The circuit
The 44\times127 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 |
|---|---|
| out 0 (63) | 111111111111111111111111111111111111111111111111111111111111111································································ |
| out 1 (95) | 1111111111111111111111111111111111111111111111111111111111111111·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1· |
| out 2 (95) | 11111111111111111111111111111111111111111111111111111111111111111··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11·· |
| out 3 (95) | 1111111111111111111111111111111111111111111111111111111111111111111····1111····1111····1111····1111····1111····1111····1111···· |
| out 4 (95) | 11111111111111111111111111111111111111111111111111111111111111111111111········11111111········11111111········11111111········ |
| out 5 (95) | 1111111111111111111111111111111111111111111111111111111111111111111111111111111················1111111111111111················ |
| out 6 (95) | 11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111································ |
| out 7 (111) | 111111111111111111111111111111111111111111111111111111111111111111·111·111·111·111·111·111·111·111·111·111·111·111·111·111·111· |
| out 8 (111) | 11111111111111111111111111111111111111111111111111111111111111111111·1·11111·1·11111·1·11111·1·11111·1·11111·1·11111·1·11111·1· |
| out 9 (111) | 111111111111111111111111111111111111111111111111111111111111111111111111·1·1·1·111111111·1·1·1·111111111·1·1·1·111111111·1·1·1· |
| out 10 (111) | 11111111111111111111111111111111111111111111111111111111111111111111111111111111·1·1·1·1·1·1·1·11111111111111111·1·1·1·1·1·1·1· |
| out 11 (111) | 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1· |
| out 12 (111) | 111111111111111111111111111111111111111111111111111111111111111111111··111111··111111··111111··111111··111111··111111··111111·· |
| out 13 (111) | 1111111111111111111111111111111111111111111111111111111111111111111111111··11··1111111111··11··1111111111··11··1111111111··11·· |
| out 14 (111) | 111111111111111111111111111111111111111111111111111111111111111111111111111111111··11··11··11··111111111111111111··11··11··11·· |
| out 15 (111) | 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111··11··11··11··11··11··11··11·· |
| out 16 (111) | 111111111111111111111111111111111111111111111111111111111111111111111111111····111111111111····111111111111····111111111111···· |
| out 17 (111) | 11111111111111111111111111111111111111111111111111111111111111111111111111111111111····1111····11111111111111111111····1111···· |
| out 18 (111) | 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111····1111····1111····1111···· |
| out 19 (111) | 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111········111111111111111111111111········ |
| out 20 (111) | 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111········11111111········ |
| out 21 (111) | 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111················ |
| out 22 (64) | ··1···1···1···1···1···1···1···1···1···1···1···1···1···1···1···1111·111·111·111·111·111·111·111·111·111·111·111·111·111·111·111· |
| out 23 (64) | ····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·1·····1·111111·1·11111·1·11111·1·11111·1·11111·1·11111·1·11111·1·11111·1· |
| out 24 (64) | ········1·1·1·1·········1·1·1·1·········1·1·1·1·········1·1·1·1111111111·1·1·1·111111111·1·1·1·111111111·1·1·1·111111111·1·1·1· |
| out 25 (64) | ················1·1·1·1·1·1·1·1·················1·1·1·1·1·1·1·111111111111111111·1·1·1·1·1·1·1·11111111111111111·1·1·1·1·1·1·1· |
| out 26 (64) | ································1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1111111111111111111111111111111111·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1· |
| out 27 (64) | ·····11······11······11······11······11······11······11······11111111··111111··111111··111111··111111··111111··111111··111111·· |
| out 28 (64) | ·········11··11··········11··11··········11··11··········11··111111111111··11··1111111111··11··1111111111··11··1111111111··11·· |
| out 29 (64) | ·················11··11··11··11··················11··11··11··11111111111111111111··11··11··11··111111111111111111··11··11··11·· |
| out 30 (64) | ·································11··11··11··11··11··11··11··111111111111111111111111111111111111··11··11··11··11··11··11··11·· |
| out 31 (64) | ···········1111············1111············1111············1111111111111111····111111111111····111111111111····111111111111···· |
| out 32 (64) | ···················1111····1111····················1111····111111111111111111111111····1111····11111111111111111111····1111···· |
| out 33 (64) | ···································1111····1111····1111····1111111111111111111111111111111111111111····1111····1111····1111···· |
| out 34 (64) | ·······················11111111························11111111111111111111111111111111········111111111111111111111111········ |
| out 35 (64) | ·······································11111111········111111111111111111111111111111111111111111111111········11111111········ |
| out 36 (64) | ···············································1111111111111111111111111111111111111111111111111111111111111111················ |
| check 37 (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 |
| check 38 (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 |
| check 39 (64) | ···1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111 |
| check 40 (64) | ·······11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111 |
| check 41 (64) | ···············1111111111111111················1111111111111111················1111111111111111················1111111111111111 |
| check 42 (64) | ·······························11111111111111111111111111111111································11111111111111111111111111111111 |
| check 43 (64) | ·······························································1111111111111111111111111111111111111111111111111111111111111111 |
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 n127Noisy T states per run.
outputs k37
distance d3Exact, proved and witnessed here.
error coefficient A_{3}2,667Faults of weight 3 in the circuit shown that pass every check and damage the output. The output error is A_{3}\,p^{3} to leading order.
wires N44
checks r7
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 d1.1226Overhead 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 width37Rank of the output rows modulo the check rows.
Output gate
22 T231 CS1615 CCZon 37 output wires
As gates: T0·T1·T2·T3·T4·T5·T6·T7·T8·T9·T10·T11·T12·T13·T14·T15·T16·T17·T18·T19·T20·T21·CS0,1·CS0,2·CS0,3·CS0,4·CS0,5·CS0,6·CS0,7·CS0,8·CS0,9·CS0,10·CS0,11·CS0,12·CS0,13·CS0,14·CS0,15·CS0,16·CS0,17·CS0,18·CS0,19·CS0,20·CS0,21·CS1,2·CS1,3·CS1,4·CS1,5·CS1,6·CS1,7·CS1,8·CS1,9·CS1,10·CS1,11·CS1,12·CS1,13·CS1,14·CS1,15·CS1,16·CS1,17·CS1,18·CS1,19·CS1,20·CS1,21·CS2,3·CS2,4·CS2,5·CS2,6·CS2,7·CS2,8·CS2,9·CS2,10·CS2,11·CS2,12·CS2,13·CS2,14·CS2,15·CS2,16·CS2,17·CS2,18·CS2,19·CS2,20·CS2,21·CS3,4·CS3,5·CS3,6·CS3,7·CS3,8·CS3,9·CS3,10·CS3,11·CS3,12·CS3,13·CS3,14·CS3,15·CS3,16·CS3,17·CS3,18·CS3,19·CS3,20·CS3,21·CS4,5·CS4,6·CS4,7·CS4,8·CS4,9·CS4,10·CS4,11·CS4,12·CS4,13·CS4,14·CS4,15·CS4,16·CS4,17·CS4,18·CS4,19·CS4,20·CS4,21·CS5,6·CS5,7·CS5,8·CS5,9·CS5,10·CS5,11·CS5,12·CS5,13·CS5,14·CS5,15·CS5,16·CS5,17·CS5,18·CS5,19·CS5,20·CS5,21·CS6,7·CS6,8·CS6,9·CS6,10·CS6,11·CS6,12·CS6,13·CS6,14·CS6,15·CS6,16·CS6,17·CS6,18·CS6,19·CS6,20·CS6,21·CS7,8·CS7,9·CS7,10·CS7,11·CS7,12·CS7,13·CS7,14·CS7,15·CS7,16·CS7,17·CS7,18·CS7,19·CS7,20·CS7,21·CS8,9·CS8,10·CS8,11·CS8,12·CS8,13·CS8,14·CS8,15·CS8,16·CS8,17·CS8,18·CS8,19·CS8,20·CS8,21·CS9,10·CS9,11·CS9,12·CS9,13·CS9,14·CS9,15·CS9,16·CS9,17·CS9,18·CS9,19·CS9,20·CS9,21·CS10,11·CS10,12·CS10,13·CS10,14·CS10,15·CS10,16·CS10,17·CS10,18·CS10,19·CS10,20·CS10,21·CS11,12·CS11,13·CS11,14·CS11,15·CS11,16·CS11,17·CS11,18·CS11,19·CS11,20·CS11,21·CS12,13·CS12,14·CS12,15·CS12,16·CS12,17·CS12,18·CS12,19·CS12,20·CS12,21·CS13,14·CS13,15·CS13,16·CS13,17·CS13,18·CS13,19·CS13,20·CS13,21·CS14,15·CS14,16·CS14,17·CS14,18·CS14,19·CS14,20·CS14,21·CS15,16·CS15,17·CS15,18·CS15,19·CS15,20·CS15,21·CS16,17·CS16,18·CS16,19·CS16,20·CS16,21·CS17,18·CS17,19·CS17,20·CS17,21·CS18,19·CS18,20·CS18,21·CS19,20·CS19,21·CS20,21·CCZ0,1,2·CCZ0,1,3·CCZ0,1,4·CCZ0,1,5·CCZ0,1,6·CCZ0,1,7·CCZ0,1,8·CCZ0,1,9·CCZ0,1,10·CCZ0,1,11·CCZ0,1,12·CCZ0,1,13·CCZ0,1,14·CCZ0,1,15·CCZ0,1,16·CCZ0,1,17·CCZ0,1,18·CCZ0,1,19·CCZ0,1,20·CCZ0,1,21·CCZ0,2,3·CCZ0,2,4·CCZ0,2,5·CCZ0,2,6·CCZ0,2,7·CCZ0,2,8·CCZ0,2,9·CCZ0,2,10·CCZ0,2,11·CCZ0,2,12·CCZ0,2,13·CCZ0,2,14·CCZ0,2,15·CCZ0,2,16·CCZ0,2,17·CCZ0,2,18·CCZ0,2,19·CCZ0,2,20·CCZ0,2,21·CCZ0,3,4·CCZ0,3,5·CCZ0,3,6·CCZ0,3,7·CCZ0,3,8·CCZ0,3,9·CCZ0,3,10·CCZ0,3,11·CCZ0,3,12·CCZ0,3,13·CCZ0,3,14·CCZ0,3,15·CCZ0,3,16·CCZ0,3,17·CCZ0,3,18·CCZ0,3,19·CCZ0,3,20·CCZ0,3,21·CCZ0,4,5·CCZ0,4,6·CCZ0,4,7·CCZ0,4,8·CCZ0,4,9·CCZ0,4,10·CCZ0,4,11·CCZ0,4,12·CCZ0,4,13·CCZ0,4,14·CCZ0,4,15·CCZ0,4,16·CCZ0,4,17·CCZ0,4,18·CCZ0,4,19·CCZ0,4,20·CCZ0,4,21·CCZ0,5,6·CCZ0,5,7·CCZ0,5,8·CCZ0,5,9·CCZ0,5,10·CCZ0,5,11·CCZ0,5,12·CCZ0,5,13·CCZ0,5,14·CCZ0,5,15·CCZ0,5,16·CCZ0,5,17·CCZ0,5,18·CCZ0,5,19·CCZ0,5,20·CCZ0,5,21·CCZ0,6,7·CCZ0,6,8·CCZ0,6,9·CCZ0,6,10·CCZ0,6,11·CCZ0,6,12·CCZ0,6,13·CCZ0,6,14·CCZ0,6,15·CCZ0,6,16·CCZ0,6,17·CCZ0,6,18·CCZ0,6,19·CCZ0,6,20·CCZ0,6,21·CCZ0,7,8·CCZ0,7,9·CCZ0,7,10·CCZ0,7,11·CCZ0,7,12·CCZ0,7,13·CCZ0,7,14·CCZ0,7,15·CCZ0,7,16·CCZ0,7,17·CCZ0,7,18·CCZ0,7,19·CCZ0,7,20·CCZ0,7,21·CCZ0,8,9·CCZ0,8,10·CCZ0,8,11·CCZ0,8,12·CCZ0,8,13·CCZ0,8,14·CCZ0,8,15·CCZ0,8,16·CCZ0,8,17·CCZ0,8,18·CCZ0,8,19·CCZ0,8,20·CCZ0,8,21·CCZ0,9,10·CCZ0,9,11·CCZ0,9,12·CCZ0,9,13·CCZ0,9,14·CCZ0,9,15·CCZ0,9,16·CCZ0,9,17·CCZ0,9,18·CCZ0,9,19·CCZ0,9,20·CCZ0,9,21·CCZ0,10,11·CCZ0,10,12·CCZ0,10,13·CCZ0,10,14·CCZ0,10,15·CCZ0,10,16·CCZ0,10,17·CCZ0,10,18·CCZ0,10,19·CCZ0,10,20·CCZ0,10,21·CCZ0,11,12·CCZ0,11,13·CCZ0,11,14·CCZ0,11,15·CCZ0,11,16·CCZ0,11,17·CCZ0,11,18·CCZ0,11,19·CCZ0,11,20·CCZ0,11,21·CCZ0,12,13·CCZ0,12,14·CCZ0,12,15·CCZ0,12,16·CCZ0,12,17·CCZ0,12,18·CCZ0,12,19·CCZ0,12,20·CCZ0,12,21·CCZ0,13,14·CCZ0,13,15·CCZ0,13,16·CCZ0,13,17·CCZ0,13,18·CCZ0,13,19·CCZ0,13,20·CCZ0,13,21·CCZ0,14,15·CCZ0,14,16·CCZ0,14,17·CCZ0,14,18·CCZ0,14,19·CCZ0,14,20·CCZ0,14,21·CCZ0,15,16·CCZ0,15,17·CCZ0,15,18·CCZ0,15,19·CCZ0,15,20·CCZ0,15,21·CCZ0,16,17·CCZ0,16,18·CCZ0,16,19·CCZ0,16,20·CCZ0,16,21·CCZ0,17,18·CCZ0,17,19·CCZ0,17,20·CCZ0,17,21·CCZ0,18,19·CCZ0,18,20·CCZ0,18,21·CCZ0,19,20·CCZ0,19,21·CCZ0,20,21·CCZ1,2,3·CCZ1,2,4·CCZ1,2,5·CCZ1,2,6·CCZ1,2,7·CCZ1,2,8·CCZ1,2,9·CCZ1,2,10·CCZ1,2,11·CCZ1,2,12·CCZ1,2,13·CCZ1,2,14·CCZ1,2,15·CCZ1,2,16·CCZ1,2,17·CCZ1,2,18·CCZ1,2,19·CCZ1,2,20·CCZ1,2,21·CCZ1,3,4·CCZ1,3,5·CCZ1,3,6·CCZ1,3,7·CCZ1,3,8·CCZ1,3,9·CCZ1,3,10·CCZ1,3,11·CCZ1,3,12·CCZ1,3,13·CCZ1,3,14·CCZ1,3,15·CCZ1,3,16·CCZ1,3,17·CCZ1,3,18·CCZ1,3,19·CCZ1,3,20·CCZ1,3,21·CCZ1,4,5·CCZ1,4,6·CCZ1,4,7·CCZ1,4,8·CCZ1,4,9·CCZ1,4,10·CCZ1,4,11·CCZ1,4,12·CCZ1,4,13·CCZ1,4,14·CCZ1,4,15·CCZ1,4,16·CCZ1,4,17·CCZ1,4,18·CCZ1,4,19·CCZ1,4,20·CCZ1,4,21·CCZ1,5,6·CCZ1,5,7·CCZ1,5,8·CCZ1,5,9·CCZ1,5,10·CCZ1,5,11·CCZ1,5,12·CCZ1,5,13·CCZ1,5,14·CCZ1,5,15·CCZ1,5,16·CCZ1,5,17·CCZ1,5,18·CCZ1,5,19·CCZ1,5,20·CCZ1,5,21·CCZ1,6,7·CCZ1,6,8·CCZ1,6,9·CCZ1,6,10·CCZ1,6,11·CCZ1,6,12·CCZ1,6,13·CCZ1,6,14·CCZ1,6,15·CCZ1,6,16·CCZ1,6,17·CCZ1,6,18·CCZ1,6,19·CCZ1,6,20·CCZ1,6,21·CCZ1,7,8·CCZ1,7,9·CCZ1,7,10·CCZ1,7,11·CCZ1,7,12·CCZ1,7,13·CCZ1,7,14·CCZ1,7,15·CCZ1,7,16·CCZ1,7,17·CCZ1,7,18·CCZ1,7,19·CCZ1,7,20·CCZ1,7,21·CCZ1,8,9·CCZ1,8,10·CCZ1,8,11·CCZ1,8,12·CCZ1,8,13·CCZ1,8,14·CCZ1,8,15·CCZ1,8,16·CCZ1,8,17·CCZ1,8,18·CCZ1,8,19·CCZ1,8,20·CCZ1,8,21·CCZ1,9,10·CCZ1,9,11·CCZ1,9,12·CCZ1,9,13·CCZ1,9,14·CCZ1,9,15·CCZ1,9,16·CCZ1,9,17·CCZ1,9,18·CCZ1,9,19·CCZ1,9,20·CCZ1,9,21·CCZ1,10,11·CCZ1,10,12·CCZ1,10,13·CCZ1,10,14·CCZ1,10,15·CCZ1,10,16·CCZ1,10,17·CCZ1,10,18·CCZ1,10,19·CCZ1,10,20·CCZ1,10,21·CCZ1,11,12·CCZ1,11,13·CCZ1,11,14·CCZ1,11,15·CCZ1,11,16·CCZ1,11,17·CCZ1,11,18·CCZ1,11,19·CCZ1,11,20·CCZ1,11,21·CCZ1,12,13·CCZ1,12,14·CCZ1,12,15·CCZ1,12,16·CCZ1,12,17·CCZ1,12,18·CCZ1,12,19·CCZ1,12,20·CCZ1,12,21·CCZ1,13,14·CCZ1,13,15·CCZ1,13,16·CCZ1,13,17·CCZ1,13,18·CCZ1,13,19·CCZ1,13,20·CCZ1,13,21·CCZ1,14,15·CCZ1,14,16·CCZ1,14,17·CCZ1,14,18·CCZ1,14,19·CCZ1,14,20·CCZ1,14,21·CCZ1,15,16·CCZ1,15,17·CCZ1,15,18·CCZ1,15,19·CCZ1,15,20·CCZ1,15,21·CCZ1,16,17·CCZ1,16,18·CCZ1,16,19·CCZ1,16,20·CCZ1,16,21·CCZ1,17,18·CCZ1,17,19·CCZ1,17,20·CCZ1,17,21·CCZ1,18,19·CCZ1,18,20·CCZ1,18,21·CCZ1,19,20·CCZ1,19,21·CCZ1,20,21·CCZ2,3,4·CCZ2,3,5·CCZ2,3,6·CCZ2,3,7·CCZ2,3,8·CCZ2,3,9·CCZ2,3,10·CCZ2,3,11·CCZ2,3,12·CCZ2,3,13·CCZ2,3,14·CCZ2,3,15·CCZ2,3,16·CCZ2,3,17·CCZ2,3,18·CCZ2,3,19·CCZ2,3,20·CCZ2,3,21·CCZ2,4,5·CCZ2,4,6·CCZ2,4,7·CCZ2,4,8·CCZ2,4,9·CCZ2,4,10·CCZ2,4,11·CCZ2,4,12·CCZ2,4,13·CCZ2,4,14·CCZ2,4,15·CCZ2,4,16·CCZ2,4,17·CCZ2,4,18·CCZ2,4,19·CCZ2,4,20·CCZ2,4,21·CCZ2,5,6·CCZ2,5,7·CCZ2,5,8·CCZ2,5,9·CCZ2,5,10·CCZ2,5,11·CCZ2,5,12·CCZ2,5,13·CCZ2,5,14·CCZ2,5,15·CCZ2,5,16·CCZ2,5,17·CCZ2,5,18·CCZ2,5,19·CCZ2,5,20·CCZ2,5,21·CCZ2,6,7·CCZ2,6,8·CCZ2,6,9·CCZ2,6,10·CCZ2,6,11·CCZ2,6,12·CCZ2,6,13·CCZ2,6,14·CCZ2,6,15·CCZ2,6,16·CCZ2,6,17·CCZ2,6,18·CCZ2,6,19·CCZ2,6,20·CCZ2,6,21·CCZ2,7,8·CCZ2,7,9·CCZ2,7,10·CCZ2,7,11·CCZ2,7,12·CCZ2,7,13·CCZ2,7,14·CCZ2,7,15·CCZ2,7,16·CCZ2,7,17·CCZ2,7,18·CCZ2,7,19·CCZ2,7,20·CCZ2,7,21·CCZ2,8,9·CCZ2,8,10·CCZ2,8,11·CCZ2,8,12·CCZ2,8,13·CCZ2,8,14·CCZ2,8,15·CCZ2,8,16·CCZ2,8,17·CCZ2,8,18·CCZ2,8,19·CCZ2,8,20·CCZ2,8,21·CCZ2,9,10·CCZ2,9,11·CCZ2,9,12·CCZ2,9,13·CCZ2,9,14·CCZ2,9,15·CCZ2,9,16·CCZ2,9,17·CCZ2,9,18·CCZ2,9,19·CCZ2,9,20·CCZ2,9,21·CCZ2,10,11·CCZ2,10,12·CCZ2,10,13·CCZ2,10,14·CCZ2,10,15·CCZ2,10,16·CCZ2,10,17·CCZ2,10,18·CCZ2,10,19·CCZ2,10,20·CCZ2,10,21·CCZ2,11,12·CCZ2,11,13·CCZ2,11,14·CCZ2,11,15·CCZ2,11,16·CCZ2,11,17·CCZ2,11,18·CCZ2,11,19·CCZ2,11,20·CCZ2,11,21·CCZ2,12,13·CCZ2,12,14·CCZ2,12,15·CCZ2,12,16·CCZ2,12,17·CCZ2,12,18·CCZ2,12,19·CCZ2,12,20·CCZ2,12,21·CCZ2,13,14·CCZ2,13,15·CCZ2,13,16·CCZ2,13,17·CCZ2,13,18·CCZ2,13,19·CCZ2,13,20·CCZ2,13,21·CCZ2,14,15·CCZ2,14,16·CCZ2,14,17·CCZ2,14,18·CCZ2,14,19·CCZ2,14,20·CCZ2,14,21·CCZ2,15,16·CCZ2,15,17·CCZ2,15,18·CCZ2,15,19·CCZ2,15,20·CCZ2,15,21·CCZ2,16,17·CCZ2,16,18·CCZ2,16,19·CCZ2,16,20·CCZ2,16,21·CCZ2,17,18·CCZ2,17,19·CCZ2,17,20·CCZ2,17,21·CCZ2,18,19·CCZ2,18,20·CCZ2,18,21·CCZ2,19,20·CCZ2,19,21·CCZ2,20,21·CCZ3,4,5·CCZ3,4,6·CCZ3,4,7·CCZ3,4,8·CCZ3,4,9·CCZ3,4,10·CCZ3,4,11·CCZ3,4,12·CCZ3,4,13·CCZ3,4,14·CCZ3,4,15·CCZ3,4,16·CCZ3,4,17·CCZ3,4,18·CCZ3,4,19·CCZ3,4,20·CCZ3,4,21·CCZ3,5,6·CCZ3,5,7·CCZ3,5,8·CCZ3,5,9·CCZ3,5,10·CCZ3,5,11·CCZ3,5,12·CCZ3,5,13·CCZ3,5,14·CCZ3,5,15·CCZ3,5,16·CCZ3,5,17·CCZ3,5,18·CCZ3,5,19·CCZ3,5,20·CCZ3,5,21·CCZ3,6,7·CCZ3,6,8·CCZ3,6,9·CCZ3,6,10·CCZ3,6,11·CCZ3,6,12·CCZ3,6,13·CCZ3,6,14·CCZ3,6,15·CCZ3,6,16·CCZ3,6,17·CCZ3,6,18·CCZ3,6,19·CCZ3,6,20·CCZ3,6,21·CCZ3,7,8·CCZ3,7,9·CCZ3,7,10·CCZ3,7,11·CCZ3,7,12·CCZ3,7,13·CCZ3,7,14·CCZ3,7,15·CCZ3,7,16·CCZ3,7,17·CCZ3,7,18·CCZ3,7,19·CCZ3,7,20·CCZ3,7,21·CCZ3,8,9·CCZ3,8,10·CCZ3,8,11·CCZ3,8,12·CCZ3,8,13·CCZ3,8,14·CCZ3,8,15·CCZ3,8,16·CCZ3,8,17·CCZ3,8,18·CCZ3,8,19·CCZ3,8,20·CCZ3,8,21·CCZ3,9,10·CCZ3,9,11·CCZ3,9,12·CCZ3,9,13·CCZ3,9,14·CCZ3,9,15·CCZ3,9,16·CCZ3,9,17·CCZ3,9,18·CCZ3,9,19·CCZ3,9,20·CCZ3,9,21·CCZ3,10,11·CCZ3,10,12·CCZ3,10,13·CCZ3,10,14·CCZ3,10,15·CCZ3,10,16·CCZ3,10,17·CCZ3,10,18·CCZ3,10,19·CCZ3,10,20·CCZ3,10,21·CCZ3,11,12·CCZ3,11,13·CCZ3,11,14·CCZ3,11,15·CCZ3,11,16·CCZ3,11,17·CCZ3,11,18·CCZ3,11,19·CCZ3,11,20·CCZ3,11,21·CCZ3,12,13·CCZ3,12,14·CCZ3,12,15·CCZ3,12,16·CCZ3,12,17·CCZ3,12,18·CCZ3,12,19·CCZ3,12,20·CCZ3,12,21·CCZ3,13,14·CCZ3,13,15·CCZ3,13,16·CCZ3,13,17·CCZ3,13,18·CCZ3,13,19·CCZ3,13,20·CCZ3,13,21·CCZ3,14,15·CCZ3,14,16·CCZ3,14,17·CCZ3,14,18·CCZ3,14,19·CCZ3,14,20·CCZ3,14,21·CCZ3,15,16·CCZ3,15,17·CCZ3,15,18·CCZ3,15,19·CCZ3,15,20·CCZ3,15,21·CCZ3,16,17·CCZ3,16,18·CCZ3,16,19·CCZ3,16,20·CCZ3,16,21·CCZ3,17,18·CCZ3,17,19·CCZ3,17,20·CCZ3,17,21·CCZ3,18,19·CCZ3,18,20·CCZ3,18,21·CCZ3,19,20·CCZ3,19,21·CCZ3,20,21·CCZ4,5,6·CCZ4,5,7·CCZ4,5,8·CCZ4,5,9·CCZ4,5,10·CCZ4,5,11·CCZ4,5,12·CCZ4,5,13·CCZ4,5,14·CCZ4,5,15·CCZ4,5,16·CCZ4,5,17·CCZ4,5,18·CCZ4,5,19·CCZ4,5,20·CCZ4,5,21·CCZ4,6,7·CCZ4,6,8·CCZ4,6,9·CCZ4,6,10·CCZ4,6,11·CCZ4,6,12·CCZ4,6,13·CCZ4,6,14·CCZ4,6,15·CCZ4,6,16·CCZ4,6,17·CCZ4,6,18·CCZ4,6,19·CCZ4,6,20·CCZ4,6,21·CCZ4,7,8·CCZ4,7,9·CCZ4,7,10·CCZ4,7,11·CCZ4,7,12·CCZ4,7,13·CCZ4,7,14·CCZ4,7,15·CCZ4,7,16·CCZ4,7,17·CCZ4,7,18·CCZ4,7,19·CCZ4,7,20·CCZ4,7,21·CCZ4,8,9·CCZ4,8,10·CCZ4,8,11·CCZ4,8,12·CCZ4,8,13·CCZ4,8,14·CCZ4,8,15·CCZ4,8,16·CCZ4,8,17·CCZ4,8,18·CCZ4,8,19·CCZ4,8,20·CCZ4,8,21·CCZ4,9,10·CCZ4,9,11·CCZ4,9,12·CCZ4,9,13·CCZ4,9,14·CCZ4,9,15·CCZ4,9,16·CCZ4,9,17·CCZ4,9,18·CCZ4,9,19·CCZ4,9,20·CCZ4,9,21·CCZ4,10,11·CCZ4,10,12·CCZ4,10,13·CCZ4,10,14·CCZ4,10,15·CCZ4,10,16·CCZ4,10,17·CCZ4,10,18·CCZ4,10,19·CCZ4,10,20·CCZ4,10,21·CCZ4,11,12·CCZ4,11,13·CCZ4,11,14·CCZ4,11,15·CCZ4,11,16·CCZ4,11,17·CCZ4,11,18·CCZ4,11,19·CCZ4,11,20·CCZ4,11,21·CCZ4,12,13·CCZ4,12,14·CCZ4,12,15·CCZ4,12,16·CCZ4,12,17·CCZ4,12,18·CCZ4,12,19·CCZ4,12,20·CCZ4,12,21·CCZ4,13,14·CCZ4,13,15·CCZ4,13,16·CCZ4,13,17·CCZ4,13,18·CCZ4,13,19·CCZ4,13,20·CCZ4,13,21·CCZ4,14,15·CCZ4,14,16·CCZ4,14,17·CCZ4,14,18·CCZ4,14,19·CCZ4,14,20·CCZ4,14,21·CCZ4,15,16·CCZ4,15,17·CCZ4,15,18·CCZ4,15,19·CCZ4,15,20·CCZ4,15,21·CCZ4,16,17·CCZ4,16,18·CCZ4,16,19·CCZ4,16,20·CCZ4,16,21·CCZ4,17,18·CCZ4,17,19·CCZ4,17,20·CCZ4,17,21·CCZ4,18,19·CCZ4,18,20·CCZ4,18,21·CCZ4,19,20·CCZ4,19,21·CCZ4,20,21·CCZ5,6,7·CCZ5,6,8·CCZ5,6,9·CCZ5,6,10·CCZ5,6,11·CCZ5,6,12·CCZ5,6,13·CCZ5,6,14·CCZ5,6,15·CCZ5,6,16·CCZ5,6,17·CCZ5,6,18·CCZ5,6,19·CCZ5,6,20·CCZ5,6,21·CCZ5,7,8·CCZ5,7,9·CCZ5,7,10·CCZ5,7,11·CCZ5,7,12·CCZ5,7,13·CCZ5,7,14·CCZ5,7,15·CCZ5,7,16·CCZ5,7,17·CCZ5,7,18·CCZ5,7,19·CCZ5,7,20·CCZ5,7,21·CCZ5,8,9·CCZ5,8,10·CCZ5,8,11·CCZ5,8,12·CCZ5,8,13·CCZ5,8,14·CCZ5,8,15·CCZ5,8,16·CCZ5,8,17·CCZ5,8,18·CCZ5,8,19·CCZ5,8,20·CCZ5,8,21·CCZ5,9,10·CCZ5,9,11·CCZ5,9,12·CCZ5,9,13·CCZ5,9,14·CCZ5,9,15·CCZ5,9,16·CCZ5,9,17·CCZ5,9,18·CCZ5,9,19·CCZ5,9,20·CCZ5,9,21·CCZ5,10,11·CCZ5,10,12·CCZ5,10,13·CCZ5,10,14·CCZ5,10,15·CCZ5,10,16·CCZ5,10,17·CCZ5,10,18·CCZ5,10,19·CCZ5,10,20·CCZ5,10,21·CCZ5,11,12·CCZ5,11,13·CCZ5,11,14·CCZ5,11,15·CCZ5,11,16·CCZ5,11,17·CCZ5,11,18·CCZ5,11,19·CCZ5,11,20·CCZ5,11,21·CCZ5,12,13·CCZ5,12,14·CCZ5,12,15·CCZ5,12,16·CCZ5,12,17·CCZ5,12,18·CCZ5,12,19·CCZ5,12,20·CCZ5,12,21·CCZ5,13,14·CCZ5,13,15·CCZ5,13,16·CCZ5,13,17·CCZ5,13,18·CCZ5,13,19·CCZ5,13,20·CCZ5,13,21·CCZ5,14,15·CCZ5,14,16·CCZ5,14,17·CCZ5,14,18·CCZ5,14,19·CCZ5,14,20·CCZ5,14,21·CCZ5,15,16·CCZ5,15,17·CCZ5,15,18·CCZ5,15,19·CCZ5,15,20·CCZ5,15,21·CCZ5,16,17·CCZ5,16,18·CCZ5,16,19·CCZ5,16,20·CCZ5,16,21·CCZ5,17,18·CCZ5,17,19·CCZ5,17,20·CCZ5,17,21·CCZ5,18,19·CCZ5,18,20·CCZ5,18,21·CCZ5,19,20·CCZ5,19,21·CCZ5,20,21·CCZ6,7,8·CCZ6,7,9·CCZ6,7,10·CCZ6,7,11·CCZ6,7,12·CCZ6,7,13·CCZ6,7,14·CCZ6,7,15·CCZ6,7,16·CCZ6,7,17·CCZ6,7,18·CCZ6,7,19·CCZ6,7,20·CCZ6,7,21·CCZ6,8,9·CCZ6,8,10·CCZ6,8,11·CCZ6,8,12·CCZ6,8,13·CCZ6,8,14·CCZ6,8,15·CCZ6,8,16·CCZ6,8,17·CCZ6,8,18·CCZ6,8,19·CCZ6,8,20·CCZ6,8,21·CCZ6,9,10·CCZ6,9,11·CCZ6,9,12·CCZ6,9,13·CCZ6,9,14·CCZ6,9,15·CCZ6,9,16·CCZ6,9,17·CCZ6,9,18·CCZ6,9,19·CCZ6,9,20·CCZ6,9,21·CCZ6,10,11·CCZ6,10,12·CCZ6,10,13·CCZ6,10,14·CCZ6,10,15·CCZ6,10,16·CCZ6,10,17·CCZ6,10,18·CCZ6,10,19·CCZ6,10,20·CCZ6,10,21·CCZ6,11,12·CCZ6,11,13·CCZ6,11,14·CCZ6,11,15·CCZ6,11,16·CCZ6,11,17·CCZ6,11,18·CCZ6,11,19·CCZ6,11,20·CCZ6,11,21·CCZ6,12,13·CCZ6,12,14·CCZ6,12,15·CCZ6,12,16·CCZ6,12,17·CCZ6,12,18·CCZ6,12,19·CCZ6,12,20·CCZ6,12,21·CCZ6,13,14·CCZ6,13,15·CCZ6,13,16·CCZ6,13,17·CCZ6,13,18·CCZ6,13,19·CCZ6,13,20·CCZ6,13,21·CCZ6,14,15·CCZ6,14,16·CCZ6,14,17·CCZ6,14,18·CCZ6,14,19·CCZ6,14,20·CCZ6,14,21·CCZ6,15,16·CCZ6,15,17·CCZ6,15,18·CCZ6,15,19·CCZ6,15,20·CCZ6,15,21·CCZ6,16,17·CCZ6,16,18·CCZ6,16,19·CCZ6,16,20·CCZ6,16,21·CCZ6,17,18·CCZ6,17,19·CCZ6,17,20·CCZ6,17,21·CCZ6,18,19·CCZ6,18,20·CCZ6,18,21·CCZ6,19,20·CCZ6,19,21·CCZ6,20,21·CCZ7,8,9·CCZ7,8,10·CCZ7,8,11·CCZ7,8,12·CCZ7,8,13·CCZ7,8,14·CCZ7,8,15·CCZ7,8,16·CCZ7,8,17·CCZ7,8,18·CCZ7,8,19·CCZ7,8,20·CCZ7,8,21·CCZ7,9,10·CCZ7,9,11·CCZ7,9,12·CCZ7,9,13·CCZ7,9,14·CCZ7,9,15·CCZ7,9,16·CCZ7,9,17·CCZ7,9,18·CCZ7,9,19·CCZ7,9,20·CCZ7,9,21·CCZ7,10,11·CCZ7,10,12·CCZ7,10,13·CCZ7,10,14·CCZ7,10,15·CCZ7,10,16·CCZ7,10,17·CCZ7,10,18·CCZ7,10,19·CCZ7,10,20·CCZ7,10,21·CCZ7,11,12·CCZ7,11,13·CCZ7,11,14·CCZ7,11,15·CCZ7,11,16·CCZ7,11,17·CCZ7,11,18·CCZ7,11,19·CCZ7,11,20·CCZ7,11,21·CCZ7,12,13·CCZ7,12,14·CCZ7,12,15·CCZ7,12,16·CCZ7,12,17·CCZ7,12,18·CCZ7,12,19·CCZ7,12,20·CCZ7,12,21·CCZ7,13,14·CCZ7,13,15·CCZ7,13,16·CCZ7,13,17·CCZ7,13,18·CCZ7,13,19·CCZ7,13,20·CCZ7,13,21·CCZ7,14,15·CCZ7,14,16·CCZ7,14,17·CCZ7,14,18·CCZ7,14,19·CCZ7,14,20·CCZ7,14,21·CCZ7,15,16·CCZ7,15,17·CCZ7,15,18·CCZ7,15,19·CCZ7,15,20·CCZ7,15,21·CCZ7,16,17·CCZ7,16,18·CCZ7,16,19·CCZ7,16,20·CCZ7,16,36·CCZ7,17,18·CCZ7,17,19·CCZ7,17,21·CCZ7,17,35·CCZ7,18,20·CCZ7,18,21·CCZ7,18,34·CCZ7,19,20·CCZ7,19,21·CCZ7,19,33·CCZ7,20,21·CCZ7,20,32·CCZ7,21,31·CCZ7,31,36·CCZ7,32,35·CCZ7,33,34·CCZ8,9,10·CCZ8,9,11·CCZ8,9,12·CCZ8,9,13·CCZ8,9,14·CCZ8,9,15·CCZ8,9,16·CCZ8,9,17·CCZ8,9,18·CCZ8,9,19·CCZ8,9,20·CCZ8,9,21·CCZ8,10,11·CCZ8,10,12·CCZ8,10,13·CCZ8,10,14·CCZ8,10,15·CCZ8,10,16·CCZ8,10,17·CCZ8,10,18·CCZ8,10,19·CCZ8,10,20·CCZ8,10,21·CCZ8,11,12·CCZ8,11,13·CCZ8,11,14·CCZ8,11,15·CCZ8,11,16·CCZ8,11,17·CCZ8,11,18·CCZ8,11,19·CCZ8,11,20·CCZ8,11,21·CCZ8,12,13·CCZ8,12,14·CCZ8,12,15·CCZ8,12,16·CCZ8,12,17·CCZ8,12,18·CCZ8,12,19·CCZ8,12,20·CCZ8,12,21·CCZ8,13,14·CCZ8,13,15·CCZ8,13,16·CCZ8,13,17·CCZ8,13,18·CCZ8,13,19·CCZ8,13,20·CCZ8,13,36·CCZ8,14,15·CCZ8,14,16·CCZ8,14,17·CCZ8,14,18·CCZ8,14,19·CCZ8,14,21·CCZ8,14,35·CCZ8,15,16·CCZ8,15,17·CCZ8,15,18·CCZ8,15,20·CCZ8,15,21·CCZ8,15,34·CCZ8,16,17·CCZ8,16,18·CCZ8,16,19·CCZ8,16,20·CCZ8,16,21·CCZ8,17,18·CCZ8,17,19·CCZ8,17,20·CCZ8,17,21·CCZ8,18,19·CCZ8,18,20·CCZ8,18,21·CCZ8,19,20·CCZ8,19,21·CCZ8,19,30·CCZ8,20,21·CCZ8,20,29·CCZ8,21,28·CCZ8,28,36·CCZ8,29,35·CCZ8,30,34·CCZ9,10,11·CCZ9,10,12·CCZ9,10,13·CCZ9,10,14·CCZ9,10,15·CCZ9,10,16·CCZ9,10,17·CCZ9,10,18·CCZ9,10,19·CCZ9,10,20·CCZ9,10,21·CCZ9,11,12·CCZ9,11,13·CCZ9,11,14·CCZ9,11,15·CCZ9,11,16·CCZ9,11,17·CCZ9,11,18·CCZ9,11,19·CCZ9,11,20·CCZ9,11,21·CCZ9,12,13·CCZ9,12,14·CCZ9,12,15·CCZ9,12,16·CCZ9,12,17·CCZ9,12,18·CCZ9,12,19·CCZ9,12,20·CCZ9,12,36·CCZ9,13,14·CCZ9,13,15·CCZ9,13,16·CCZ9,13,17·CCZ9,13,18·CCZ9,13,19·CCZ9,13,20·CCZ9,13,21·CCZ9,14,15·CCZ9,14,16·CCZ9,14,17·CCZ9,14,19·CCZ9,14,20·CCZ9,14,21·CCZ9,14,33·CCZ9,15,16·CCZ9,15,18·CCZ9,15,19·CCZ9,15,20·CCZ9,15,21·CCZ9,15,32·CCZ9,16,17·CCZ9,16,18·CCZ9,16,19·CCZ9,16,20·CCZ9,16,21·CCZ9,17,18·CCZ9,17,19·CCZ9,17,20·CCZ9,17,21·CCZ9,17,30·CCZ9,18,19·CCZ9,18,20·CCZ9,18,21·CCZ9,18,29·CCZ9,19,20·CCZ9,19,21·CCZ9,20,21·CCZ9,21,27·CCZ9,27,36·CCZ9,29,33·CCZ9,30,32·CCZ10,11,12·CCZ10,11,13·CCZ10,11,14·CCZ10,11,15·CCZ10,11,16·CCZ10,11,17·CCZ10,11,18·CCZ10,11,19·CCZ10,11,20·CCZ10,11,21·CCZ10,12,13·CCZ10,12,14·CCZ10,12,15·CCZ10,12,16·CCZ10,12,17·CCZ10,12,18·CCZ10,12,19·CCZ10,12,21·CCZ10,12,35·CCZ10,13,14·CCZ10,13,15·CCZ10,13,16·CCZ10,13,17·CCZ10,13,19·CCZ10,13,20·CCZ10,13,21·CCZ10,13,33·CCZ10,14,15·CCZ10,14,16·CCZ10,14,17·CCZ10,14,18·CCZ10,14,19·CCZ10,14,20·CCZ10,14,21·CCZ10,15,17·CCZ10,15,18·CCZ10,15,19·CCZ10,15,20·CCZ10,15,21·CCZ10,15,31·CCZ10,16,17·CCZ10,16,18·CCZ10,16,19·CCZ10,16,20·CCZ10,16,21·CCZ10,16,30·CCZ10,17,18·CCZ10,17,19·CCZ10,17,20·CCZ10,17,21·CCZ10,18,19·CCZ10,18,20·CCZ10,18,21·CCZ10,18,28·CCZ10,19,20·CCZ10,19,21·CCZ10,20,21·CCZ10,20,27·CCZ10,27,35·CCZ10,28,33·CCZ10,30,31·CCZ11,12,13·CCZ11,12,14·CCZ11,12,15·CCZ11,12,16·CCZ11,12,17·CCZ11,12,18·CCZ11,12,20·CCZ11,12,21·CCZ11,12,34·CCZ11,13,14·CCZ11,13,15·CCZ11,13,16·CCZ11,13,18·CCZ11,13,19·CCZ11,13,20·CCZ11,13,21·CCZ11,13,32·CCZ11,14,15·CCZ11,14,17·CCZ11,14,18·CCZ11,14,19·CCZ11,14,20·CCZ11,14,21·CCZ11,14,31·CCZ11,15,16·CCZ11,15,17·CCZ11,15,18·CCZ11,15,19·CCZ11,15,20·CCZ11,15,21·CCZ11,16,17·CCZ11,16,18·CCZ11,16,19·CCZ11,16,20·CCZ11,16,21·CCZ11,16,29·CCZ11,17,18·CCZ11,17,19·CCZ11,17,20·CCZ11,17,21·CCZ11,17,28·CCZ11,18,19·CCZ11,18,20·CCZ11,18,21·CCZ11,19,20·CCZ11,19,21·CCZ11,19,27·CCZ11,20,21·CCZ11,27,34·CCZ11,28,32·CCZ11,29,31·CCZ12,13,14·CCZ12,13,15·CCZ12,13,16·CCZ12,13,17·CCZ12,13,18·CCZ12,13,19·CCZ12,13,20·CCZ12,13,21·CCZ12,14,15·CCZ12,14,16·CCZ12,14,17·CCZ12,14,18·CCZ12,14,19·CCZ12,14,20·CCZ12,14,21·CCZ12,15,16·CCZ12,15,17·CCZ12,15,18·CCZ12,15,19·CCZ12,15,20·CCZ12,15,21·CCZ12,16,17·CCZ12,16,18·CCZ12,16,19·CCZ12,16,20·CCZ12,16,21·CCZ12,17,18·CCZ12,17,19·CCZ12,17,20·CCZ12,17,21·CCZ12,18,19·CCZ12,18,20·CCZ12,18,21·CCZ12,19,20·CCZ12,19,21·CCZ12,19,26·CCZ12,20,21·CCZ12,20,25·CCZ12,21,24·CCZ12,24,36·CCZ12,25,35·CCZ12,26,34·CCZ13,14,15·CCZ13,14,16·CCZ13,14,17·CCZ13,14,18·CCZ13,14,19·CCZ13,14,20·CCZ13,14,21·CCZ13,15,16·CCZ13,15,17·CCZ13,15,18·CCZ13,15,19·CCZ13,15,20·CCZ13,15,21·CCZ13,16,17·CCZ13,16,18·CCZ13,16,19·CCZ13,16,20·CCZ13,16,21·CCZ13,17,18·CCZ13,17,19·CCZ13,17,20·CCZ13,17,21·CCZ13,17,26·CCZ13,18,19·CCZ13,18,20·CCZ13,18,21·CCZ13,18,25·CCZ13,19,20·CCZ13,19,21·CCZ13,20,21·CCZ13,21,23·CCZ13,23,36·CCZ13,25,33·CCZ13,26,32·CCZ14,15,16·CCZ14,15,17·CCZ14,15,18·CCZ14,15,19·CCZ14,15,20·CCZ14,15,21·CCZ14,16,17·CCZ14,16,18·CCZ14,16,19·CCZ14,16,20·CCZ14,16,21·CCZ14,16,26·CCZ14,17,18·CCZ14,17,19·CCZ14,17,20·CCZ14,17,21·CCZ14,18,19·CCZ14,18,20·CCZ14,18,21·CCZ14,18,24·CCZ14,19,20·CCZ14,19,21·CCZ14,20,21·CCZ14,20,23·CCZ14,23,35·CCZ14,24,33·CCZ14,26,31·CCZ15,16,17·CCZ15,16,18·CCZ15,16,19·CCZ15,16,20·CCZ15,16,21·CCZ15,16,25·CCZ15,17,18·CCZ15,17,19·CCZ15,17,20·CCZ15,17,21·CCZ15,17,24·CCZ15,18,19·CCZ15,18,20·CCZ15,18,21·CCZ15,19,20·CCZ15,19,21·CCZ15,19,23·CCZ15,20,21·CCZ15,23,34·CCZ15,24,32·CCZ15,25,31·CCZ16,17,18·CCZ16,17,19·CCZ16,17,20·CCZ16,17,21·CCZ16,18,19·CCZ16,18,20·CCZ16,18,21·CCZ16,19,20·CCZ16,19,21·CCZ16,20,21·CCZ16,21,22·CCZ16,22,36·CCZ16,25,30·CCZ16,26,29·CCZ17,18,19·CCZ17,18,20·CCZ17,18,21·CCZ17,19,20·CCZ17,19,21·CCZ17,20,21·CCZ17,20,22·CCZ17,22,35·CCZ17,24,30·CCZ17,26,28·CCZ18,19,20·CCZ18,19,21·CCZ18,19,22·CCZ18,20,21·CCZ18,22,34·CCZ18,24,29·CCZ18,25,28·CCZ19,20,21·CCZ19,22,33·CCZ19,23,30·CCZ19,26,27·CCZ20,22,32·CCZ20,23,29·CCZ20,25,27·CCZ21,22,31·CCZ21,23,28·CCZ21,24,27
As monomials: 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(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
With every rotation a \mathrm{T}=\mathrm{diag}(1,e^{i\pi/4}), the rotations deposit the gate above times a diagonal Clifford. Undo it after the rotations and before the checks are measured, with 22 single-wire gates: \mathrm{S} on output 0, \mathrm{S} on output 1, \mathrm{S} on output 2, \mathrm{S} on output 3, \mathrm{S} on output 4, \mathrm{S} on output 5, \mathrm{S} on output 6, \mathrm{S} on output 7, \mathrm{S} on output 8, \mathrm{S} on output 9, \mathrm{S} on output 10, \mathrm{S} on output 11, \mathrm{S} on output 12, \mathrm{S} on output 13, \mathrm{S} on output 14, \mathrm{S} on output 15, \mathrm{S} on output 16, \mathrm{S} on output 17, \mathrm{S} on output 18, \mathrm{S} on output 19, \mathrm{S} on output 20, \mathrm{S} on output 21. The circuit's action is then exactly the gate above, and an error-free run passes every check with certainty.
These Clifford gates cannot be avoided. No choice of \mathrm{T}, \mathrm{T}^3, \mathrm{T}^5 or \mathrm{T}^\dagger for each rotation removes them; the catalogue proves this by showing that the linear system over \mathbb{Z}_4 the powers would have to solve has no solution.
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/127.37.3.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-127.37.3.a.txt", "w") as out:
out.write("\n".join(lines) + "\n")
Distance
d=3, exact. No fault of weight below 3 is both undetectable and damaging. One of weight 3 is.
Exactly 2,667 faults of weight 3 are undetectable and damaging, so A_{3}=2{,}667. If each rotation fails independently with probability p, an accepted run has a damaged output with probability A_{3}\,p^{3}+O(p^{4}).
A damaging fault of weight 3 sits on columns [0,1,2].
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 the 48 < n < 128 AI campaigns, from punctured r=7 simplex parents with free-row appending; a verified witness, not a maximum
Found by: AI search: punctured r=7 simplex parents
Discovery: AI search
1 recorded source
[
{
"regime": "AI search: punctured r=7 simplex parents",
"file": "magic-states-AI/campaign_48to128/found/n127_k37_web.json",
"label": "8",
"N": 44,
"d": 3,
"d_is_exact": true,
"origin": null,
"provenance": "n127_k37_web.json[0]"
}
]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/127.37.3.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[741].