48,68+29,50,83+29,51,82+29,52,81+29,53,80+29,58,73+29,59,72+29,60,71+29,61,70+29,62,69+29,63,68+29,64,83+29,65,82+29,66,81+29,67,80+29,68,73+29,68,79+29,68,82+29,68,83+29,69,72+29,69,78+29,69,81+29,69,83+29,70,71+29,70,77+29,70,80+29,70,83+29,71,76+29,71,81+29,71,82+29,72,75+29,72,80+29,72,82+29,73,74+29,73,80+29,73,81+30,34,83+30,36,79+30,37,78+30,38,77+30,40,73+30,41,72+30,42,71+30,46,67+30,47,66+30,48,65+30,49,83+30,51,79+30,52,78+30,53,77+30,55,73+30,56,72+30,57,71+30,61,67+30,62,66+30,63,65+30,64,83+30,65,73+30,65,79+30,65,82+30,65,83+30,66,72+30,66,78+30,66,81+30,66,83+30,67,71+30,67,77+30,67,80+30,67,83+30,68,79+30,69,78+30,70,77+30,71,76+30,71,78+30,71,79+30,72,75+30,72,77+30,72,79+30,73,74+30,73,77+30,73,78+31,34,82+31,35,79+31,37,76+31,38,75+31,39,73+31,41,70+31,42,69+31,44,67+31,45,66+31,48,64+31,49,82+31,50,79+31,52,76+31,53,75+31,54,73+31,56,70+31,57,69+31,59,67+31,60,66+31,63,64+31,64,73+31,64,79+31,64,82+31,64,83+31,65,82+31,66,70+31,66,76+31,66,81+31,66,82+31,67,69+31,67,75+31,67,80+31,67,82+31,68,79+31,69,76+31,69,78+31,69,79+31,70,75+31,70,77+31,70,79+31,71,76+31,72,75+31,73,74+31,73,75+31,73,76+32,34,81+32,35,78+32,36,76+32,38,74+32,39,72+32,40,70+32,42,68+32,43,67+32,45,65+32,47,64+32,49,81+32,50,78+32,51,76+32,53,74+32,54,72+32,55,70+32,57,68+32,58,67+32,60,65+32,62,64+32,64,72+32,64,78+32,64,81+32,64,83+32,65,70+32,65,76+32,65,81+32,65,82+32,66,81+32,67,68+32,67,74+32,67,80+32,67,81+32,68,76+32,68,78+32,68,79+32,69,78+32,70,74+32,70,77+32,70,78+32,71,76+32,72,74+32,72,75+32,72,76+32,73,74+33,34,80+33,35,77+33,36,75+33,37,74+33,39,71+33,40,69+33,41,68+33,43,66+33,44,65+33,46,64+33,49,80+33,50,77+33,51,75+33,52,74+33,54,71+33,55,69+33,56,68+33,58,66+33,59,65+33,61,64+33,64,71+33,64,77+33,64,80+33,64,83+33,65,69+33,65,75+33,65,80+33,65,82+33,66,68+33,66,74+33,66,80+33,66,81+33,67,80+33,68,75+33,68,77+33,68,79+33,69,74+33,69,77+33,69,78+33,70,77+33,71,74+33,71,75+33,71,76+33,72,75+33,73,74+34,43,63+34,44,62+34,45,61+34,46,60+34,47,59+34,48,58+34,50,83+34,51,82+34,52,81+34,53,80+34,54,83+34,55,82+34,56,81+34,57,80+34,58,63+34,58,73+34,58,79+34,58,82+34,58,83+34,59,62+34,59,72+34,59,78+34,59,81+34,59,83+34,60,61+34,60,71+34,60,77+34,60,80+34,60,83+34,61,70+34,61,76+34,61,81+34,61,82+34,62,69+34,62,75+34,62,80+34,62,82+34,63,68+34,63,74+34,63,80+34,63,81+35,40,63+35,41,62+35,42,61+35,46,57+35,47,56+35,48,55+35,49,83+35,51,79+35,52,78+35,53,77+35,54,83+35,55,63+35,55,73+35,55,79+35,55,82+35,55,83+35,56,62+35,56,72+35,56,78+35,56,81+35,56,83+35,57,61+35,57,71+35,57,77+35,57,80+35,57,83+35,58,79+35,59,78+35,60,77+35,61,67+35,61,76+35,61,78+35,61,79+35,62,66+35,62,75+35,62,77+35,62,79+35,63,65+35,63,74+35,63,77+35,63,78+36,39,63+36,41,60+36,42,59+36,44,57+36,45,56+36,48,54+36,49,82+36,50,79+36,52,76+36,53,75+36,54,63+36,54,73+36,54,79+36,54,82+36,54,83+36,55,82+36,56,60+36,56,70+36,56,76+36,56,81+36,56,82+36,57,59+36,57,69+36,57,75+36,57,80+36,57,82+36,58,79+36,59,67+36,59,76+36,59,78+36,59,79+36,60,66+36,60,75+36,60,77+36,60,79+36,61,76+36,62,75+36,63,64+36,63,74+36,63,75+36,63,76+37,39,62+37,40,60+37,42,58+37,43,57+37,45,55+37,47,54+37,49,81+37,50,78+37,51,76+37,53,74+37,54,62+37,54,72+37,54,78+37,54,81+37,54,83+37,55,60+37,55,70+37,55,76+37,55,81+37,55,82+37,56,81+37,57,58+37,57,68+37,57,74+37,57,80+37,57,81+37,58,67+37,58,76+37,58,78+37,58,79+37,59,78+37,60,65+37,60,74+37,60,77+37,60,78+37,61,76+37,62,64+37,62,74+37,62,75+37,62,76+37,63,74+38,39,61+38,40,59+38,41,58+38,43,56+38,44,55+38,46,54+38,49,80+38,50,77+38,51,75+38,52,74+38,54,61+38,54,71+38,54,77+38,54,80+38,54,83+38,55,59+38,55,69+38,55,75+38,55,80+38,55,82+38,56,58+38,56,68+38,56,74+38,56,80+38,56,81+38,57,80+38,58,66+38,58,75+38,58,77+38,58,79+38,59,65+38,59,74+38,59,77+38,59,78+38,60,77+38,61,64+38,61,74+38,61,75+38,61,76+38,62,75+38,63,74+39,46,53+39,47,52+39,48,51+39,49,83+39,50,83+39,51,63+39,51,73+39,51,79+39,51,82+39,51,83+39,52,62+39,52,72+39,52,78+39,52,81+39,52,83+39,53,61+39,53,71+39,53,77+39,53,80+39,53,83+39,55,73+39,56,72+39,57,71+39,58,73+39,59,72+39,60,71+39,61,67+39,61,70+39,61,72+39,61,73+39,62,66+39,62,69+39,62,71+39,62,73+39,63,65+39,63,68+39,63,71+39,63,72+40,44,53+40,45,52+40,48,50+40,49,82+40,50,63+40,50,73+40,50,79+40,50,82+40,50,83+40,51,82+40,52,60+40,52,70+40,52,76+40,52,81+40,52,82+40,53,59+40,53,69+40,53,75+40,53,80+40,53,82+40,54,73+40,56,70+40,57,69+40,58,73+40,59,67+40,59,70+40,59,72+40,59,73+40,60,66+40,60,69+40,60,71+40,60,73+40,61,70+40,62,69+40,63,64+40,63,68+40,63,69+40,63,70+41,43,53+41,45,51+41,47,50+41,49,81+41,50,62+41,50,72+41,50,78+41,50,81+41,50,83+41,51,60+41,51,70+41,51,76+41,51,81+41,51,82+41,52,81+41,53,58+41,53,68+41,53,74+41,53,80+41,53,81+41,54,72+41,55,70+41,57,68+41,58,67+41,58,70+41,58,72+41,58,73+41,59,72+41,60,65+41,60,68+41,60,71+41,60,72+41,61,70+41,62,64+41,62,68+41,62,69+41,62,70+41,63,68+42,43,52+42,44,51+42,46,50+42,49,80+42,50,61+42,50,71+42,50,77+42,50,80+42,50,83+42,51,59+42,51,69+42,51,75+42,51,80+42,51,82+42,52,58+42,52,68+42,52,74+42,52,80+42,52,81+42,53,80+42,54,71+42,55,69+42,56,68+42,58,66+42,58,69+42,58,71+42,58,73+42,59,65+42,59,68+42,59,71+42,59,72+42,60,71+42,61,64+42,61,68+42,61,69+42,61,70+42,62,69+42,63,68+43,48,49+43,49,63+43,49,73+43,49,79+43,49,82+43,49,83+43,50,79+43,51,79+43,52,57+43,52,67+43,52,76+43,52,78+43,52,79+43,53,56+43,53,66+43,53,75+43,53,77+43,53,79+43,54,73+43,55,73+43,56,67+43,56,70+43,56,72+43,56,73+43,57,66+43,57,69+43,57,71+43,57,73+43,59,67+43,60,66+43,61,67+43,62,66+43,63,64+43,63,65+43,63,66+43,63,67+44,47,49+44,49,62+44,49,72+44,49,78+44,49,81+44,49,83+44,50,78+44,51,57+44,51,67+44,51,76+44,51,78+44,51,79+44,52,78+44,53,55+44,53,65+44,53,74+44,53,77+44,53,78+44,54,72+44,55,67+44,55,70+44,55,72+44,55,73+44,56,72+44,57,65+44,57,68+44,57,71+44,57,72+44,58,67+44,60,65+44,61,67+44,62,64+44,62,65+44,62,66+44,62,67+44,63,65+45,46,49+45,49,61+45,49,71+45,49,77+45,49,80+45,49,83+45,50,77+45,51,56+45,51,66+45,51,75+45,51,77+45,51,79+45,52,55+45,52,65+45,52,74+45,52,77+45,52,78+45,53,77+45,54,71+45,55,66+45,55,69+45,55,71+45,55,73+45,56,65+45,56,68+45,56,71+45,56,72+45,57,71+45,58,66+45,59,65+45,61,64+45,61,65+45,61,66+45,61,67+45,62,66+45,63,65+46,49,60+46,49,70+46,49,76+46,49,81+46,49,82+46,50,57+46,50,67+46,50,76+46,50,78+46,50,79+46,51,76+46,52,76+46,53,54+46,53,64+46,53,74+46,53,75+46,53,76+46,54,67+46,54,70+46,54,72+46,54,73+46,55,70+46,56,70+46,57,64+46,57,68+46,57,69+46,57,70+46,58,67+46,59,67+46,60,64+46,60,65+46,60,66+46,60,67+46,62,64+46,63,64+47,49,59+47,49,69+47,49,75+47,49,80+47,49,82+47,50,56+47,50,66+47,50,75+47,50,77+47,50,79+47,51,75+47,52,54+47,52,64+47,52,74+47,52,75+47,52,76+47,53,75+47,54,66+47,54,69+47,54,71+47,54,73+47,55,69+47,56,64+47,56,68+47,56,69+47,56,70+47,57,69+47,58,66+47,59,64+47,59,65+47,59,66+47,59,67+47,60,66+47,61,64+47,63,64+48,49,58+48,49,68+48,49,74+48,49,80+48,49,81+48,50,55+48,50,65+48,50,74+48,50,77+48,50,78+48,51,54+48,51,64+48,51,74+48,51,75+48,51,76+48,52,74+48,53,74+48,54,65+48,54,68+48,54,71+48,54,72+48,55,64+48,55,68+48,55,69+48,55,70+48,56,68+48,57,68+48,58,64+48,58,65+48,58,66+48,58,67+48,59,65+48,60,65+48,61,64+48,62,64 (output wires per term, separated by +)
This row stands for every gate that a CNOT circuit and S, Z and CZ gates on the outputs can reach from this one. They all prepare the same magic state. The wires are shown in the labelling of the stored circuit.
Clifford correction
None. With every rotation a \mathrm{T}=\mathrm{diag}(1,e^{i\pi/4}), the circuit's action is exactly the gate above, and an error-free run passes every check with certainty.
Rotations are numbered by matrix column, from 0. The correction comes from the columns by the weight expansion of Bravyi and Haah (2012). verify_catalog.py re-derives it, checks the corrected circuit's phase input by input against the gate, and on small circuits simulates it gate by gate.
Convert the output with CNOT and S
A CNOT circuit on the outputs, followed by S, Z and CZ gates, turns this factory's gate into other gates that prepare the same magic state. Type a target gate to check whether it is reachable. If it is, you get the circuit, checked on every basis state.
Or list every gate it reaches. The list is built in your browser and saved as a text file, one gate per line.
Python code that writes the same list
import json, urllib.request
NAMES = {1: "T", 2: "CS", 3: "CCZ"}
def equivalent_gates(k, terms, limit=1_000_000):
"""Every gate a CNOT circuit and S, Z, CZ gates reach from this one.
A gate is its set of terms mod Clifford: {a} is T, {a,b} is CS and
{a,b,c} is CCZ on those outputs. CNOT c->t sends x_t to
x_c + x_t - 2 x_c x_t, so a term on t but not c adds the term with t
replaced by c, and, below CCZ, the term with c added.
"""
start = frozenset(frozenset(m) for m in terms)
seen, queue = {start}, [start]
for gate in queue:
for c in range(k):
for t in range(k):
if c == t:
continue
new = set(gate)
for m in gate:
if t in m and c not in m:
new ^= {(m - {t}) | {c}}
if len(m) < 3:
new ^= {m | {c}}
new = frozenset(new)
if new not in seen:
seen.add(new)
queue.append(new)
if len(seen) > limit:
raise RuntimeError(f"more than {limit} equivalent gates")
return seen
def name(gate, k):
sep = "," if k > 10 else ""
terms = sorted((sorted(m) for m in gate), key=lambda m: (len(m), m))
return "·".join(NAMES[len(m)] + sep.join(map(str, m)) for m in terms) or "I"
url = "https://shubhamj810.github.io/magic-state-factory-repo/data/factories/256.84.4.a.json"
record = json.load(urllib.request.urlopen(url))
k = record["parameters"]["k"]
gates = equivalent_gates(k, record["gate"]["monomials"])
lines = sorted(name(g, k) for g in gates)
print(len(lines), "equivalent gates")
with open("equivalent-gates-256.84.4.a.txt", "w") as out:
out.write("\n".join(lines) + "\n")
Distance
d=4, exact. No fault of weight below 4 is both undetectable and damaging. One of weight 4 is.
Exactly 690,880 faults of weight 4 are undetectable and damaging, so A_{4}=690{,}880. If each rotation fails independently with probability p, an accepted run has a damaged output with probability A_{4}\,p^{4}+O(p^{5}).
A damaging fault of weight 4 sits on columns [0,1,2,3].
A fault is a set of faulty columns. It is undetectable when their check parts cancel, and damaging when their output parts do not.
Original source
S. P. Jain, A. Wills, and S. Singh, "Symmetry and AI-assisted discovery of magic-state factories," arXiv:2610.06535 (2026). arxiv.org/abs/2610.06535
Provenance
found by an AI search campaign; a verified witness, not a maximum
Found by: AI search
Discovery: AI search
1 recorded source
[
{
"regime": "AI search",
"file": "/private/tmp/claude-501/-Users-shubhamjain-Documents-Claude-remote-magic-states-AI/25937cc1-0367-40c1-acc9-031067b831d4/scratchpad/msai_for_merge.jsonl",
"label": "msAI:49c31ede38ccccc5",
"N": 93,
"d": 4,
"d_is_exact": true,
"origin": "magic-states-AI/master_catalogue/master_catalogue.jsonl",
"provenance": "magic-states-AI/master_catalogue/master_catalogue.jsonl",
"notes": "gamma_campaign"
}
]Use it in code
Load this factory and re-derive every number from its circuit with the catalogue's own verifier.
# git clone https://github.com/shubhamj810/magic-state-factory-repo
# cd magic-state-factory-repo/master_catalog
import json, urllib.request
import verify_catalog as VC
url = "https://shubhamj810.github.io/magic-state-factory-repo/data/factories/256.84.4.a.json"
record = json.load(urllib.request.urlopen(url))
columns = record["circuit"]["columns"]
k, N = record["parameters"]["k"], record["parameters"]["N"]
facts, problems = VC.derive(columns, k, N) # gate, T-count, degree, ...
distance = VC.measure_distance(columns, k, N) # exhaustive fault search
print(facts["gate_human"], distance["d_exact"] or distance["d_at_least"],
problems or "verified")
Raw record
Show the JSON this page is built from
Source: master_catalog.json, factories[927].