{"id":"48.5.3.e","row":259,"stored":{"file":"master_catalog.json","row":259,"construction":null,"columns_sha256":null},"parameters":{"n":48,"k":5,"d":3,"N":11,"r":6,"level":3},"gate":{"string":"01+023+124","human":"CS01\u00b7CCZ023\u00b7CCZ124","monomials":[[0,1],[0,2,3],[1,2,4]],"sk_key":[[0,1],[0,2,3],[1,2,4]],"sk_fingerprint":"d68ede3be2bcf02b","sk_canonical_frame":true,"relabelled_into_canonical_frame":true},"circuit":{"columns":[[1,4,9],[3,5,9],[0,2,4,6,9],[1,2,3,5,6,9],[2,3,4,7,9],[2,5,7,9],[3,4,6,7,9],[0,5,6,7,9],[0,1,2,3,8,9],[1,2,4,5,8,9],[0,3,6,8,9],[0,4,5,6,8,9],[0,7,8,9],[1,3,4,5,7,8,9],[2,6,7,8,9],[1,2,3,4,5,6,7,8,9],[0,3,4,10],[0,1,2,3,5,10],[1,3,4,6,10],[1,2,3,5,6,10],[1,2,7,10],[0,1,4,5,7,10],[2,6,7,10],[0,1,4,5,6,7,10],[0,3,8,10],[1,2,3,4,5,8,10],[3,6,8,10],[0,2,3,4,5,6,8,10],[1,2,4,7,8,10],[1,5,7,8,10],[1,2,4,6,7,8,10],[5,6,7,8,10],[0,9,10],[0,5,9,10],[0,1,6,9,10],[5,6,9,10],[1,7,9,10],[0,1,5,7,9,10],[1,6,7,9,10],[5,6,7,9,10],[8,9,10],[1,5,8,9,10],[0,6,8,9,10],[5,6,8,9,10],[0,1,7,8,9,10],[5,7,8,9,10],[6,7,8,9,10],[5,6,7,8,9,10]]},"distance":{"d":3,"is_exact":true,"upper":3,"witness":[0,16,32],"A_d":248,"certified":null},"clifford":{"kind":"required","correction":{"S":[[0,3],[1,1]],"CZ":[[0,3],[0,4],[0,7],[1,2],[1,4],[1,8]]},"rotation_powers":null},"metrics":{"gamma":2.0587454935679026,"gamma_t":1.3410606756416914,"gamma_rho":null,"gamma_claim":2.0587454935679026,"gamma_t_claim":1.3410606756416914,"gamma_rho_claim":null,"v_ex":null,"rate":0.10416666666666667,"t_count":11,"poly_degree":2,"effective_width":5},"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 40","N":11,"d":3,"d_is_exact":true,"origin":"exhaustive classification of generalised triorthogonal protocols through length 54 (Wills, Jain and Singh)","provenance":"Pareto point 40 of the length-54 classification: output class Q5_0000000000142000 (CS45CCZ135CCZ234), (n, S, d_Z) = (48, 11, 3), leading error coefficient 248; the outputs are its 5 logical rows"}]},"references":[{"key":"gong2026magic","short":"Gong et al. (2026)","full":"A. Gong, C. A. Pattison, P. Rall, and A. Wills, \"Magic State Distillation via Codes over Binary Extension Fields,\" arXiv:2608.09727 (2026).","url":"https://arxiv.org/abs/2608.09727","date":"2026-08-10"}]}