{"id":"212.40.3.a","row":858,"stored":{"file":"master_catalog.json","row":858,"construction":null,"columns_sha256":null},"parameters":{"n":212,"k":40,"d":3,"N":54,"r":14,"level":3},"gate":{"string":"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","human":"T0\u00b7T1\u00b7T2\u00b7T3\u00b7T4\u00b7T5\u00b7T6\u00b7T7\u00b7T8\u00b7T9\u00b7T10\u00b7T11\u00b7T12\u00b7T13\u00b7T14\u00b7T15\u00b7T16\u00b7T17\u00b7T18\u00b7T19\u00b7T20\u00b7T21\u00b7T22\u00b7T23\u00b7T24\u00b7T25\u00b7T26\u00b7T27\u00b7T28\u00b7T29\u00b7T30\u00b7T31\u00b7T32\u00b7T33\u00b7T34\u00b7T35\u00b7T36\u00b7T37\u00b7T38\u00b7T39","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]],"sk_key":[[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]],"sk_fingerprint":"c116f6e2037fc6d7","sk_canonical_frame":true,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[21,47],[18,40,41,47,48],[0,47,48,49],[2,4,8,9,10,11,12,17,20,22,24,26,30,35,39,43,44,46,48,49,51,52],[0,3,4,5,6,7,9,16,19,21,22,23,30,31,32,33,35,36,37,38,39,43,48,50,51,52],[14,47,48,49,50],[25,47,51],[34,40,41,44,47,48,51],[3,4,5,10,12,14,15,18,19,21,22,23,25,26,29,30,31,32,33,35,37,38,39,41,42,43,46,49,52],[0,4,6,7,8,11,17,18,22,24,28,30,34,38,39,42,43,44,46,51,52,53],[26,40,43,44,47,50,51],[0,3,9,13,15,16,18,20,21,26,28,29,35,36,38,41,42,44,46,47,48,50,53],[9,40,42,43,44,47,49,50,51],[3,47,48,49,50,51],[14,42,43,45,47,48,49,50],[0,1,3,4,6,7,8,10,11,12,15,16,19,20,21,23,24,26,27,28,31,32,37,40,43,44,46,47,50,52,53],[33,40,41,42,45],[0,4,5,8,13,15,17,18,19,21,27,28,30,36,38,41,42,43,45,47,48,49,50,53],[30,40,41,42,43,45,47,48,49,50,52],[0,3,4,15,16,19,21,23,26,27,30,32,35,37,38,40,41,44,46,50,52],[3,5,13,15,17,18,19,20,23,26,29,30,31,37,38,41,42,44,45,48,50,52],[31,40,42,44,45],[32,47,48,49,51,52],[0,1,2,4,6,7,9,10,12,14,15,16,18,20,23,24,25,26,27,28,33,34,37,41,42,46,50,52,53],[11,47,48,50,51,52],[1,2,3,8,9,15,16,18,19,20,21,24,25,28,29,31,32,34,35,36,38,40,42,46,52,53],[10,40,46],[0,1,4,5,6,7,9,11,13,15,16,17,19,21,25,29,32,33,35,36,37,38,39,41,42,47,48,52,53],[12,40,42,46,47,49,53],[0,1,3,4,8,9,14,18,20,21,23,24,25,26,28,29,31,35,36,38,39,40,41,44,46,48,52],[36,40,43,46,47,50,53],[38,40,41,43,46,47,48,50,53],[20,47,51,53],[16,40,41,44,46,47,48,51,53],[35,47,49,51,53],[0,1,2,4,7,8,12,20,23,24,26,28,31,34,39,40,43,44,47,49,50,51,52],[0,1,2,3,4,5,7,10,11,13,14,17,18,19,20,28,29,32,33,34,35,36,37,38,39,41,42,46,47,48,51,52],[6,40,41,42,43,44,46,47,48,49,50,51,53],[5,47,52,53],[4,47,48,52,53],[27,47,49,52,53],[0,1,2,4,7,8,12,20,23,24,26,28,31,34,39,44,45,46,47,49,50],[0,1,3,4,5,7,8,9,13,14,15,18,19,21,25,28,29,34,35,36,38,41,43,46,47,48,49,50,52,53],[0,1,2,3,6,9,10,15,16,18,19,20,21,22,24,26,27,28,32,34,35,37,38,41,44,49,51],[0,1,3,4,5,7,8,11,18,19,20,22,24,25,26,28,29,33,34,38,39,40,41,43,44,45,46,47,48,49,51],[4,5,6,7,9,10,11,14,15,16,19,21,22,27,31,33,34,36,37,38,39,40,42,43,46,47,49,50,51],[22,47,48,51,52,53],[3,5,8,9,12,13,16,18,21,23,24,26,28,29,32,34,35,36,38,43,44,45,47,49,50,52],[39,40,41,42,44,45,46,47,48,49,51,52,53],[3,8,9,15,16,17,18,19,20,21,27,28,31,32,37,43,44,45,46,47,48,50,52],[37,47,48,50,51,52,53],[23,40,42,43,44,45,46],[3,4,5,8,9,18,19,21,23,29,31,32,35,36,38,42,44,46,48,52,53],[21,40,47],[18,47,48],[13,40,42],[0,40,41,42,47,48,49],[2,4,8,9,10,11,12,17,20,22,24,26,30,35,39,40,44,46,48,49,51,52],[0,3,4,5,6,7,9,16,19,21,22,23,30,31,32,33,35,36,37,38,39,40,41,48,50,51,52],[2,40,42,43],[25,40,44,47,51],[34,47,48,51],[3,4,5,10,12,14,15,18,19,21,22,23,25,26,29,30,31,32,33,35,37,38,39,40,41,43,44,46,49,52],[0,4,6,7,8,11,17,18,22,24,28,30,34,38,39,40,41,43,46,51,52,53],[26,47,50,51],[0,3,9,13,15,16,18,20,21,26,28,29,35,36,38,40,42,43,46,47,48,50,53],[9,47,49,50,51],[3,40,41,42,43,44,47,48,49,50,51],[1,40,45],[0,1,3,4,6,7,8,10,11,12,15,16,19,20,21,23,24,26,27,28,31,32,37,42,43,44,45,46,47,50,52,53],[19,40,43,45],[8,40,41,43,45],[0,4,5,8,13,15,17,18,19,21,27,28,30,36,38,40,41,47,48,49,50,53],[30,47,48,49,50,52],[0,3,4,15,16,19,21,23,26,27,30,32,35,37,38,41,45,46,50,52],[3,5,13,15,17,18,19,20,23,26,29,30,31,37,38,40,42,48,50,52],[32,40,41,42,44,45,47,48,49,51,52],[0,1,2,4,6,7,9,10,12,14,15,16,18,20,23,24,25,26,27,28,33,34,37,40,41,42,43,44,45,46,50,52,53],[11,40,41,43,44,45,47,48,50,51,52],[1,2,3,8,9,15,16,18,19,20,21,24,25,28,29,31,32,34,35,36,38,43,44,45,46,52,53],[24,40,41,42,43,44,45],[0,1,4,5,6,7,9,11,13,15,16,17,19,21,25,29,32,33,35,36,37,38,39,40,42,46,47,48,52,53],[12,47,49,53],[0,1,3,4,8,9,14,18,20,21,23,24,25,26,28,29,31,35,36,38,39,42,44,48,52],[36,47,50,53],[38,47,48,50,53],[15,40,42,43,46],[28,40,41,42,43,46],[20,40,44,46,47,51,53],[16,47,48,51,53],[35,40,42,44,46,47,49,51,53],[29,40,41,42,44,46],[0,1,2,4,7,8,12,20,23,24,26,28,31,34,39,46,47,49,50,51,52],[7,40,41,43,44,46],[0,1,2,3,4,5,7,10,11,13,14,17,18,19,20,28,29,32,33,34,35,36,37,38,39,40,41,43,44,47,48,51,52],[6,47,48,49,50,51,53],[5,40,45,46,47,52,53],[4,40,41,45,46,47,48,52,53],[27,40,42,45,46,47,49,52,53],[17,40,41,42,45,46],[0,1,2,4,7,8,12,20,23,24,26,28,31,34,39,40,43,44,47,49,50],[0,1,3,4,5,7,8,9,13,14,15,18,19,21,25,28,29,34,35,36,38,40,45,47,48,49,50,52,53],[0,1,2,3,6,9,10,15,16,18,19,20,21,22,24,26,27,28,32,34,35,37,38,40,41,42,43,44,45,46,49,51],[0,1,3,4,5,7,8,11,18,19,20,22,24,25,26,28,29,33,34,38,39,42,44,47,48,49,51],[4,5,6,7,9,10,11,14,15,16,19,21,22,27,31,33,34,36,37,38,39,42,43,44,45,47,49,50,51],[22,40,41,44,45,46,47,48,51,52,53],[3,5,8,9,12,13,16,18,21,23,24,26,28,29,32,34,35,36,38,40,42,43,46,47,49,50,52],[39,47,48,49,51,52,53],[3,8,9,15,16,17,18,19,20,21,27,28,31,32,37,40,47,48,50,52],[37,40,41,43,44,45,46,47,48,50,51,52,53],[3,4,5,8,9,18,19,21,23,29,31,32,35,36,38,40,41,43,45,48,52,53],[18,40,41],[13,47,49],[2,4,8,9,10,11,12,17,20,22,24,26,30,35,39,43,44,46,47,48,49,50,51,52],[0,3,4,5,6,7,9,16,19,21,22,23,30,31,32,33,35,36,37,38,39,43,47,51,52],[2,47,49,50],[34,40,41,44],[3,4,5,10,12,14,15,18,19,21,22,23,25,26,29,30,31,32,33,35,37,38,39,41,42,43,46,47,51,52],[0,4,6,7,8,11,17,18,22,24,28,30,34,38,39,42,43,44,46,47,48,49,52,53],[26,40,43,44],[0,3,9,13,15,16,18,20,21,26,28,29,35,36,38,41,42,44,46,51,53],[9,40,42,43,44],[1,47,52],[14,42,43,45,49,50,52],[0,1,3,4,6,7,8,10,11,12,15,16,19,20,21,23,24,26,27,28,31,32,37,40,43,44,46,49,50,53],[33,40,41,42,45,47,48,49,52],[19,47,50,52],[8,47,48,50,52],[0,4,5,8,13,15,17,18,19,21,27,28,30,36,38,41,42,43,45,48,52,53],[30,40,41,42,43,45],[0,3,4,15,16,19,21,23,26,27,30,32,35,37,38,40,41,44,46,47,50,51],[3,5,13,15,17,18,19,20,23,26,29,30,31,37,38,41,42,44,45,47,50,51],[31,40,42,44,45,47,49,51,52],[0,1,2,4,6,7,9,10,12,14,15,16,18,20,23,24,25,26,27,28,33,34,37,41,42,46,47,51,53],[1,2,3,8,9,15,16,18,19,20,21,24,25,28,29,31,32,34,35,36,38,40,42,46,47,49,50,51,53],[24,47,48,49,50,51,52],[10,40,46,47,53],[0,1,4,5,6,7,9,11,13,15,16,17,19,21,25,29,32,33,35,36,37,38,39,41,42,52],[12,40,42,46],[0,1,3,4,8,9,14,18,20,21,23,24,25,26,28,29,31,35,36,38,39,40,41,44,46,47,49,52,53],[36,40,43,46],[38,40,41,43,46],[15,47,49,50,53],[28,47,48,49,50,53],[16,40,41,44,46],[29,47,48,49,51,53],[7,47,48,50,51,53],[0,1,2,3,4,5,7,10,11,13,14,17,18,19,20,28,29,32,33,34,35,36,37,38,39,41,42,46,48,49,50,52,53],[6,40,41,42,43,44,46],[17,47,48,49,52,53],[0,1,2,4,7,8,12,20,23,24,26,28,31,34,39,44,45,46,49,52,53],[0,1,3,4,5,7,8,9,13,14,15,18,19,21,25,28,29,34,35,36,38,41,43,46,49],[0,1,2,3,6,9,10,15,16,18,19,20,21,22,24,26,27,28,32,34,35,37,38,41,44,47,50,51,52,53],[0,1,3,4,5,7,8,11,18,19,20,22,24,25,26,28,29,33,34,38,39,40,41,43,44,45,46,50,51,52,53],[4,5,6,7,9,10,11,14,15,16,19,21,22,27,31,33,34,36,37,38,39,40,42,43,46,49,50,52,53],[3,5,8,9,12,13,16,18,21,23,24,26,28,29,32,34,35,36,38,43,44,45,50,51,53],[39,40,41,42,44,45,46],[3,8,9,15,16,17,18,19,20,21,27,28,31,32,37,43,44,45,46,48,51,53],[23,40,42,43,44,45,46,47,49,50,51,52,53],[3,4,5,8,9,18,19,21,23,29,31,32,35,36,38,42,44,46,47,49,50,51],[21,40],[13,40,42,47,49],[0,40,41,42],[2,4,8,9,10,11,12,17,20,22,24,26,30,35,39,40,44,46,47,48,49,50,51,52],[0,3,4,5,6,7,9,16,19,21,22,23,30,31,32,33,35,36,37,38,39,40,41,47,51,52],[2,40,42,43,47,49,50],[14,40,41,42,43],[25,40,44],[3,4,5,10,12,14,15,18,19,21,22,23,25,26,29,30,31,32,33,35,37,38,39,40,41,43,44,46,47,51,52],[0,4,6,7,8,11,17,18,22,24,28,30,34,38,39,40,41,43,46,47,48,49,52,53],[0,3,9,13,15,16,18,20,21,26,28,29,35,36,38,40,42,43,46,51,53],[3,40,41,42,43,44],[1,40,45,47,52],[14,40,41,42,43,49,50,52],[0,1,3,4,6,7,8,10,11,12,15,16,19,20,21,23,24,26,27,28,31,32,37,42,43,44,45,46,49,50,53],[33,47,48,49,52],[19,40,43,45,47,50,52],[8,40,41,43,45,47,48,50,52],[0,4,5,8,13,15,17,18,19,21,27,28,30,36,38,40,41,48,52,53],[0,3,4,15,16,19,21,23,26,27,30,32,35,37,38,41,45,46,47,50,51],[3,5,13,15,17,18,19,20,23,26,29,30,31,37,38,40,42,47,50,51],[31,47,49,51,52],[32,40,41,42,44,45],[0,1,2,4,6,7,9,10,12,14,15,16,18,20,23,24,25,26,27,28,33,34,37,40,41,42,43,44,45,46,47,51,53],[11,40,41,43,44,45],[1,2,3,8,9,15,16,18,19,20,21,24,25,28,29,31,32,34,35,36,38,43,44,45,46,47,49,50,51,53],[24,40,41,42,43,44,45,47,48,49,50,51,52],[10,47,53],[0,1,4,5,6,7,9,11,13,15,16,17,19,21,25,29,32,33,35,36,37,38,39,40,42,46,52],[0,1,3,4,8,9,14,18,20,21,23,24,25,26,28,29,31,35,36,38,39,42,44,47,49,52,53],[15,40,42,43,46,47,49,50,53],[28,40,41,42,43,46,47,48,49,50,53],[20,40,44,46],[35,40,42,44,46],[29,40,41,42,44,46,47,48,49,51,53],[0,1,2,4,7,8,12,20,23,24,26,28,31,34,39,46,49,52,53],[7,40,41,43,44,46,47,48,50,51,53],[0,1,2,3,4,5,7,10,11,13,14,17,18,19,20,28,29,32,33,34,35,36,37,38,39,40,41,43,44,48,49,50,52,53],[5,40,45,46],[4,40,41,45,46],[27,40,42,45,46],[17,40,41,42,45,46,47,48,49,52,53],[0,1,3,4,5,7,8,9,13,14,15,18,19,21,25,28,29,34,35,36,38,40,45,49],[0,1,2,3,6,9,10,15,16,18,19,20,21,22,24,26,27,28,32,34,35,37,38,40,41,42,43,44,45,46,47,50,51,52,53],[0,1,3,4,5,7,8,11,18,19,20,22,24,25,26,28,29,33,34,38,39,42,44,50,51,52,53],[4,5,6,7,9,10,11,14,15,16,19,21,22,27,31,33,34,36,37,38,39,42,43,44,45,49,50,52,53],[22,40,41,44,45,46],[3,5,8,9,12,13,16,18,21,23,24,26,28,29,32,34,35,36,38,40,42,43,46,50,51,53],[3,8,9,15,16,17,18,19,20,21,27,28,31,32,37,40,48,51,53],[37,40,41,43,44,45,46],[23,47,49,50,51,52,53],[3,4,5,8,9,18,19,21,23,29,31,32,35,36,38,40,41,43,45,47,49,50,51]]},"distance":{"d":3,"is_exact":true,"upper":3,"witness":[0,53,160],"A_d":128,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,2],[1,2],[2,2],[3,1],[4,2],[5,1],[6,1],[7,2],[8,2],[9,1],[10,1],[11,1],[12,2],[13,1],[14,2],[15,1],[16,1],[17,1],[18,1],[19,1],[20,2],[21,1],[22,1],[23,2],[24,2],[25,1],[26,2],[27,1],[28,2],[29,1],[30,1],[31,2],[32,1],[33,1],[34,2],[35,1],[36,1],[37,1],[38,1],[39,2],[40,2],[41,1],[42,1],[43,2],[44,1],[47,1],[48,1],[49,2],[50,1],[52,1],[53,1]],"CZ":[[0,1],[0,2],[0,4],[0,7],[0,8],[0,12],[0,20],[0,23],[0,24],[0,26],[0,28],[0,31],[0,34],[0,39],[0,40],[0,43],[0,44],[0,49],[0,52],[0,53],[1,2],[1,4],[1,7],[1,8],[1,12],[1,20],[1,23],[1,24],[1,26],[1,28],[1,31],[1,34],[1,39],[1,40],[1,43],[1,44],[1,49],[1,52],[1,53],[2,4],[2,7],[2,8],[2,12],[2,20],[2,23],[2,24],[2,26],[2,28],[2,31],[2,34],[2,39],[2,40],[2,43],[2,44],[2,49],[2,52],[2,53],[4,7],[4,8],[4,12],[4,20],[4,23],[4,24],[4,26],[4,28],[4,31],[4,34],[4,39],[4,40],[4,43],[4,44],[4,49],[4,52],[4,53],[7,8],[7,12],[7,20],[7,23],[7,24],[7,26],[7,28],[7,31],[7,34],[7,39],[7,40],[7,43],[7,44],[7,49],[7,52],[7,53],[8,12],[8,20],[8,23],[8,24],[8,26],[8,28],[8,31],[8,34],[8,39],[8,40],[8,43],[8,44],[8,49],[8,52],[8,53],[12,20],[12,23],[12,24],[12,26],[12,28],[12,31],[12,34],[12,39],[12,40],[12,43],[12,44],[12,49],[12,52],[12,53],[14,40],[14,41],[14,42],[14,43],[14,47],[14,48],[14,49],[14,50],[20,23],[20,24],[20,26],[20,28],[20,31],[20,34],[20,39],[20,40],[20,43],[20,44],[20,49],[20,52],[20,53],[23,24],[23,26],[23,28],[23,31],[23,34],[23,39],[23,40],[23,43],[23,44],[23,49],[23,52],[23,53],[24,26],[24,28],[24,31],[24,34],[24,39],[24,40],[24,43],[24,44],[24,49],[24,52],[24,53],[26,28],[26,31],[26,34],[26,39],[26,40],[26,43],[26,44],[26,49],[26,52],[26,53],[28,31],[28,34],[28,39],[28,40],[28,43],[28,44],[28,49],[28,52],[28,53],[31,34],[31,39],[31,40],[31,43],[31,44],[31,49],[31,52],[31,53],[34,39],[34,40],[34,43],[34,44],[34,49],[34,52],[34,53],[39,40],[39,43],[39,44],[39,49],[39,52],[39,53],[40,41],[40,42],[40,44],[40,47],[40,48],[40,50],[40,52],[40,53],[41,42],[41,43],[41,47],[41,48],[41,49],[41,50],[42,43],[42,47],[42,48],[42,49],[42,50],[43,44],[43,47],[43,48],[43,50],[43,52],[43,53],[44,49],[44,52],[44,53],[47,48],[47,49],[47,50],[48,49],[48,50],[49,50],[49,52],[49,53],[52,53]]},"rotation_powers":[[0,7],[1,7],[3,3],[4,3],[5,5],[6,3],[7,7],[8,7],[9,3],[10,7],[11,7],[13,5],[14,7],[15,5],[17,3],[18,7],[19,3],[20,7],[21,3],[22,5],[28,3],[29,7],[30,5],[31,5],[32,3],[33,5],[34,7],[36,3],[37,3],[38,7],[39,5],[40,3],[41,7],[53,3],[55,5],[57,7],[60,3],[63,3],[64,3],[66,3],[71,3],[72,3],[75,3],[76,3],[77,3],[81,5],[82,3],[84,3],[86,3],[90,3],[92,3],[93,3],[94,3],[96,3],[97,3],[99,3],[100,3],[101,3],[102,3],[105,3],[106,3],[109,3],[112,7],[113,3],[114,5],[115,3],[117,3],[119,3],[122,3],[125,3],[126,3],[128,3],[130,3],[132,3],[137,3],[138,3],[141,3],[142,3],[144,3],[147,3],[150,3],[151,3],[153,3],[156,3],[159,3],[160,3],[162,3],[163,3],[164,3],[166,3],[167,3],[170,3],[171,3],[178,3],[181,3],[183,3],[184,3],[186,3],[187,3],[188,3],[189,3],[190,3],[191,3],[193,3],[194,3],[195,3],[197,3],[198,3],[203,3],[204,3],[207,3],[210,3],[211,3]]},"metrics":{"gamma":1.518012166585085,"gamma_t":null,"gamma_rho":1.518012166585085,"gamma_claim":1.518012166585085,"gamma_t_claim":null,"gamma_rho_claim":1.518012166585085,"v_ex":40,"rate":0.18867924528301888,"t_count":null,"poly_degree":null,"effective_width":40},"provenance":{"regimes":["AI search (gamma frontier): contraction of an existing code"],"discovery":"AI search","strongest_claim":"a paired-column contraction of an existing triorthogonal code, in the pure-T distillation-exponent release (gamma = log(n/k)/log d, n < 1000); a verified witness, not a maximum -- only the release's own frontier points are the lowest gamma it knows","sources":[{"regime":"AI search (gamma frontier): contraction of an existing code","file":"gamma_frontier_release/protocols/paired_column_contraction_212_40_3_e37afa333176.json","label":"paired_column_contraction_212_40_3_e37afa333176","N":54,"d":3,"d_is_exact":true,"origin":"Existing triorthogonal code with physical paired-column contraction","provenance":"gamma frontier release record paired_column_contraction_212_40_3_e37afa333176 (source catalogue id paired_column_contraction_212_40_3_e37afa333176); construction: Existing triorthogonal code with physical paired-column contraction; matrix e37afa33317662c7...; the release states d=3 (exact), gamma=1.518012166585085 (exact)","notes":"matrix gamma_frontier_release/matrices/e37afa33317662c727796a31e3e5c765943e9752fc69d867a0cf5ab1bf7e3380.matrix"}]},"references":[{"key":"jain2026symmetry","short":"Jain et al. (2026)","full":"S. P. Jain, A. Wills, and S. Singh, \"Symmetry and AI-assisted discovery of magic-state factories,\" arXiv:2610.06535 (2026).","url":"https://arxiv.org/abs/2610.06535","date":"2026-10-05"}]}