{"id":"47.3.3.b","row":220,"stored":{"file":"master_catalog.json","row":220,"construction":null,"columns_sha256":null},"parameters":{"n":47,"k":3,"d":3,"N":9,"r":6,"level":3},"gate":{"string":"0+01+012","human":"T0\u00b7CS01\u00b7CCZ012","monomials":[[0],[0,1],[0,1,2]],"sk_key":[[0],[0,1],[0,1,2]],"sk_fingerprint":"5811d0137e61f139","sk_canonical_frame":true,"relabelled_into_canonical_frame":true},"circuit":{"columns":[[1,2,3],[1,4],[2,3,4],[1,2,5],[3,5],[2,4,5],[0,1,3,4,5],[1,2,6],[1,2,3,6],[1,2,4,6],[0,1,2,3,4,6],[0,1,2,5,6],[1,2,3,5,6],[1,2,4,5,6],[1,2,3,4,5,6],[7],[0,1,2,3,7],[0,1,4,7],[2,3,4,7],[1,2,5,7],[1,3,5,7],[0,2,4,5,7],[0,3,4,5,7],[1,6,7],[0,3,6,7],[0,4,6,7],[1,3,4,6,7],[0,1,5,6,7],[0,1,3,5,6,7],[4,5,6,7],[3,4,5,6,7],[0,1,8],[0,1,3,8],[4,8],[0,3,4,8],[0,5,8],[0,1,3,5,8],[0,1,4,5,8],[3,4,5,8],[1,6,8],[0,3,6,8],[1,4,6,8],[3,4,6,8],[0,5,6,8],[3,5,6,8],[4,5,6,8],[3,4,5,6,8]]},"distance":{"d":3,"is_exact":true,"upper":3,"witness":[0,5,6],"A_d":253,"certified":null},"clifford":{"kind":"required","correction":{"S":[[0,3],[1,3]],"CZ":[[0,3],[0,5],[1,4],[1,6],[1,8]]},"rotation_powers":null},"metrics":{"gamma":2.504555375379736,"gamma_t":2.0395818546618094,"gamma_rho":null,"gamma_claim":2.504555375379736,"gamma_t_claim":2.0395818546618094,"gamma_rho_claim":null,"v_ex":null,"rate":0.06382978723404255,"t_count":5,"poly_degree":2,"effective_width":3},"provenance":{"regimes":["exhaustive classification n<=54 (Pareto point)"],"discovery":"pre-existing","strongest_claim":"a Pareto point of the exhaustive classification of generalised triorthogonal protocols through length 54 (classification/length54/): no protocol with n <= 54, the same CNOT+S output class and the same exact distance has n and S both no larger and one of them smaller, where S counts every matrix row -- the N of this table","sources":[{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/length54/pareto_frontier.json","label":"protocol 13","N":9,"d":3,"d_is_exact":true,"origin":"exhaustive classification of generalised triorthogonal protocols through length 54 (Wills, Jain and Singh)","provenance":"Pareto point 13 of the length-54 classification: output class Q3_0000000000000068 (T1CS12CS13), (n, S, d_Z) = (47, 9, 3), leading error coefficient 253; the outputs are its 3 logical rows"}]},"references":[{"key":"wills2026classification","short":"Wills et al. (2026)","full":"A. Wills, S. P. Jain, and S. Singh, \"Classification of Generalised Triorthogonal Codes through Length 54,\" arXiv:2609.30860 (2026).","url":"https://arxiv.org/abs/2609.30860","date":"2026-09-25"}]}