{"id":"212.10.2.a","row":856,"stored":{"file":"master_catalog.json","row":856,"construction":null,"columns_sha256":null},"parameters":{"n":212,"k":10,"d":2,"N":12,"r":2,"level":3},"gate":{"string":"012+013+014+015+016+017+018+019+023+024+025+026+027+028+029+034+035+036+037+038+039+045+046+047+048+049+056+057+058+059+067+068+069+078+079+089+123+124+125+126+127+128+129+134+135+136+137+138+139+145+146+147+148+149+156+157+158+159+167+168+169+178+179+189+234+235+236+237+238+239+245+246+247+248+249+256+257+258+259+267+268+269+278+279+289+345+346+347+348+349+356+357+358+359+367+368+369+378+379+389+456+457+458+459+467+468+469+478+479+489+567+568+569+578+579+589+678+679+689+789","human":"CCZ012\u00b7CCZ013\u00b7CCZ014\u00b7CCZ015\u00b7CCZ016\u00b7CCZ017\u00b7CCZ018\u00b7CCZ019\u00b7CCZ023\u00b7CCZ024\u00b7CCZ025\u00b7CCZ026\u00b7CCZ027\u00b7CCZ028\u00b7CCZ029\u00b7CCZ034\u00b7CCZ035\u00b7CCZ036\u00b7CCZ037\u00b7CCZ038\u00b7CCZ039\u00b7CCZ045\u00b7CCZ046\u00b7CCZ047\u00b7CCZ048\u00b7CCZ049\u00b7CCZ056\u00b7CCZ057\u00b7CCZ058\u00b7CCZ059\u00b7CCZ067\u00b7CCZ068\u00b7CCZ069\u00b7CCZ078\u00b7CCZ079\u00b7CCZ089\u00b7CCZ123\u00b7CCZ124\u00b7CCZ125\u00b7CCZ126\u00b7CCZ127\u00b7CCZ128\u00b7CCZ129\u00b7CCZ134\u00b7CCZ135\u00b7CCZ136\u00b7CCZ137\u00b7CCZ138\u00b7CCZ139\u00b7CCZ145\u00b7CCZ146\u00b7CCZ147\u00b7CCZ148\u00b7CCZ149\u00b7CCZ156\u00b7CCZ157\u00b7CCZ158\u00b7CCZ159\u00b7CCZ167\u00b7CCZ168\u00b7CCZ169\u00b7CCZ178\u00b7CCZ179\u00b7CCZ189\u00b7CCZ234\u00b7CCZ235\u00b7CCZ236\u00b7CCZ237\u00b7CCZ238\u00b7CCZ239\u00b7CCZ245\u00b7CCZ246\u00b7CCZ247\u00b7CCZ248\u00b7CCZ249\u00b7CCZ256\u00b7CCZ257\u00b7CCZ258\u00b7CCZ259\u00b7CCZ267\u00b7CCZ268\u00b7CCZ269\u00b7CCZ278\u00b7CCZ279\u00b7CCZ289\u00b7CCZ345\u00b7CCZ346\u00b7CCZ347\u00b7CCZ348\u00b7CCZ349\u00b7CCZ356\u00b7CCZ357\u00b7CCZ358\u00b7CCZ359\u00b7CCZ367\u00b7CCZ368\u00b7CCZ369\u00b7CCZ378\u00b7CCZ379\u00b7CCZ389\u00b7CCZ456\u00b7CCZ457\u00b7CCZ458\u00b7CCZ459\u00b7CCZ467\u00b7CCZ468\u00b7CCZ469\u00b7CCZ478\u00b7CCZ479\u00b7CCZ489\u00b7CCZ567\u00b7CCZ568\u00b7CCZ569\u00b7CCZ578\u00b7CCZ579\u00b7CCZ589\u00b7CCZ678\u00b7CCZ679\u00b7CCZ689\u00b7CCZ789","monomials":[[0,1,2],[0,1,3],[0,1,4],[0,1,5],[0,1,6],[0,1,7],[0,1,8],[0,1,9],[0,2,3],[0,2,4],[0,2,5],[0,2,6],[0,2,7],[0,2,8],[0,2,9],[0,3,4],[0,3,5],[0,3,6],[0,3,7],[0,3,8],[0,3,9],[0,4,5],[0,4,6],[0,4,7],[0,4,8],[0,4,9],[0,5,6],[0,5,7],[0,5,8],[0,5,9],[0,6,7],[0,6,8],[0,6,9],[0,7,8],[0,7,9],[0,8,9],[1,2,3],[1,2,4],[1,2,5],[1,2,6],[1,2,7],[1,2,8],[1,2,9],[1,3,4],[1,3,5],[1,3,6],[1,3,7],[1,3,8],[1,3,9],[1,4,5],[1,4,6],[1,4,7],[1,4,8],[1,4,9],[1,5,6],[1,5,7],[1,5,8],[1,5,9],[1,6,7],[1,6,8],[1,6,9],[1,7,8],[1,7,9],[1,8,9],[2,3,4],[2,3,5],[2,3,6],[2,3,7],[2,3,8],[2,3,9],[2,4,5],[2,4,6],[2,4,7],[2,4,8],[2,4,9],[2,5,6],[2,5,7],[2,5,8],[2,5,9],[2,6,7],[2,6,8],[2,6,9],[2,7,8],[2,7,9],[2,8,9],[3,4,5],[3,4,6],[3,4,7],[3,4,8],[3,4,9],[3,5,6],[3,5,7],[3,5,8],[3,5,9],[3,6,7],[3,6,8],[3,6,9],[3,7,8],[3,7,9],[3,8,9],[4,5,6],[4,5,7],[4,5,8],[4,5,9],[4,6,7],[4,6,8],[4,6,9],[4,7,8],[4,7,9],[4,8,9],[5,6,7],[5,6,8],[5,6,9],[5,7,8],[5,7,9],[5,8,9],[6,7,8],[6,7,9],[6,8,9],[7,8,9]],"sk_key":null,"sk_fingerprint":"23f08dfafb270e97","sk_canonical_frame":false,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[0,1,2,3,4,5,6,7,10],[0,1,2,3,4,5,6,7,11],[0,1,2,3,4,5,6,8,10],[0,1,2,3,4,5,6,8,11],[0,1,2,3,4,5,6,9,10],[0,1,2,3,4,5,6,9,11],[0,1,2,3,4,5,6,10,11],[0,1,2,3,4,5,7,8,10],[0,1,2,3,4,5,7,8,11],[0,1,2,3,4,5,7,9,10],[0,1,2,3,4,5,7,9,11],[0,1,2,3,4,5,7,10,11],[0,1,2,3,4,5,8,9,10],[0,1,2,3,4,5,8,9,11],[0,1,2,3,4,5,8,10,11],[0,1,2,3,4,5,9,10,11],[0,1,2,3,4,6,7,8,10],[0,1,2,3,4,6,7,8,11],[0,1,2,3,4,6,7,9,10],[0,1,2,3,4,6,7,9,11],[0,1,2,3,4,6,7,10,11],[0,1,2,3,4,6,8,9,10],[0,1,2,3,4,6,8,9,11],[0,1,2,3,4,6,8,10,11],[0,1,2,3,4,6,9,10,11],[0,1,2,3,4,7,8,9,10],[0,1,2,3,4,7,8,9,11],[0,1,2,3,4,7,8,10,11],[0,1,2,3,4,7,9,10,11],[0,1,2,3,4,8,9,10,11],[0,1,2,3,5,6,7,8,10],[0,1,2,3,5,6,7,8,11],[0,1,2,3,5,6,7,9,10],[0,1,2,3,5,6,7,9,11],[0,1,2,3,5,6,7,10,11],[0,1,2,3,5,6,8,9,10],[0,1,2,3,5,6,8,9,11],[0,1,2,3,5,6,8,10,11],[0,1,2,3,5,6,9,10,11],[0,1,2,3,5,7,8,9,10],[0,1,2,3,5,7,8,9,11],[0,1,2,3,5,7,8,10,11],[0,1,2,3,5,7,9,10,11],[0,1,2,3,5,8,9,10,11],[0,1,2,3,6,7,8,9,10],[0,1,2,3,6,7,8,9,11],[0,1,2,3,6,7,8,10,11],[0,1,2,3,6,7,9,10,11],[0,1,2,3,6,8,9,10,11],[0,1,2,3,7,8,9,10,11],[0,1,2,4,5,6,7,8,10],[0,1,2,4,5,6,7,8,11],[0,1,2,4,5,6,7,9,10],[0,1,2,4,5,6,7,9,11],[0,1,2,4,5,6,7,10,11],[0,1,2,4,5,6,8,9,10],[0,1,2,4,5,6,8,9,11],[0,1,2,4,5,6,8,10,11],[0,1,2,4,5,6,9,10,11],[0,1,2,4,5,7,8,9,10],[0,1,2,4,5,7,8,9,11],[0,1,2,4,5,7,8,10,11],[0,1,2,4,5,7,9,10,11],[0,1,2,4,5,8,9,10,11],[0,1,2,4,6,7,8,9,10],[0,1,2,4,6,7,8,9,11],[0,1,2,4,6,7,8,10,11],[0,1,2,4,6,7,9,10,11],[0,1,2,4,6,8,9,10,11],[0,1,2,4,7,8,9,10,11],[0,1,2,5,6,7,8,9,10],[0,1,2,5,6,7,8,9,11],[0,1,2,5,6,7,8,10,11],[0,1,2,5,6,7,9,10,11],[0,1,2,5,6,8,9,10,11],[0,1,2,5,7,8,9,10,11],[0,1,2,6,7,8,9,10,11],[0,1,3,4,5,6,7,8,10],[0,1,3,4,5,6,7,8,11],[0,1,3,4,5,6,7,9,10],[0,1,3,4,5,6,7,9,11],[0,1,3,4,5,6,7,10,11],[0,1,3,4,5,6,8,9,10],[0,1,3,4,5,6,8,9,11],[0,1,3,4,5,6,8,10,11],[0,1,3,4,5,6,9,10,11],[0,1,3,4,5,7,8,9,10],[0,1,3,4,5,7,8,9,11],[0,1,3,4,5,7,8,10,11],[0,1,3,4,5,7,9,10,11],[0,1,3,4,5,8,9,10,11],[0,1,3,4,6,7,8,9,10],[0,1,3,4,6,7,8,9,11],[0,1,3,4,6,7,8,10,11],[0,1,3,4,6,7,9,10,11],[0,1,3,4,6,8,9,10,11],[0,1,3,4,7,8,9,10,11],[0,1,3,5,6,7,8,9,10],[0,1,3,5,6,7,8,9,11],[0,1,3,5,6,7,8,10,11],[0,1,3,5,6,7,9,10,11],[0,1,3,5,6,8,9,10,11],[0,1,3,5,7,8,9,10,11],[0,1,3,6,7,8,9,10,11],[0,1,4,5,6,7,8,9,10],[0,1,4,5,6,7,8,9,11],[0,1,4,5,6,7,8,10,11],[0,1,4,5,6,7,9,10,11],[0,1,4,5,6,8,9,10,11],[0,1,4,5,7,8,9,10,11],[0,1,4,6,7,8,9,10,11],[0,1,5,6,7,8,9,10,11],[0,2,3,4,5,6,7,8,10],[0,2,3,4,5,6,7,8,11],[0,2,3,4,5,6,7,9,10],[0,2,3,4,5,6,7,9,11],[0,2,3,4,5,6,7,10,11],[0,2,3,4,5,6,8,9,10],[0,2,3,4,5,6,8,9,11],[0,2,3,4,5,6,8,10,11],[0,2,3,4,5,6,9,10,11],[0,2,3,4,5,7,8,9,10],[0,2,3,4,5,7,8,9,11],[0,2,3,4,5,7,8,10,11],[0,2,3,4,5,7,9,10,11],[0,2,3,4,5,8,9,10,11],[0,2,3,4,6,7,8,9,10],[0,2,3,4,6,7,8,9,11],[0,2,3,4,6,7,8,10,11],[0,2,3,4,6,7,9,10,11],[0,2,3,4,6,8,9,10,11],[0,2,3,4,7,8,9,10,11],[0,2,3,5,6,7,8,9,10],[0,2,3,5,6,7,8,9,11],[0,2,3,5,6,7,8,10,11],[0,2,3,5,6,7,9,10,11],[0,2,3,5,6,8,9,10,11],[0,2,3,5,7,8,9,10,11],[0,2,3,6,7,8,9,10,11],[0,2,4,5,6,7,8,9,10],[0,2,4,5,6,7,8,9,11],[0,2,4,5,6,7,8,10,11],[0,2,4,5,6,7,9,10,11],[0,2,4,5,6,8,9,10,11],[0,2,4,5,7,8,9,10,11],[0,2,4,6,7,8,9,10,11],[0,2,5,6,7,8,9,10,11],[0,3,4,5,6,7,8,9,10],[0,3,4,5,6,7,8,9,11],[0,3,4,5,6,7,8,10,11],[0,3,4,5,6,7,9,10,11],[0,3,4,5,6,8,9,10,11],[0,3,4,5,7,8,9,10,11],[0,3,4,6,7,8,9,10,11],[0,3,5,6,7,8,9,10,11],[0,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,10],[1,2,3,4,5,6,7,8,11],[1,2,3,4,5,6,7,9,10],[1,2,3,4,5,6,7,9,11],[1,2,3,4,5,6,7,10,11],[1,2,3,4,5,6,8,9,10],[1,2,3,4,5,6,8,9,11],[1,2,3,4,5,6,8,10,11],[1,2,3,4,5,6,9,10,11],[1,2,3,4,5,7,8,9,10],[1,2,3,4,5,7,8,9,11],[1,2,3,4,5,7,8,10,11],[1,2,3,4,5,7,9,10,11],[1,2,3,4,5,8,9,10,11],[1,2,3,4,6,7,8,9,10],[1,2,3,4,6,7,8,9,11],[1,2,3,4,6,7,8,10,11],[1,2,3,4,6,7,9,10,11],[1,2,3,4,6,8,9,10,11],[1,2,3,4,7,8,9,10,11],[1,2,3,5,6,7,8,9,10],[1,2,3,5,6,7,8,9,11],[1,2,3,5,6,7,8,10,11],[1,2,3,5,6,7,9,10,11],[1,2,3,5,6,8,9,10,11],[1,2,3,5,7,8,9,10,11],[1,2,3,6,7,8,9,10,11],[1,2,4,5,6,7,8,9,10],[1,2,4,5,6,7,8,9,11],[1,2,4,5,6,7,8,10,11],[1,2,4,5,6,7,9,10,11],[1,2,4,5,6,8,9,10,11],[1,2,4,5,7,8,9,10,11],[1,2,4,6,7,8,9,10,11],[1,2,5,6,7,8,9,10,11],[1,3,4,5,6,7,8,9,10],[1,3,4,5,6,7,8,9,11],[1,3,4,5,6,7,8,10,11],[1,3,4,5,6,7,9,10,11],[1,3,4,5,6,8,9,10,11],[1,3,4,5,7,8,9,10,11],[1,3,4,6,7,8,9,10,11],[1,3,5,6,7,8,9,10,11],[1,4,5,6,7,8,9,10,11],[2,3,4,5,6,7,8,9,10],[2,3,4,5,6,7,8,9,11],[2,3,4,5,6,7,8,10,11],[2,3,4,5,6,7,9,10,11],[2,3,4,5,6,8,9,10,11],[2,3,4,5,7,8,9,10,11],[2,3,4,6,7,8,9,10,11],[2,3,5,6,7,8,9,10,11],[2,4,5,6,7,8,9,10,11],[3,4,5,6,7,8,9,10,11],[10],[11]]},"distance":{"d":2,"is_exact":true,"upper":2,"witness":[0,2],"A_d":9210,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,2],[1,2],[2,2],[3,2],[4,2],[5,2],[6,2],[7,2],[8,2],[9,2],[10,1],[11,1]],"CZ":[]},"rotation_powers":[[4,5],[7,5],[16,5],[30,5],[50,5],[77,5],[112,5],[156,5],[210,3],[211,3]]},"metrics":{"gamma":4.405992359675837,"gamma_t":null,"gamma_rho":null,"gamma_claim":4.405992359675837,"gamma_t_claim":null,"gamma_rho_claim":null,"v_ex":null,"rate":0.04716981132075472,"t_count":null,"poly_degree":null,"effective_width":10},"provenance":{"regimes":["borrowed-identity search: two-group"],"discovery":"pre-existing","strongest_claim":"found by the two-group borrowed-identity search of Singh, Gidney and Jones (arXiv:2606.28518; borrowed_identities/): an identity circuit symmetric within its output and check blocks, with its output-only gates removed; a verified witness, not a maximum","sources":[{"regime":"borrowed-identity search: two-group","file":"borrowed_identities/circuits/large_unified_l3.json","label":"lu-1078","N":12,"d":2,"d_is_exact":true,"origin":"https://github.com/shraggy/Magic_state_factory_search at ae9fbf8, outputs/large_unified_factories_l3.csv line 1078","provenance":"larger unified search l=3, two-group s_total=8,s_O=1"}]},"references":[{"key":"singh2026borrowed","short":"Singh et al. (2026)","full":"S. Singh, C. Gidney, and C. Jones, \"Borrowed Identities: Malleable Distillation Factories and a Unified Numerical Search,\" arXiv:2606.28518 (2026).","url":"https://arxiv.org/abs/2606.28518","date":"2026-06-26"}]}