{"id":"106.5.2.a","row":679,"stored":{"file":"master_catalog.json","row":679,"construction":null,"columns_sha256":null},"parameters":{"n":106,"k":5,"d":2,"N":10,"r":5,"level":3},"gate":{"string":"01+02+03+04+12+13+14+23+24+34+012+013+014+023+024+034+123+124+134+234","human":"CS01\u00b7CS02\u00b7CS03\u00b7CS04\u00b7CS12\u00b7CS13\u00b7CS14\u00b7CS23\u00b7CS24\u00b7CS34\u00b7CCZ012\u00b7CCZ013\u00b7CCZ014\u00b7CCZ023\u00b7CCZ024\u00b7CCZ034\u00b7CCZ123\u00b7CCZ124\u00b7CCZ134\u00b7CCZ234","monomials":[[0,1],[0,2],[0,3],[0,4],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4],[0,1,2],[0,1,3],[0,1,4],[0,2,3],[0,2,4],[0,3,4],[1,2,3],[1,2,4],[1,3,4],[2,3,4]],"sk_key":[[0,1],[0,1,2],[0,1,3],[0,1,4],[0,2],[0,2,3],[0,2,4],[0,3],[0,3,4],[0,4],[1,2],[1,2,3],[1,2,4],[1,3],[1,3,4],[1,4],[2,3],[2,3,4],[2,4],[3,4]],"sk_fingerprint":"0b698deadfd13dda","sk_canonical_frame":true,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[5],[6],[7],[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],[5,6,7,8,9],[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],[1,5,6],[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],[2,5,6],[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],[3,5,6],[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],[0,5,6,7,8],[0,5,6,7,9],[0,5,6,8,9],[0,5,7,8,9],[0,6,7,8,9],[1,5,6,7,8],[1,5,6,7,9],[1,5,6,8,9],[1,5,7,8,9],[1,6,7,8,9],[2,5,6,7,8],[2,5,6,7,9],[2,5,6,8,9],[2,5,7,8,9],[2,6,7,8,9],[3,5,6,7,8],[3,5,6,7,9],[3,5,6,8,9],[3,5,7,8,9],[3,6,7,8,9],[4,5,6,7,8],[4,5,6,7,9],[4,5,6,8,9],[4,5,7,8,9],[4,6,7,8,9],[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],[0,1,2,3,4,6,7],[0,1,2,3,4,6,8],[0,1,2,3,4,6,9],[0,1,2,3,4,7,8],[0,1,2,3,4,7,9],[0,1,2,3,4,8,9],[0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,9],[0,1,2,3,4,5,6,8,9],[0,1,2,3,4,5,7,8,9],[0,1,2,3,4,6,7,8,9]]},"distance":{"d":2,"is_exact":true,"upper":2,"witness":[16,26],"A_d":225,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,1],[1,1],[2,1],[3,1],[4,1]],"CZ":[]},"rotation_powers":[[0,5],[1,5],[16,7],[26,7],[36,7],[46,7],[56,7],[98,3],[99,3],[100,3],[101,3],[102,3],[103,3]]},"metrics":{"gamma":4.405992359675837,"gamma_t":4.142957953842044,"gamma_rho":null,"gamma_claim":4.405992359675837,"gamma_t_claim":4.142957953842044,"gamma_rho_claim":null,"v_ex":null,"rate":0.04716981132075472,"t_count":6,"poly_degree":2,"effective_width":5},"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-row132-two-group","N":10,"d":2,"d_is_exact":true,"origin":"https://github.com/shraggy/Magic_state_factory_search at cae4982, outputs/factory_catalogue_l3.csv row 132","provenance":"two-group l=3 n=10 k=5 s_total=2 s_O=4 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"}]}