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and Albert (arXiv:2408.12752), Table II: [[103,1,19]] quantum QR code (extended QR [104,52,20]) doubled onto [[805,1,17]]; the doubling theorem of their Sec. III gives d >= min(d_sd, d_tri + 2) = 19, with d_sd at least the classical distance less one (their Lemma 2.4)"}},"clifford":{"kind":"powers","correction":{"S":[[0,3],[252,1],[253,2],[254,3],[256,1],[257,2]],"CZ":[[0,252],[0,254],[0,256],[252,253],[252,254],[252,255],[252,256],[252,257],[254,255],[254,256],[254,257],[256,257]]},"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],[158,7],[159,7],[160,7],[161,7],[162,7],[163,7],[164,7],[165,7],[166,7],[167,7],[168,7],[169,7],[170,7],[171,7],[172,7],[173,7],[174,7],[175,7],[176,7],[177,7],[178,7],[179,7],[180,7],[181,7],[182,7],[183,7],[184,7],[185,7],[186,7],[187,7],[188,7],[189,7],[190,7],[191,7],[192,7],[193,7],[194,7],[195,7],[196,7],[197,7],[198,7],[199,7],[200,7],[201,7],[202,7],[203,7],[204,7],[205,7],[412,7],[413,7],[414,7],[415,7],[416,7],[417,7],[418,7],[419,7],[420,7],[421,7],[422,7],[423,7],[424,7],[425,7],[426,7],[427,7],[428,7],[429,7],[430,7],[431,7],[432,7],[433,7],[434,7],[435,7],[436,7],[437,7],[438,7],[439,7],[440,7],[441,7],[442,7],[443,7],[444,7],[445,7],[446,7],[447,7],[448,7],[449,7],[450,7],[451,7],[452,7],[453,7],[454,7],[455,7],[456,7],[457,7],[458,7],[459,7],[460,7],[461,7],[462,7],[463,7],[464,7],[465,7],[466,7],[467,7],[468,7],[469,7],[470,7],[471,7],[472,7],[473,7],[474,7],[475,7],[476,7],[477,7],[478,7],[479,7],[480,7],[481,7],[482,7],[483,7],[484,7],[485,7],[486,7],[487,7],[488,7],[489,7],[490,7],[491,7],[492,7],[493,7],[494,7],[495,7],[496,7],[497,7],[498,7],[499,7],[500,7],[501,7],[502,7],[503,7],[504,7],[505,7],[506,7],[507,7],[508,7],[509,7],[510,7],[511,7],[512,7],[513,7],[514,7],[515,7],[516,7],[517,7],[518,7],[519,7],[520,7],[521,7],[522,7],[523,7],[524,7],[525,7],[526,7],[527,7],[528,7],[529,7],[530,7],[531,7],[532,7],[533,7],[534,7],[535,7],[536,7],[537,7],[538,7],[539,7],[540,7],[541,7],[542,7],[543,7],[544,7],[545,7],[546,7],[547,7],[548,7],[549,7],[550,7],[551,7],[552,7],[553,7],[554,7],[555,7],[556,7],[557,7],[558,7],[559,7],[560,7],[561,7],[562,7],[563,7],[564,7],[565,7],[566,7],[567,7],[568,7],[569,7],[728,7],[729,7],[730,7],[731,7],[732,7],[733,7],[734,7],[735,7],[736,7],[737,7],[738,7],[739,7],[740,7],[741,7],[742,7],[743,7],[744,7],[745,7],[746,7],[747,7],[748,7],[749,7],[750,7],[751,7],[752,7],[753,7],[754,7],[755,7],[756,7],[757,7],[758,7],[759,7],[760,7],[761,7],[762,7],[763,7],[764,7],[765,7],[766,7],[767,7],[768,7],[769,7],[770,7],[771,7],[772,7],[773,7],[774,7],[775,7],[776,7],[777,7],[778,7],[779,7],[780,7],[781,7],[782,7],[783,7],[784,7],[785,7],[786,7],[787,7],[788,7],[789,7],[790,7],[791,7],[792,7],[793,7],[794,7],[795,7],[796,7],[797,7],[798,7],[799,7],[800,7],[801,7],[802,7],[803,7],[804,7],[805,7],[806,7],[807,7],[808,7],[809,7],[810,7],[811,7],[812,7],[813,7],[814,7],[815,7],[816,7],[817,7],[818,7],[819,7],[820,7],[821,7],[916,7],[917,7],[918,7],[919,7],[920,7],[921,7],[922,7],[923,7],[924,7],[925,7],[926,7],[927,7],[928,7],[929,7],[930,7],[931,7],[932,7],[933,7],[934,7],[935,7],[936,7],[937,7],[938,7],[939,7],[940,7],[941,7],[942,7],[943,7],[944,7],[945,7],[946,7],[947,7],[948,7],[949,7],[950,7],[951,7],[952,7],[953,7],[954,7],[955,7],[956,7],[957,7],[958,7],[959,7],[960,7],[961,7],[962,7],[963,7],[964,7],[965,7],[966,7],[967,7],[968,7],[969,7],[970,7],[971,7],[972,7],[973,7],[974,7],[975,7],[976,7],[977,7],[978,7],[979,7],[980,7],[981,7],[982,7],[983,7],[984,7],[985,7],[986,7],[987,7],[988,7],[989,7],[990,7],[991,7],[992,7],[993,7],[994,7],[995,7],[996,7],[997,7],[998,7],[999,7],[1000,7],[1001,7],[1002,7],[1003,7],[1004,7],[1005,7],[1006,7],[1007,7],[1008,7],[1009,7],[1010,7]]},"metrics":{"gamma":3.8613973236046344,"gamma_t":3.8613973236046344,"gamma_rho":3.8613973236046344,"gamma_claim":2.3497499075287775,"gamma_t_claim":2.3497499075287775,"gamma_rho_claim":2.3497499075287775,"v_ex":1,"rate":0.0009891196834817012,"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-1011","N":258,"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":"[[103,1,19]] quantum QR code (extended QR [104,52,20]) doubled onto [[805,1,17]] (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"}]}