{"id":"762.4.2.a","row":1218,"stored":{"file":"master_catalog.json","row":1218,"construction":null,"columns_sha256":null},"parameters":{"n":762,"k":4,"d":2,"N":11,"r":7,"level":3},"gate":{"string":"01+02+03+12+13+23+012+013+023+123","human":"CS01\u00b7CS02\u00b7CS03\u00b7CS12\u00b7CS13\u00b7CS23\u00b7CCZ012\u00b7CCZ013\u00b7CCZ023\u00b7CCZ123","monomials":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3],[0,1,2],[0,1,3],[0,2,3],[1,2,3]],"sk_key":[[0,1],[0,1,2],[0,1,3],[0,2],[0,2,3],[0,3],[1,2],[1,2,3],[1,3],[2,3]],"sk_fingerprint":"4fa0e4de56a3871a","sk_canonical_frame":true,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[4],[5],[6],[7],[8],[9],[10],[4,5],[4,6],[4,7],[4,8],[4,9],[4,10],[5,6],[5,7],[5,8],[5,9],[5,10],[6,7],[6,8],[6,9],[6,10],[7,8],[7,9],[7,10],[8,9],[8,10],[9,10],[4,5,6],[4,5,7],[4,5,8],[4,5,9],[4,5,10],[4,6,7],[4,6,8],[4,6,9],[4,6,10],[4,7,8],[4,7,9],[4,7,10],[4,8,9],[4,8,10],[4,9,10],[5,6,7],[5,6,8],[5,6,9],[5,6,10],[5,7,8],[5,7,9],[5,7,10],[5,8,9],[5,8,10],[5,9,10],[6,7,8],[6,7,9],[6,7,10],[6,8,9],[6,8,10],[6,9,10],[7,8,9],[7,8,10],[7,9,10],[8,9,10],[4,5,6,7],[4,5,6,8],[4,5,6,9],[4,5,6,10],[4,5,7,8],[4,5,7,9],[4,5,7,10],[4,5,8,9],[4,5,8,10],[4,5,9,10],[4,6,7,8],[4,6,7,9],[4,6,7,10],[4,6,8,9],[4,6,8,10],[4,6,9,10],[4,7,8,9],[4,7,8,10],[4,7,9,10],[4,8,9,10],[5,6,7,8],[5,6,7,9],[5,6,7,10],[5,6,8,9],[5,6,8,10],[5,6,9,10],[5,7,8,9],[5,7,8,10],[5,7,9,10],[5,8,9,10],[6,7,8,9],[6,7,8,10],[6,7,9,10],[6,8,9,10],[7,8,9,10],[4,5,6,7,8],[4,5,6,7,9],[4,5,6,7,10],[4,5,6,8,9],[4,5,6,8,10],[4,5,6,9,10],[4,5,7,8,9],[4,5,7,8,10],[4,5,7,9,10],[4,5,8,9,10],[4,6,7,8,9],[4,6,7,8,10],[4,6,7,9,10],[4,6,8,9,10],[4,7,8,9,10],[5,6,7,8,9],[5,6,7,8,10],[5,6,7,9,10],[5,6,8,9,10],[5,7,8,9,10],[6,7,8,9,10],[4,5,6,7,8,9],[4,5,6,7,8,10],[4,5,6,7,9,10],[4,5,6,8,9,10],[4,5,7,8,9,10],[4,6,7,8,9,10],[5,6,7,8,9,10],[4,5,6,7,8,9,10],[0,4],[0,5],[0,6],[0,7],[0,8],[0,9],[0,10],[1,4],[1,5],[1,6],[1,7],[1,8],[1,9],[1,10],[2,4],[2,5],[2,6],[2,7],[2,8],[2,9],[2,10],[3,4],[3,5],[3,6],[3,7],[3,8],[3,9],[3,10],[0,4,5],[0,4,6],[0,4,7],[0,4,8],[0,4,9],[0,4,10],[0,5,6],[0,5,7],[0,5,8],[0,5,9],[0,5,10],[0,6,7],[0,6,8],[0,6,9],[0,6,10],[0,7,8],[0,7,9],[0,7,10],[0,8,9],[0,8,10],[0,9,10],[1,4,5],[1,4,6],[1,4,7],[1,4,8],[1,4,9],[1,4,10],[1,5,6],[1,5,7],[1,5,8],[1,5,9],[1,5,10],[1,6,7],[1,6,8],[1,6,9],[1,6,10],[1,7,8],[1,7,9],[1,7,10],[1,8,9],[1,8,10],[1,9,10],[2,4,5],[2,4,6],[2,4,7],[2,4,8],[2,4,9],[2,4,10],[2,5,6],[2,5,7],[2,5,8],[2,5,9],[2,5,10],[2,6,7],[2,6,8],[2,6,9],[2,6,10],[2,7,8],[2,7,9],[2,7,10],[2,8,9],[2,8,10],[2,9,10],[3,4,5],[3,4,6],[3,4,7],[3,4,8],[3,4,9],[3,4,10],[3,5,6],[3,5,7],[3,5,8],[3,5,9],[3,5,10],[3,6,7],[3,6,8],[3,6,9],[3,6,10],[3,7,8],[3,7,9],[3,7,10],[3,8,9],[3,8,10],[3,9,10],[0,4,5,6],[0,4,5,7],[0,4,5,8],[0,4,5,9],[0,4,5,10],[0,4,6,7],[0,4,6,8],[0,4,6,9],[0,4,6,10],[0,4,7,8],[0,4,7,9],[0,4,7,10],[0,4,8,9],[0,4,8,10],[0,4,9,10],[0,5,6,7],[0,5,6,8],[0,5,6,9],[0,5,6,10],[0,5,7,8],[0,5,7,9],[0,5,7,10],[0,5,8,9],[0,5,8,10],[0,5,9,10],[0,6,7,8],[0,6,7,9],[0,6,7,10],[0,6,8,9],[0,6,8,10],[0,6,9,10],[0,7,8,9],[0,7,8,10],[0,7,9,10],[0,8,9,10],[1,4,5,6],[1,4,5,7],[1,4,5,8],[1,4,5,9],[1,4,5,10],[1,4,6,7],[1,4,6,8],[1,4,6,9],[1,4,6,10],[1,4,7,8],[1,4,7,9],[1,4,7,10],[1,4,8,9],[1,4,8,10],[1,4,9,10],[1,5,6,7],[1,5,6,8],[1,5,6,9],[1,5,6,10],[1,5,7,8],[1,5,7,9],[1,5,7,10],[1,5,8,9],[1,5,8,10],[1,5,9,10],[1,6,7,8],[1,6,7,9],[1,6,7,10],[1,6,8,9],[1,6,8,10],[1,6,9,10],[1,7,8,9],[1,7,8,10],[1,7,9,10],[1,8,9,10],[2,4,5,6],[2,4,5,7],[2,4,5,8],[2,4,5,9],[2,4,5,10],[2,4,6,7],[2,4,6,8],[2,4,6,9],[2,4,6,10],[2,4,7,8],[2,4,7,9],[2,4,7,10],[2,4,8,9],[2,4,8,10],[2,4,9,10],[2,5,6,7],[2,5,6,8],[2,5,6,9],[2,5,6,10],[2,5,7,8],[2,5,7,9],[2,5,7,10],[2,5,8,9],[2,5,8,10],[2,5,9,10],[2,6,7,8],[2,6,7,9],[2,6,7,10],[2,6,8,9],[2,6,8,10],[2,6,9,10],[2,7,8,9],[2,7,8,10],[2,7,9,10],[2,8,9,10],[3,4,5,6],[3,4,5,7],[3,4,5,8],[3,4,5,9],[3,4,5,10],[3,4,6,7],[3,4,6,8],[3,4,6,9],[3,4,6,10],[3,4,7,8],[3,4,7,9],[3,4,7,10],[3,4,8,9],[3,4,8,10],[3,4,9,10],[3,5,6,7],[3,5,6,8],[3,5,6,9],[3,5,6,10],[3,5,7,8],[3,5,7,9],[3,5,7,10],[3,5,8,9],[3,5,8,10],[3,5,9,10],[3,6,7,8],[3,6,7,9],[3,6,7,10],[3,6,8,9],[3,6,8,10],[3,6,9,10],[3,7,8,9],[3,7,8,10],[3,7,9,10],[3,8,9,10],[0,4,5,6,7],[0,4,5,6,8],[0,4,5,6,9],[0,4,5,6,10],[0,4,5,7,8],[0,4,5,7,9],[0,4,5,7,10],[0,4,5,8,9],[0,4,5,8,10],[0,4,5,9,10],[0,4,6,7,8],[0,4,6,7,9],[0,4,6,7,10],[0,4,6,8,9],[0,4,6,8,10],[0,4,6,9,10],[0,4,7,8,9],[0,4,7,8,10],[0,4,7,9,10],[0,4,8,9,10],[0,5,6,7,8],[0,5,6,7,9],[0,5,6,7,10],[0,5,6,8,9],[0,5,6,8,10],[0,5,6,9,10],[0,5,7,8,9],[0,5,7,8,10],[0,5,7,9,10],[0,5,8,9,10],[0,6,7,8,9],[0,6,7,8,10],[0,6,7,9,10],[0,6,8,9,10],[0,7,8,9,10],[1,4,5,6,7],[1,4,5,6,8],[1,4,5,6,9],[1,4,5,6,10],[1,4,5,7,8],[1,4,5,7,9],[1,4,5,7,10],[1,4,5,8,9],[1,4,5,8,10],[1,4,5,9,10],[1,4,6,7,8],[1,4,6,7,9],[1,4,6,7,10],[1,4,6,8,9],[1,4,6,8,10],[1,4,6,9,10],[1,4,7,8,9],[1,4,7,8,10],[1,4,7,9,10],[1,4,8,9,10],[1,5,6,7,8],[1,5,6,7,9],[1,5,6,7,10],[1,5,6,8,9],[1,5,6,8,10],[1,5,6,9,10],[1,5,7,8,9],[1,5,7,8,10],[1,5,7,9,10],[1,5,8,9,10],[1,6,7,8,9],[1,6,7,8,10],[1,6,7,9,10],[1,6,8,9,10],[1,7,8,9,10],[2,4,5,6,7],[2,4,5,6,8],[2,4,5,6,9],[2,4,5,6,10],[2,4,5,7,8],[2,4,5,7,9],[2,4,5,7,10],[2,4,5,8,9],[2,4,5,8,10],[2,4,5,9,10],[2,4,6,7,8],[2,4,6,7,9],[2,4,6,7,10],[2,4,6,8,9],[2,4,6,8,10],[2,4,6,9,10],[2,4,7,8,9],[2,4,7,8,10],[2,4,7,9,10],[2,4,8,9,10],[2,5,6,7,8],[2,5,6,7,9],[2,5,6,7,10],[2,5,6,8,9],[2,5,6,8,10],[2,5,6,9,10],[2,5,7,8,9],[2,5,7,8,10],[2,5,7,9,10],[2,5,8,9,10],[2,6,7,8,9],[2,6,7,8,10],[2,6,7,9,10],[2,6,8,9,10],[2,7,8,9,10],[3,4,5,6,7],[3,4,5,6,8],[3,4,5,6,9],[3,4,5,6,10],[3,4,5,7,8],[3,4,5,7,9],[3,4,5,7,10],[3,4,5,8,9],[3,4,5,8,10],[3,4,5,9,10],[3,4,6,7,8],[3,4,6,7,9],[3,4,6,7,10],[3,4,6,8,9],[3,4,6,8,10],[3,4,6,9,10],[3,4,7,8,9],[3,4,7,8,10],[3,4,7,9,10],[3,4,8,9,10],[3,5,6,7,8],[3,5,6,7,9],[3,5,6,7,10],[3,5,6,8,9],[3,5,6,8,10],[3,5,6,9,10],[3,5,7,8,9],[3,5,7,8,10],[3,5,7,9,10],[3,5,8,9,10],[3,6,7,8,9],[3,6,7,8,10],[3,6,7,9,10],[3,6,8,9,10],[3,7,8,9,10],[0,4,5,6,7,8],[0,4,5,6,7,9],[0,4,5,6,7,10],[0,4,5,6,8,9],[0,4,5,6,8,10],[0,4,5,6,9,10],[0,4,5,7,8,9],[0,4,5,7,8,10],[0,4,5,7,9,10],[0,4,5,8,9,10],[0,4,6,7,8,9],[0,4,6,7,8,10],[0,4,6,7,9,10],[0,4,6,8,9,10],[0,4,7,8,9,10],[0,5,6,7,8,9],[0,5,6,7,8,10],[0,5,6,7,9,10],[0,5,6,8,9,10],[0,5,7,8,9,10],[0,6,7,8,9,10],[1,4,5,6,7,8],[1,4,5,6,7,9],[1,4,5,6,7,10],[1,4,5,6,8,9],[1,4,5,6,8,10],[1,4,5,6,9,10],[1,4,5,7,8,9],[1,4,5,7,8,10],[1,4,5,7,9,10],[1,4,5,8,9,10],[1,4,6,7,8,9],[1,4,6,7,8,10],[1,4,6,7,9,10],[1,4,6,8,9,10],[1,4,7,8,9,10],[1,5,6,7,8,9],[1,5,6,7,8,10],[1,5,6,7,9,10],[1,5,6,8,9,10],[1,5,7,8,9,10],[1,6,7,8,9,10],[2,4,5,6,7,8],[2,4,5,6,7,9],[2,4,5,6,7,10],[2,4,5,6,8,9],[2,4,5,6,8,10],[2,4,5,6,9,10],[2,4,5,7,8,9],[2,4,5,7,8,10],[2,4,5,7,9,10],[2,4,5,8,9,10],[2,4,6,7,8,9],[2,4,6,7,8,10],[2,4,6,7,9,10],[2,4,6,8,9,10],[2,4,7,8,9,10],[2,5,6,7,8,9],[2,5,6,7,8,10],[2,5,6,7,9,10],[2,5,6,8,9,10],[2,5,7,8,9,10],[2,6,7,8,9,10],[3,4,5,6,7,8],[3,4,5,6,7,9],[3,4,5,6,7,10],[3,4,5,6,8,9],[3,4,5,6,8,10],[3,4,5,6,9,10],[3,4,5,7,8,9],[3,4,5,7,8,10],[3,4,5,7,9,10],[3,4,5,8,9,10],[3,4,6,7,8,9],[3,4,6,7,8,10],[3,4,6,7,9,10],[3,4,6,8,9,10],[3,4,7,8,9,10],[3,5,6,7,8,9],[3,5,6,7,8,10],[3,5,6,7,9,10],[3,5,6,8,9,10],[3,5,7,8,9,10],[3,6,7,8,9,10],[0,4,5,6,7,8,9],[0,4,5,6,7,8,10],[0,4,5,6,7,9,10],[0,4,5,6,8,9,10],[0,4,5,7,8,9,10],[0,4,6,7,8,9,10],[0,5,6,7,8,9,10],[1,4,5,6,7,8,9],[1,4,5,6,7,8,10],[1,4,5,6,7,9,10],[1,4,5,6,8,9,10],[1,4,5,7,8,9,10],[1,4,6,7,8,9,10],[1,5,6,7,8,9,10],[2,4,5,6,7,8,9],[2,4,5,6,7,8,10],[2,4,5,6,7,9,10],[2,4,5,6,8,9,10],[2,4,5,7,8,9,10],[2,4,6,7,8,9,10],[2,5,6,7,8,9,10],[3,4,5,6,7,8,9],[3,4,5,6,7,8,10],[3,4,5,6,7,9,10],[3,4,5,6,8,9,10],[3,4,5,7,8,9,10],[3,4,6,7,8,9,10],[3,5,6,7,8,9,10],[0,4,5,6,7,8,9,10],[1,4,5,6,7,8,9,10],[2,4,5,6,7,8,9,10],[3,4,5,6,7,8,9,10],[0,1,2,3,4],[0,1,2,3,5],[0,1,2,3,6],[0,1,2,3,7],[0,1,2,3,8],[0,1,2,3,9],[0,1,2,3,10],[0,1,2,3,4,5],[0,1,2,3,4,6],[0,1,2,3,4,7],[0,1,2,3,4,8],[0,1,2,3,4,9],[0,1,2,3,4,10],[0,1,2,3,5,6],[0,1,2,3,5,7],[0,1,2,3,5,8],[0,1,2,3,5,9],[0,1,2,3,5,10],[0,1,2,3,6,7],[0,1,2,3,6,8],[0,1,2,3,6,9],[0,1,2,3,6,10],[0,1,2,3,7,8],[0,1,2,3,7,9],[0,1,2,3,7,10],[0,1,2,3,8,9],[0,1,2,3,8,10],[0,1,2,3,9,10],[0,1,2,3,4,5,6],[0,1,2,3,4,5,7],[0,1,2,3,4,5,8],[0,1,2,3,4,5,9],[0,1,2,3,4,5,10],[0,1,2,3,4,6,7],[0,1,2,3,4,6,8],[0,1,2,3,4,6,9],[0,1,2,3,4,6,10],[0,1,2,3,4,7,8],[0,1,2,3,4,7,9],[0,1,2,3,4,7,10],[0,1,2,3,4,8,9],[0,1,2,3,4,8,10],[0,1,2,3,4,9,10],[0,1,2,3,5,6,7],[0,1,2,3,5,6,8],[0,1,2,3,5,6,9],[0,1,2,3,5,6,10],[0,1,2,3,5,7,8],[0,1,2,3,5,7,9],[0,1,2,3,5,7,10],[0,1,2,3,5,8,9],[0,1,2,3,5,8,10],[0,1,2,3,5,9,10],[0,1,2,3,6,7,8],[0,1,2,3,6,7,9],[0,1,2,3,6,7,10],[0,1,2,3,6,8,9],[0,1,2,3,6,8,10],[0,1,2,3,6,9,10],[0,1,2,3,7,8,9],[0,1,2,3,7,8,10],[0,1,2,3,7,9,10],[0,1,2,3,8,9,10],[0,1,2,3,4,5,6,7],[0,1,2,3,4,5,6,8],[0,1,2,3,4,5,6,9],[0,1,2,3,4,5,6,10],[0,1,2,3,4,5,7,8],[0,1,2,3,4,5,7,9],[0,1,2,3,4,5,7,10],[0,1,2,3,4,5,8,9],[0,1,2,3,4,5,8,10],[0,1,2,3,4,5,9,10],[0,1,2,3,4,6,7,8],[0,1,2,3,4,6,7,9],[0,1,2,3,4,6,7,10],[0,1,2,3,4,6,8,9],[0,1,2,3,4,6,8,10],[0,1,2,3,4,6,9,10],[0,1,2,3,4,7,8,9],[0,1,2,3,4,7,8,10],[0,1,2,3,4,7,9,10],[0,1,2,3,4,8,9,10],[0,1,2,3,5,6,7,8],[0,1,2,3,5,6,7,9],[0,1,2,3,5,6,7,10],[0,1,2,3,5,6,8,9],[0,1,2,3,5,6,8,10],[0,1,2,3,5,6,9,10],[0,1,2,3,5,7,8,9],[0,1,2,3,5,7,8,10],[0,1,2,3,5,7,9,10],[0,1,2,3,5,8,9,10],[0,1,2,3,6,7,8,9],[0,1,2,3,6,7,8,10],[0,1,2,3,6,7,9,10],[0,1,2,3,6,8,9,10],[0,1,2,3,7,8,9,10],[0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,9],[0,1,2,3,4,5,6,7,10],[0,1,2,3,4,5,6,8,9],[0,1,2,3,4,5,6,8,10],[0,1,2,3,4,5,6,9,10],[0,1,2,3,4,5,7,8,9],[0,1,2,3,4,5,7,8,10],[0,1,2,3,4,5,7,9,10],[0,1,2,3,4,5,8,9,10],[0,1,2,3,4,6,7,8,9],[0,1,2,3,4,6,7,8,10],[0,1,2,3,4,6,7,9,10],[0,1,2,3,4,6,8,9,10],[0,1,2,3,4,7,8,9,10],[0,1,2,3,5,6,7,8,9],[0,1,2,3,5,6,7,8,10],[0,1,2,3,5,6,7,9,10],[0,1,2,3,5,6,8,9,10],[0,1,2,3,5,7,8,9,10],[0,1,2,3,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9],[0,1,2,3,4,5,6,7,8,10],[0,1,2,3,4,5,6,7,9,10],[0,1,2,3,4,5,6,8,9,10],[0,1,2,3,4,5,7,8,9,10],[0,1,2,3,4,6,7,8,9,10],[0,1,2,3,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10]]},"distance":{"d":2,"is_exact":true,"upper":2,"witness":[0,127],"A_d":1905,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,1],[1,1],[2,1],[3,1]],"CZ":[]},"rotation_powers":[[0,3],[127,7],[134,7],[141,7],[148,7],[748,3],[749,3],[750,3],[751,3],[752,3],[753,3],[754,3],[755,3],[756,3],[757,3],[758,3],[759,3],[760,3],[761,3]]},"metrics":{"gamma":7.573647187493322,"gamma_t":7.25171909260596,"gamma_rho":null,"gamma_claim":7.573647187493322,"gamma_t_claim":7.25171909260596,"gamma_rho_claim":null,"v_ex":null,"rate":0.005249343832020997,"t_count":5,"poly_degree":2,"effective_width":4},"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/circuits_l3.json","label":"l3-row120-two-group","N":11,"d":2,"d_is_exact":true,"origin":"https://github.com/shraggy/Magic_state_factory_search at cae4982, outputs/factory_catalogue_l3.csv row 120","provenance":"two-group l=3 n=11 k=4 s_total=1 s_O=3 s_S=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"}]}