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and Albert (arXiv:2408.12752), Table II: [[79,1,15]] quantum QR code (extended QR [80,40,16]) doubled onto [[283,1,11]]; the doubling theorem of their Sec. III gives d >= min(d_sd, d_tri + 2) = 13, with d_sd at least the classical distance less one (their Lemma 2.4)"}},"clifford":{"kind":"powers","correction":{"S":[[111,1],[112,2],[113,3]],"CZ":[[0,111],[0,113],[111,112],[111,113]]},"rotation_powers":[[0,7],[1,7],[2,7],[3,7],[4,7],[5,7],[6,7],[7,7],[8,7],[9,7],[10,7],[11,7],[12,7],[13,7],[14,7],[15,7],[16,7],[17,7],[18,7],[19,7],[20,7],[21,7],[22,7],[23,7],[24,7],[25,7],[26,7],[27,7],[28,7],[29,7],[30,7],[31,7],[32,7],[33,7],[34,7],[35,7],[36,7],[37,7],[38,7],[39,7],[40,7],[41,7],[42,7],[43,7],[44,7],[45,7],[46,7],[47,7],[48,7],[49,7],[50,7],[51,7],[52,7],[53,7],[54,7],[55,7],[56,7],[57,7],[58,7],[59,7],[60,7],[61,7],[62,7],[63,7],[64,7],[65,7],[66,7],[67,7],[68,7],[69,7],[70,7],[71,7],[72,7],[73,7],[74,7],[75,7],[76,7],[77,7],[78,7],[79,7],[80,7],[81,7],[82,7],[83,7],[84,7],[85,7],[86,7],[87,7],[88,7],[89,7],[90,7],[91,7],[92,7],[93,7],[94,7],[95,7],[96,7],[97,7],[98,7],[99,7],[100,7],[101,7],[102,7],[103,7],[104,7],[105,7],[106,7],[107,7],[108,7],[109,7],[110,7],[111,7],[112,7],[113,7],[114,7],[115,7],[116,7],[117,7],[118,7],[119,7],[120,7],[121,7],[122,7],[123,7],[124,7],[125,7],[126,7],[127,7],[128,7],[129,7],[130,7],[131,7],[132,7],[133,7],[134,7],[135,7],[136,7],[137,7],[138,7],[139,7],[140,7],[141,7],[142,7],[143,7],[144,7],[145,7],[146,7],[147,7],[148,7],[149,7],[150,7],[151,7],[152,7],[153,7],[154,7],[155,7],[156,7],[157,7],[252,7],[253,7],[254,7],[255,7],[256,7],[257,7],[258,7],[259,7],[260,7],[261,7],[262,7],[263,7],[264,7],[265,7],[266,7],[267,7],[268,7],[269,7],[270,7],[271,7],[272,7],[273,7],[274,7],[275,7],[276,7],[277,7],[278,7],[279,7],[280,7],[281,7],[282,7],[283,7],[284,7],[285,7],[286,7],[287,7],[288,7],[289,7],[290,7],[291,7],[292,7],[293,7],[294,7],[295,7],[296,7],[297,7],[298,7],[299,7],[300,7],[301,7],[302,7],[303,7],[304,7],[305,7],[306,7],[307,7],[308,7],[309,7],[310,7],[311,7],[312,7],[313,7],[314,7],[315,7],[316,7],[317,7],[318,7],[319,7],[320,7],[321,7],[322,7],[323,7],[324,7],[325,7],[326,7],[327,7],[328,7],[329,7],[330,7],[331,7],[332,7],[333,7],[334,7],[335,7],[336,7],[337,7],[338,7],[339,7],[340,7],[341,7],[342,7],[343,7],[344,7],[345,7]]},"metrics":{"gamma":3.3983606505343005,"gamma_t":3.3983606505343005,"gamma_rho":3.3983606505343005,"gamma_claim":2.3739435079813873,"gamma_t_claim":2.3739435079813873,"gamma_rho_claim":2.3739435079813873,"v_ex":1,"rate":0.0022675736961451248,"t_count":1,"poly_degree":1,"effective_width":1},"provenance":{"regimes":["Jain-Albert doubling: weak triply even family"],"discovery":"pre-existing","strongest_claim":"a code of the weak triply even family of Jain and Albert (arXiv:2408.12752, Table II; transversal_t_codes/): quadratic-residue CSS codes doubled onto [[95,1,7]], rebuilt here from the paper's construction, where T on some qubits and T-dagger on the rest is a logical T; a verified witness, not a maximum","sources":[{"regime":"Jain-Albert doubling: weak triply even family","file":"transversal_t_codes/factories.json","label":"JA-441","N":114,"d":6,"d_is_exact":false,"origin":"S. P. Jain and V. V. Albert, \"Transversal Clifford and T-gate codes of short length and high distance\", IEEE J. Sel. Areas Inf. Theory 6, 127 (2025), arXiv:2408.12752","provenance":"[[79,1,15]] quantum QR code (extended QR [80,40,16]) doubled onto [[283,1,11]] (Table II)"}]},"references":[{"key":"jain2025transversal","short":"Jain & Albert (2025)","full":"S. P. Jain and V. V. Albert, \"Transversal Clifford and T-gate codes of short length and high distance,\" IEEE J. Sel. Areas Inf. Theory 6, 127-137 (2025); arXiv:2408.12752.","url":"https://doi.org/10.1109/JSAIT.2025.3570832","date":"2025-05-16"}]}