{"id":"51.5.3.h","row":285,"stored":{"file":"master_catalog.json","row":285,"construction":null,"columns_sha256":null},"parameters":{"n":51,"k":5,"d":3,"N":12,"r":7,"level":3},"gate":{"string":"0+12+012+034+134","human":"T0\u00b7CS12\u00b7CCZ012\u00b7CCZ034\u00b7CCZ134","monomials":[[0],[1,2],[0,1,2],[0,3,4],[1,3,4]],"sk_key":[[0],[0,1,2],[0,3,4],[1,2],[1,3,4]],"sk_fingerprint":"78d653c9252d16c6","sk_canonical_frame":true,"relabelled_into_canonical_frame":true},"circuit":{"columns":[[1,2,4,5,6],[1,2,3,4,7],[2,3,4,5,6,7],[0,1,2,3,8],[0,1,4,6,8],[2,4,5,7,8],[0,2,3,4,6,7,8],[1,2,3,4,9],[1,4,6,9],[1,2,3,5,7,9],[2,6,7,9],[1,8,9],[0,2,5,6,8,9],[0,1,2,7,8,9],[1,2,5,6,7,8,9],[2,3,4,10],[1,2,3,4,5,6,10],[1,2,3,7,10],[0,1,2,3,4,5,6,7,10],[0,2,3,8,10],[1,2,3,4,5,8,10],[2,3,4,5,9,10],[0,2,3,5,6,9,10],[2,3,7,9,10],[0,2,3,4,5,6,7,9,10],[0,1,2,5,6,11],[0,1,7,11],[0,3,5,7,11],[3,4,6,7,11],[0,3,8,11],[0,1,2,4,5,8,11],[0,1,5,6,8,11],[3,5,6,7,8,11],[3,4,9,11],[1,2,4,5,9,11],[1,4,5,6,9,11],[3,7,9,11],[0,1,5,7,9,11],[0,4,6,7,9,11],[1,2,5,6,8,9,11],[5,6,7,8,9,11],[1,10,11],[5,10,11],[0,4,6,10,11],[5,6,7,10,11],[0,1,4,5,8,10,11],[7,8,10,11],[6,9,10,11],[5,6,7,9,10,11],[8,9,10,11],[7,8,9,10,11]]},"distance":{"d":3,"is_exact":true,"upper":3,"witness":[0,1,2],"A_d":184,"certified":null},"clifford":{"kind":"required","correction":{"S":[[0,3],[2,3],[3,1],[4,1],[5,3],[8,2],[10,2],[11,3]],"CZ":[[0,1],[0,2],[0,5],[0,6],[0,8],[0,9],[0,10],[0,11],[1,5],[1,6],[1,8],[1,9],[1,10],[1,11],[2,4],[2,8],[2,10],[3,5],[3,8],[3,10],[3,11],[4,8],[5,8],[5,10],[5,11],[8,10],[8,11],[10,11]]},"rotation_powers":null},"metrics":{"gamma":2.1139284024446385,"gamma_t":1.6861126624481935,"gamma_rho":null,"gamma_claim":2.1139284024446385,"gamma_t_claim":1.6861126624481935,"gamma_rho_claim":null,"v_ex":null,"rate":0.09803921568627451,"t_count":8,"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 57","N":12,"d":3,"d_is_exact":true,"origin":"exhaustive classification of generalised triorthogonal protocols through length 54 (Wills, Jain and Singh)","provenance":"Pareto point 57 of the length-54 classification: output class Q5_0000000001084c00 (T^(tensor 2) tensor (T1CCZ123)), (n, S, d_Z) = (51, 12, 3), leading error coefficient 184; the outputs are its 5 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"}]}