The circuit
The 98\times511 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 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 |
|---|---|
| out 0 (7) | 1111111························································································································································································································································································································································································································································································································ |
| out 1 (7) | ········1········1········1········1········1········1········1································································································································································································································································································································································································································································ |
| out 2 (7) | ················1················1················1················1················1················1················1········································································································································································································································································································································································································ |
| out 3 (7) | ························1·················1······1····························1······1·············1························1·································································································································································································································································································································································································· |
| out 4 (7) | ································1································1································1································1································1································1································1························································································································································································································································································ |
| out 5 (7) | ········································1········································1········································1·················1······················1··························································1······················1········································································································································································································································································· |
| out 6 (7) | ················································1·································1··············1····················································1··············1·····························1················································1·········································································································································································································································································· |
| out 7 (7) | ························································1·················1······································1···································1························1·············································1······1··························································································································································································································································································· |
| out 8 (7) | ································································1································································1································································1································································1································································1································································1································································1························································ |
| out 9 (7) | ········································································1········································································1········································································1········································································1········································································1········································································1········································································1 |
| out 10 (7) | ················································································1················································································1················································································1·································1··············································1··················································································································1··············································1························· |
| out 11 (7) | ························································································1························································································1························································1·················································································1······································1··········································1······································································1········································· |
| out 12 (7) | ································································································1·································································1······························1····································································································1······························1·····························································1································································································1·························· |
| out 13 (7) | ········································································································1·················································································1······················1············································································1······················································································1·············································1········································1·················································· |
| out 14 (7) | ················································································································1·································1··············································································1···································································1················································1·····························································································1··············1··························································· |
| out 15 (7) | ························································································································1·················1······································································································1···································1························································································1·············································1······································································1··························· |
| out 16 (7) | ···························1····································································································1···························1····································································································1···························1····································································································1···························1································································································ |
| out 17 (7) | ···························································1············································································1···········································1····························································································1···························1············································································································1···········1························································································ |
| out 18 (7) | ···························································································1····················································1···························································1····················································································1···························································································1····················································1···························································1················ |
| out 19 (7) | ···························································································································1····························1···········································································1············································································1···························1····························································································1···········································1········································ |
| out 20 (7) | ····························································1······························································································1····1································································································································································1····························································1······························································································1····1···························· |
| out 21 (7) | ····················1···················································································································································1··················1··········································································································································1··········1···························································································································································1············1···· |
| out 22 (7) | ············································································································1···································································1··········································1··················································1··················································································1···························································································1····················1············································ |
| out 23 (7) | ····································································1···················································································································1··································································1··························································1··························································1···························1····························································1···················································· |
| out 24 (7) | ······························································································1······························································1··································1··························································································1······································1······························································1··························································································1·································· |
| out 25 (7) | ······································································································1······································································1··························1··················································································································1······························1······················································1··································································································1·········· |
| out 26 (7) | ··············································1·································································································································································1············································1·····························································································1······················1·························1····················1····························································································· |
| out 27 (7) | ······················1······················································································································································································1··········1·················································································································································1················1········································1············1············································································· |
| out 28 (7) | ·····························································································1································································································1·································1·································································1·························································································1······························1·····································1····························································· |
| out 29 (7) | ·············································································································1························1·································································································1·················································································1·························1······································································································1·····················1············································· |
| out 30 (7) | ·····························································1················································································································································1·································1·································1·························1··············································································································································································1·····1····························· |
| out 31 (7) | ·············1········································································································································································································································1·1···········1·····1······································································································································································································································1·········1··· |
| out 32 (7) | ·························1···················1·····1············································································································································································································1·························1···················1·····1·········································································································································································································· |
| out 33 (7) | ·········1·········1·········1····································································································································································································································1·····1·············1·····1·································································································································································································································· |
| out 34 (7) | ·························································1···················1·····································1····························································································································································1·························1···················································1·····1·························································································································································· |
| out 35 (7) | ·········································1·········································1·········································1··························································································································································1·························1·························1·························1························································································································································ |
| out 36 (7) | ·························································································1·························································································1·························································1··················································1·························································································1·························1·························································1················································ |
| out 37 (7) | ·········································································1·········································································1·········································································1··········································································1·························································1·························································································1·························································1········ |
| out 38 (7) | ·························································································································1···················1·····································································································1····························································1·························1···················································································································1·····1·························································· |
| out 39 (7) | ·········································································································1···················································································1·····················1····································································································1·························1···················································1·····································································································1·················· |
| out 40 (7) | ····················································································································1····································1····················································································1····································································1············1············································································································1············································1···································· |
| out 41 (7) | ····························································································1············································1············································································1····························································1····················································1····················································································1····································1···························································· |
| out 42 (7) | ····································1·························································································································1··························1······················································································································································1··································1·························································································1····························1···················· |
| out 43 (7) | ············1·························································································································································1··1·········································································································································································1····1·························································································································································1··········1· |
| out 44 (7) | ··············1·····································································································································································································1····1······································································································································1············1·····································································1······1···································································· |
| out 45 (7) | ······················································1··················································································································································1··················································1································································································1··········1··········································1··············1···························································································· |
| out 46 (7) | ······························································································································1·····1····················································································································1···················1··································································································1·························1········································································································1··········· |
| out 47 (7) | ······································································1·····································································································1····························································1···················································································1··························································1·························1································································1··········································· |
| out 48 (7) | ···········································1··············································································································1···························1··································································································1···························1··········································································1···········································1·················································································· |
| out 49 (7) | ···········1······················································································································1···········1······················································································································1···1··························································································································1···1······················································································································ |
| out 50 (7) | ···········································································································1······························································1···························1··················································································································1···························1··························································1···········································································································1·· |
| out 51 (7) | ···········································································1······································································································1···········································································1··········································1···························································1··················································1···························································1·········································· |
| out 52 (7) | ······················································································1····················································································1··············································································1·················1············································································1······································································1····················································································1········· |
| out 53 (7) | ··············································································································1····························1······················································································1·································································1····································1······························································································1····························1························································· |
| out 54 (7) | ······································1···································································································································································1································1················································································································1····························1···························1··························1·············································································· |
| out 55 (7) | ······························1············································································································································································1······1······················································································································································1··········1················································1··················1······································································ |
| out 56 (7) | ··························································································1··························································································1······················································1······························································1··········································································1··································1······································1······························································ |
| out 57 (7) | ··································································1··································································1······························································1······································································1··································································1··························································1······························································1······················································ |
| out 58 (7) | ··········································································································1·················································································1························1················································1····································································································1·············································································1······················1·············································· |
| out 59 (7) | ··················································································································1·································1················································································1························································································1············1·····························································································1··············································1······································ |
| out 60 (7) | ··························································1·················1········································1································································································································································································································································································1····················1·············································1······1······························ |
| out 61 (7) | ··································1··································1······························1······································································································································································································································································································1··································1··························1······························1······················ |
| out 62 (7) | ··········1··········1······1·········································································································································································································································································································································································································································································1····1····1········1····· |
| out 63 (7) | ··················1··················1··············1······················································································································································································································································································································································································································1··················1··········1··············1······ |
| out 64 (43) | ··········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·············· |
| out 65 (27) | ··················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······ |
| out 66 (51) | ············1·····1···1·······1···1···1·············1·············1·1·1·····························1·································································1·····1·····1·····1···········1··1··1···1···1·····1·······················1·····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······ |
| out 67 (35) | ··········1·······1····1····1·1···1·················1·················1·····························1·································1·····································1·································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 (51) | ··········1·1·········1·····1·········1···················1···········1·····1·····························1·······1·····1·············1···1·········1··········1······1·····1···············1·············1·············1···············1·············1·1···········1·1···1····································1········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 69 (51) | ······················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·················· |
| out 70 (43) | ··········1·················1·1···1···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·············· |
| out 71 (51) | ··········1···········1·····1·····1···1···················1·········1·······1·······················1·····1·······1····11·················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·············· |
| out 72 (35) | ··················1···········1·····················1·1························1··························1···1···1·····1·················1·········1·······································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 (19) | ··················1····························1····1·····1·······1·········1·····························1·················································································1··1····1··········································1······················1·······································································1·······································································1·················································1·······1·············1·1·······················1······ |
| out 74 (51) | ··········1·1·········1·····1·1···································1·1·1···································1···1·········1··············1··1···························1·····1···········1···1·······1·············1·····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············ |
| out 75 (51) | ·······1··············1···········1···1···············1···········1···1·····························1·····1·············1·····1·····1·1···1·································1·····1·········1·······1·····1···11·······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······················ |
| out 76 (51) | ··········1·1·········1·····1·1·······1···················1·········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············ |
| out 77 (59) | ··········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·················1·····························1···········1·1···1·1·······1·················1·······1···1·1···1·····1·1···········1·······························1···1···1·····1·1·····1·········1·······1······ |
| out 78 (35) | ··········1·······1·········1·1·······1·············1···············1··1··································1·······································································1·····1···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 (249) | 11·1··1·1·1·1·111·11·1·1·11··1······1·1·11··1·11··111·1·111·111·1111····1····1·1·······1·1·11·····1··1·········1·1···1·11·11·1·1····1····1··1··1111···11·1···1··1····1111··1·11··1·····111·111··1·1111··1·1····1···111·1······1··1·11·11··1··11111··1·11111111·11··11111·111·11·1··1··1····11·11·1··111111·11·1···1······1·1··11··1··1·1·1·1·1·······11·1·111··1·11·111111·1111·1··1111···1···1····1111111·1111·1·······11·····11·11·1·111·1·11·1··1·1··1···11·11·1··1·1·1111·1···1·11111·1····111111···1111111·1·1··111··1·1·· |
| out 80 (241) | 1·11·1··11·111···1···1··11··11··1·1··11··111111·····11···1···1·1··1·11··11··11····1··11···1111···111···111··11··1····1·····1·111·1·1·1····1·1·1·····11··1·1··1···1·1···1·111111·1··1····1···11··1·11·11·111·111··1··111·111·111·11·1·1·111··11···11·11··11··11······11··1····1···11·11··111·1···11·111···111·11·11·1····11···1··1··1····111·1·1·····11··11···111111··1··11···11··11·11··111·1····1··11···11··1···1···1·1111·····11··111·11··1111·11·11·111···11··11·111··1··1··111··1·11111·1···11·····1·1··1···11··1·1111···1· |
| out 81 (239) | ···1111··11·1·111···1····1··11··1··1····1·1·11··111··11·1111··1··11······1·111··11·11·1··1·11·1··1·1···1·1··1·1··1····1·11·1111·111···1····1·11··1·1·1···1··11····1111·1·1111···111·1···1··1··1··11·111··1111···1·111·1···1·1·1·1···1···1111·11··11·1····1·1·1·11·11···1·····111·····1111······1111·1111··1··1·1·11·1··1·11·11·1·11111111·1···11··1·111111·1·11·1·11·1·1·······1··1····11···1·11·11·1··11·1··1·1····11111···1·111·1··111··1··1·11·1·11··1·1·1·11··11·1·11·1·······1·······1··1·11·1··111····11·11······1·····11 |
| out 82 (253) | 11··11··1111··1·1·111···1·1·1··1·11111·1·11····1·····1·11·11·1·11·11·1·1····1·1111·····1111·11···········1111·1·111···1···111111··11····11111···1·1·11···111···11··11··1··1·11··11·1·11111·1·1·1··1·11·11·1·11·1····11·11···1···1···1·11·1·1··1·111······1·1···1·111·111····11·1111·11··11··11···1···11··11·····1··1·1····1·1···1····1··1111111·111111··11·1··11·1··11·1·····111··1···11··1·11····11·1111·1·11·1111·11··11···11·11··1·1·11·1··1111·1··11·1·1·11··1·11···1111·1·1·1·1111··1·1·1·1··1·1·11·······11·1·11·11·1··1· |
| out 83 (279) | 11·1··11·11·11···1·111···111····111·1·11·1·1··11··11·1·1·11·1·111·1··11··1111···1111··111·111111·11··1··11111·1·11·1··11··1····111····11··1·1··1····1·1···1·····11·11111···1····11·1·1···1·11·111···1·1··1·11·1··1111·11·1··11·1·111·11111·1·1··11·····11···111·111111·1·1···111·11·11·111··1···1·11··111·1·111111····1·11·1··11·11··1··1111·11··1·1··11·11111·11··1111·11·1111·11·111111·1··1·1···11·111·1·111111···111·11···1·1·····11··1·1·1······1·1·1·11·11·111111·11111····1111·11····11111···11···111··1·11·11···1·1··11 |
| out 84 (255) | 1·11·1·····1·1·11··1···1·····1111·1···111·········1····1·111·1·1·111·11·1111·1·11·1····11·1·11·1·1···1·1·1·11····11·11···111···1····1·11··1·111··111··1·1111·1··1·1·11····1·11·11·1·1111····1·1111··11·111····1···11·11·····1·1··1·1·1·11·11···1·11···11·1··1··1111111111···1·1··11111·1··11·11·1·1·111···11··11·1··1·11·1····111·1·1111·11···1··111··1·11···11···11·1··1··11·11·1··11·1111····11·1·11····11·1·1·11·1·111·1··1111··1···111··11··1····11·1··11·1·11···11·1·111·1··11··11··1···111·11····111··11····1111··1·1111· |
| out 85 (265) | 11··11·1··11···11··1·1·11····111·1111···111·1··11····1··11·111111·11····1·111·11·1······1·1··11··1··1·1··111··1···11·111···111·1·1·111·1·1··1·1·11·····1·······111··11··1···1·1··11·1····11111·11····11·1··1111111····1·1···1···1·1·111··1··1·····1·111111···1··11·11·11·11·1·1·1···111··11111·1·1··111·1··11111·11·11··1·11·1·1···1·1·····1··11····111··1·1111·1111·11111··111···11···111··111·1111···1111···1··1·1·11111··11111·111·1·1·111·1·11·111··1·1111·1·111··1····11···111····11··1··1···111·11111·11·····1·111111·1·· |
| out 86 (269) | 1·1·1·111·111·1·1··111·11·1··11111111···1··1·111·1·1·11·1·1·11·····111·11···11···11··1·1111·1111·11·····1·1111·111111·1·11··1····111·1111····1··1···1··1·111···1·1·1··1······111111·11···111·11·1·111111111·1·1·1·111111··1···11·1·1···1···1·1·1·11·1······1··1·11·111·1··1···1··111·111111·11·1··111·1······1·11·11····1··1·11··11·11·1·1··1····1111111··1···1·1··11·1·1·11···1·111111·1·11···1··11···111··1·1···11·1·111··1111···11···1··11·1·111··1111·11·····1·111·1111···1·1·111··1111·1··1·11·11····11··1111········1111· |
| out 87 (251) | ·1111··111111·1··11·1·1··1·111·1·1··1·1····111··1·1·11··1·1·1··111···111··11···11·1···1·1111·111··1·111··11··11·····11111··1··11···1·····11···1···1111·1·1·1111·····111·1·1·11····1···1······111··11··1··1··11··1····111·1···1···11111·11·1111·1··1···1·111·11·11111·····1··111··1·1111111·1····111···111··1·1·11·111····11111·1·····1··111·11···11··1·11········11··111··11·1·1···1·····1····1·1·11·1111·111·11···11·11111····1111···1··1····1·111·1·11··11·11··1·111·1···1·1·11·11··1·111·11·1···11·11·1···1··111·1··1···1··· |
| out 88 (269) | ·1111··1·111111··11·11···111·111·1·111···1·111····1····1111·11··1···1··11···111111···1··11111··1···1·11111·1··1·11··1·11111·11···1·1111·1111·11·1··1··11·111111··11·111···11·111··1·1······1··1111·111···1·1··1··1·1···1·11··11·11·1··1·1·1·11·1····11··11111·111·1·11·····1··1·1·····1··1···1··1·1··1·1····1··1·1111·1··1111·11·1·11···1111·11·11111··1····11··11··1·111·1·1·11····111··1·111·1·1·111·1··1··111····11··1·1·11111···111·11111···11111·11····111·1······1····1··1··1·1·1111111·11··1·1····1111···11·1111···1·111 |
| check 89 (256) | 1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1 |
| check 90 (256) | ·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··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··11··11 |
| check 91 (256) | ···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····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····1111····1111 |
| check 92 (256) | ·······11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111 |
| check 93 (256) | ···············1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111 |
| check 94 (256) | ·······························11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111 |
| check 95 (256) | ·······························································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111 |
| check 96 (256) | ·······························································································································11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111································································································································11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 |
| check 97 (256) | ·······························································································································································································································································1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 |
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
Output gate
As gates: \mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2}\cdot \mathrm{T}_{3}\cdot \mathrm{T}_{4}\cdot \mathrm{T}_{5}\cdot \mathrm{T}_{6}\cdot \mathrm{T}_{7}\cdot \mathrm{T}_{8}\cdot \mathrm{T}_{9}\cdot \mathrm{T}_{10}\cdot \mathrm{T}_{11}\cdot \mathrm{T}_{12}\cdot \mathrm{T}_{13}\cdot \mathrm{T}_{14}\cdot \mathrm{T}_{15}\cdot \mathrm{T}_{16}\cdot \mathrm{T}_{17}\cdot \mathrm{T}_{18}\cdot \mathrm{T}_{19}\cdot \mathrm{T}_{20}\cdot \mathrm{T}_{21}\cdot \mathrm{T}_{22}\cdot \mathrm{T}_{23}\cdot \mathrm{T}_{24}\cdot \mathrm{T}_{25}\cdot \mathrm{T}_{26}\cdot \mathrm{T}_{27}\cdot \mathrm{T}_{28}\cdot \mathrm{T}_{29}\cdot \mathrm{T}_{30}\cdot \mathrm{T}_{31}\cdot \mathrm{T}_{32}\cdot \mathrm{T}_{33}\cdot \mathrm{T}_{34}\cdot \mathrm{T}_{35}\cdot \mathrm{T}_{36}\cdot \mathrm{T}_{37}\cdot \mathrm{T}_{38}\cdot \mathrm{T}_{39}\cdot \mathrm{T}_{40}\cdot \mathrm{T}_{41}\cdot \mathrm{T}_{42}\cdot \mathrm{T}_{43}\cdot \mathrm{T}_{44}\cdot \mathrm{T}_{45}\cdot \mathrm{T}_{46}\cdot \mathrm{T}_{47}\cdot \mathrm{T}_{48}\cdot \mathrm{T}_{49}\cdot \mathrm{T}_{50}\cdot \mathrm{T}_{51}\cdot \mathrm{T}_{52}\cdot \mathrm{T}_{53}\cdot \mathrm{T}_{54}\cdot \mathrm{T}_{55}\cdot \mathrm{T}_{56}\cdot \mathrm{T}_{57}\cdot \mathrm{T}_{58}\cdot \mathrm{T}_{59}\cdot \mathrm{T}_{60}\cdot \mathrm{T}_{61}\cdot \mathrm{T}_{62}\cdot \mathrm{T}_{63}\cdot \mathrm{T}_{64}\cdot \mathrm{T}_{65}\cdot \mathrm{T}_{66}\cdot \mathrm{T}_{67}\cdot \mathrm{T}_{68}\cdot \mathrm{T}_{69}\cdot \mathrm{T}_{70}\cdot \mathrm{T}_{71}\cdot \mathrm{T}_{72}\cdot \mathrm{T}_{73}\cdot \mathrm{T}_{74}\cdot \mathrm{T}_{75}\cdot \mathrm{T}_{76}\cdot \mathrm{T}_{77}\cdot \mathrm{T}_{78}\cdot \mathrm{T}_{79}\cdot \mathrm{T}_{80}\cdot \mathrm{T}_{81}\cdot \mathrm{T}_{82}\cdot \mathrm{T}_{83}\cdot \mathrm{T}_{84}\cdot \mathrm{T}_{85}\cdot \mathrm{T}_{86}\cdot \mathrm{T}_{87}\cdot \mathrm{T}_{88}
As monomials: 0+1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38+39+40+41+42+43+44+45+46+47+48+49+50+51+52+53+54+55+56+57+58+59+60+61+62+63+64+65+66+67+68+69+70+71+72+73+74+75+76+77+78+79+80+81+82+83+84+85+86+87+88 (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 S_k-canonical frame.
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 86 single-wire gates and 173 \mathrm{CZ} 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, \mathrm{S} on output 22, \mathrm{S} on output 23 and 235 more (listed below). The circuit's action is then exactly the gate above, and an error-free run passes every check with certainty.
All 259 correction 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, \mathrm{S} on output 22, \mathrm{S} on output 23, \mathrm{S} on output 24, \mathrm{S} on output 25, \mathrm{S} on output 26, \mathrm{S} on output 27, \mathrm{S} on output 28, \mathrm{S} on output 29, \mathrm{S} on output 30, \mathrm{S} on output 31, \mathrm{S} on output 32, \mathrm{S} on output 33, \mathrm{S} on output 34, \mathrm{S} on output 35, \mathrm{S} on output 36, \mathrm{S} on output 37, \mathrm{S} on output 38, \mathrm{S} on output 39, \mathrm{S} on output 40, \mathrm{S} on output 41, \mathrm{S} on output 42, \mathrm{S} on output 43, \mathrm{S} on output 44, \mathrm{S} on output 45, \mathrm{S} on output 46, \mathrm{S} on output 47, \mathrm{S} on output 48, \mathrm{S} on output 49, \mathrm{S} on output 50, \mathrm{S} on output 51, \mathrm{S} on output 52, \mathrm{S} on output 53, \mathrm{S} on output 54, \mathrm{S} on output 55, \mathrm{S} on output 56, \mathrm{S} on output 57, \mathrm{S} on output 58, \mathrm{S} on output 59, \mathrm{S} on output 60, \mathrm{S} on output 61, \mathrm{S} on output 62, \mathrm{S} on output 63, \mathrm{S}^\dagger on output 64, \mathrm{S}^\dagger on output 65, \mathrm{S}^\dagger on output 66, \mathrm{S}^\dagger on output 67, \mathrm{S}^\dagger on output 68, \mathrm{S}^\dagger on output 69, \mathrm{S}^\dagger on output 70, \mathrm{S}^\dagger on output 71, \mathrm{S}^\dagger on output 72, \mathrm{S}^\dagger on output 73, \mathrm{S}^\dagger on output 74, \mathrm{S}^\dagger on output 75, \mathrm{S}^\dagger on output 76, \mathrm{S}^\dagger on output 77, \mathrm{S}^\dagger on output 78, \mathrm{S} on output 81, \mathrm{Z} on output 82, \mathrm{S} on output 83, \mathrm{S} on output 84, \mathrm{Z} on output 86, \mathrm{S}^\dagger on output 87, \mathrm{Z} on output 88, \mathrm{CZ} on output 64, output 82, \mathrm{CZ} on output 64, output 83, \mathrm{CZ} on output 64, output 84, \mathrm{CZ} on output 64, check 92, \mathrm{CZ} on output 64, check 94, \mathrm{CZ} on output 65, output 79, \mathrm{CZ} on output 65, output 81, \mathrm{CZ} on output 65, output 84, \mathrm{CZ} on output 65, output 85, \mathrm{CZ} on output 65, output 86, \mathrm{CZ} on output 65, output 87, \mathrm{CZ} on output 65, output 88, \mathrm{CZ} on output 65, check 94, \mathrm{CZ} on output 65, check 96, \mathrm{CZ} on output 65, check 97, \mathrm{CZ} on output 66, output 81, \mathrm{CZ} on output 66, output 84, \mathrm{CZ} on output 66, check 92, \mathrm{CZ} on output 66, check 93, \mathrm{CZ} on output 66, check 94, \mathrm{CZ} on output 66, check 97, \mathrm{CZ} on output 67, output 79, \mathrm{CZ} on output 67, output 81, \mathrm{CZ} on output 67, output 86, \mathrm{CZ} on output 67, output 87, \mathrm{CZ} on output 67, output 88, \mathrm{CZ} on output 67, check 92, \mathrm{CZ} on output 67, check 95, \mathrm{CZ} on output 67, check 96, \mathrm{CZ} on output 67, check 97, \mathrm{CZ} on output 68, output 82, \mathrm{CZ} on output 68, output 83, \mathrm{CZ} on output 68, output 87, \mathrm{CZ} on output 68, output 88, \mathrm{CZ} on output 68, check 92, \mathrm{CZ} on output 68, check 94, \mathrm{CZ} on output 68, check 96, \mathrm{CZ} on output 69, output 83, \mathrm{CZ} on output 69, output 84, \mathrm{CZ} on output 69, output 86, \mathrm{CZ} on output 69, output 87, \mathrm{CZ} on output 69, output 88, \mathrm{CZ} on output 69, check 94, \mathrm{CZ} on output 69, check 95, \mathrm{CZ} on output 70, output 81, \mathrm{CZ} on output 70, output 82, \mathrm{CZ} on output 70, output 85, \mathrm{CZ} on output 70, output 86, \mathrm{CZ} on output 70, output 87, \mathrm{CZ} on output 70, output 88, \mathrm{CZ} on output 70, check 92, \mathrm{CZ} on output 70, check 93, \mathrm{CZ} on output 70, check 97, \mathrm{CZ} on output 71, output 82, \mathrm{CZ} on output 71, output 83, \mathrm{CZ} on output 71, output 84, \mathrm{CZ} on output 71, output 85, \mathrm{CZ} on output 71, output 87, \mathrm{CZ} on output 71, output 88, \mathrm{CZ} on output 71, check 95, \mathrm{CZ} on output 71, check 96, \mathrm{CZ} on output 72, output 81, \mathrm{CZ} on output 72, output 82, \mathrm{CZ} on output 72, output 83, \mathrm{CZ} on output 72, output 85, \mathrm{CZ} on output 72, output 87, \mathrm{CZ} on output 72, output 88, \mathrm{CZ} on output 72, check 92, \mathrm{CZ} on output 72, check 93, \mathrm{CZ} on output 72, check 94, \mathrm{CZ} on output 72, check 95, \mathrm{CZ} on output 72, check 96, \mathrm{CZ} on output 72, check 97, \mathrm{CZ} on output 73, output 79, \mathrm{CZ} on output 73, output 83, \mathrm{CZ} on output 73, check 93, \mathrm{CZ} on output 73, check 96, \mathrm{CZ} on output 74, output 79, \mathrm{CZ} on output 74, output 83, \mathrm{CZ} on output 74, check 93, \mathrm{CZ} on output 74, check 96, \mathrm{CZ} on output 75, output 82, \mathrm{CZ} on output 75, output 84, \mathrm{CZ} on output 75, output 85, \mathrm{CZ} on output 75, output 86, \mathrm{CZ} on output 75, output 87, \mathrm{CZ} on output 75, output 88, \mathrm{CZ} on output 75, check 94, \mathrm{CZ} on output 76, output 79, \mathrm{CZ} on output 76, output 81, \mathrm{CZ} on output 76, output 82, \mathrm{CZ} on output 76, output 83, \mathrm{CZ} on output 76, output 84, \mathrm{CZ} on output 76, output 86, \mathrm{CZ} on output 76, output 87, \mathrm{CZ} on output 76, output 88, \mathrm{CZ} on output 76, check 94, \mathrm{CZ} on output 76, check 95, \mathrm{CZ} on output 76, check 96, \mathrm{CZ} on output 76, check 97, \mathrm{CZ} on output 78, output 81, \mathrm{CZ} on output 78, output 82, \mathrm{CZ} on output 78, output 83, \mathrm{CZ} on output 78, output 85, \mathrm{CZ} on output 78, output 87, \mathrm{CZ} on output 78, output 88, \mathrm{CZ} on output 78, check 92, \mathrm{CZ} on output 78, check 93, \mathrm{CZ} on output 78, check 94, \mathrm{CZ} on output 78, check 95, \mathrm{CZ} on output 78, check 96, \mathrm{CZ} on output 78, check 97, \mathrm{CZ} on output 79, output 82, \mathrm{CZ} on output 79, output 83, \mathrm{CZ} on output 79, output 84, \mathrm{CZ} on output 79, output 85, \mathrm{CZ} on output 79, output 86, \mathrm{CZ} on output 79, check 90, \mathrm{CZ} on output 79, check 91, \mathrm{CZ} on output 79, check 94, \mathrm{CZ} on output 79, check 97, \mathrm{CZ} on output 80, output 81, \mathrm{CZ} on output 80, output 82, \mathrm{CZ} on output 80, output 84, \mathrm{CZ} on output 80, output 86, \mathrm{CZ} on output 80, check 89, \mathrm{CZ} on output 80, check 95, \mathrm{CZ} on output 80, check 96, \mathrm{CZ} on output 81, output 83, \mathrm{CZ} on output 81, output 84, \mathrm{CZ} on output 81, output 86, \mathrm{CZ} on output 81, output 87, \mathrm{CZ} on output 81, check 89, \mathrm{CZ} on output 81, check 90, \mathrm{CZ} on output 81, check 92, \mathrm{CZ} on output 82, output 84, \mathrm{CZ} on output 82, output 88, \mathrm{CZ} on output 82, check 89, \mathrm{CZ} on output 82, check 90, \mathrm{CZ} on output 82, check 91, \mathrm{CZ} on output 82, check 92, \mathrm{CZ} on output 82, check 93, \mathrm{CZ} on output 82, check 95, \mathrm{CZ} on output 82, check 97, \mathrm{CZ} on output 83, check 89, \mathrm{CZ} on output 83, check 91, \mathrm{CZ} on output 83, check 93, \mathrm{CZ} on output 83, check 95, \mathrm{CZ} on output 83, check 96, \mathrm{CZ} on output 83, check 97, \mathrm{CZ} on output 84, output 85, \mathrm{CZ} on output 84, output 87, \mathrm{CZ} on output 84, check 91, \mathrm{CZ} on output 84, check 94, \mathrm{CZ} on output 85, output 88, \mathrm{CZ} on output 85, check 90, \mathrm{CZ} on output 85, check 91, \mathrm{CZ} on output 85, check 93, \mathrm{CZ} on output 85, check 94, \mathrm{CZ} on output 85, check 95, \mathrm{CZ} on output 85, check 96, \mathrm{CZ} on output 86, check 89, \mathrm{CZ} on output 86, check 93, \mathrm{CZ} on output 86, check 95, \mathrm{CZ} on output 86, check 96, \mathrm{CZ} on output 87, check 90, \mathrm{CZ} on output 87, check 91, \mathrm{CZ} on output 87, check 92, \mathrm{CZ} on output 87, check 93, \mathrm{CZ} on output 87, check 97, \mathrm{CZ} on output 88, check 89, \mathrm{CZ} on output 88, check 95, \mathrm{CZ} on output 88, check 97
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/511.89.3.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-511.89.3.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 43,435 faults of weight 3 are undetectable and damaging, so A_{3}=43{,}435. 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
widest pure-T frames held at n=511. DOMINATED on rate by the derived direct-sum baseline [[508,92,3]] = 4 x [[127,23,3]], rho = 508/92 = 5.5217. These are WIDTH points, explicitly NOT rate records: beating the baseline at n=511 needs k >= 93.
Found by: AI search: full-simplex pure-T frames
Discovery: AI search
1 recorded source
[
{
"regime": "AI search: full-simplex pure-T frames",
"file": "campaigns/2026-09-15_255_511_rate/found/n511_k89_d3_DOMINATED_not_a_record.json",
"label": "0",
"N": 98,
"d": 3,
"d_is_exact": true,
"origin": null,
"provenance": "magic-states-AI campaigns/2026-09-15_255_511_rate. gab64 Gabidulin scaffold at r=9 plus rows supplied by the T-R1 reduction (a partial LINE spread of the relaxed rank-6 frame), finished with 10 generic Phi-drop-1 rows (tools/hybrid.py, 15-line seed, certificate cap 91). Supersedes nothing; the campaign's own [[255,43,3]] and [[511,81,3]] predecessors remain valid factories and remain dominated."
}
]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/511.89.3.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[1110].