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and Albert (arXiv:2408.12752), Table I: [[103,1,19]] quantum QR code (extended QR [104,52,20]) doubled onto [[777,1,17]]; 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); the [[69,1,13]] input's distance is proved here by enumerating every word of SD70 through the punctured coordinate"}},"clifford":{"kind":"powers","correction":{"S":[[0,1],[52,2],[53,2],[54,2],[55,2],[59,2],[61,2],[63,2],[67,2],[68,2],[72,2],[77,2],[78,2],[79,2],[84,2],[86,2],[87,2],[88,2],[89,2],[93,2],[96,2],[99,2],[101,2],[152,2],[153,2],[154,2],[155,2],[156,2],[157,2],[158,2],[159,2],[160,2],[161,2],[162,2],[163,2],[164,2],[165,2],[166,2],[167,2],[168,2],[169,2],[170,2],[171,2],[172,2],[173,2],[174,2],[198,2],[201,2],[202,2],[205,2],[206,2],[208,2],[209,2],[212,2],[215,2],[217,2],[245,2],[247,2],[249,2]],"CZ":[[52,249],[53,249],[54,249],[55,249],[59,249],[61,249],[63,249],[67,249],[68,249],[72,249],[77,249],[78,249],[79,249],[84,249],[86,249],[87,249],[88,249],[89,249],[93,249],[96,249],[99,249],[101,249],[152,247],[153,247],[154,247],[155,247],[156,247],[157,247],[158,247],[159,247],[160,247],[161,247],[162,247],[163,247],[164,247],[165,247],[166,247],[167,247],[168,247],[169,247],[170,247],[171,247],[172,247],[173,247],[174,247],[198,245],[201,245],[202,245],[205,245],[206,245],[208,245],[209,245],[212,245],[215,245],[217,245]]},"rotation_powers":[[0,7],[1,7],[3,7],[8,7],[9,7],[12,7],[13,7],[14,7],[15,7],[17,7],[18,7],[19,7],[20,7],[26,7],[28,7],[29,7],[30,7],[31,7],[32,7],[35,7],[37,7],[42,7],[43,7],[44,7],[48,7],[49,7],[51,7],[105,3],[107,3],[108,3],[109,3],[110,3],[113,3],[114,3],[119,3],[124,3],[125,3],[126,3],[127,3],[128,3],[130,3],[136,3],[137,3],[139,3],[141,3],[142,3],[143,3],[144,3],[148,3],[149,3],[150,3],[153,3],[155,3],[156,3],[157,3],[158,3],[159,3],[160,3],[161,3],[162,3],[163,3],[164,3],[165,3],[166,3],[167,3],[168,3],[169,3],[170,3],[171,3],[172,3],[173,3],[174,3],[175,3],[176,3],[177,3],[178,3],[179,3],[180,3],[181,3],[182,3],[183,3],[184,3],[185,3],[186,3],[187,3],[188,3],[189,3],[190,3],[191,3],[192,3],[193,3],[194,3],[195,3],[196,3],[197,3],[198,3],[199,3],[200,3],[201,3],[202,3],[203,3],[204,3],[205,3],[206,3],[209,3],[212,3],[213,3],[214,3],[215,3],[216,3],[217,3],[220,3],[222,3],[225,3],[226,3],[230,3],[233,3],[237,3],[238,3],[239,3],[243,3],[246,3],[247,3],[249,3],[250,3],[252,3],[253,3],[254,3],[257,3],[307,3],[310,3],[313,3],[314,3],[315,3],[316,3],[317,3],[318,3],[321,3],[323,3],[326,3],[327,3],[331,3],[334,3],[338,3],[339,3],[340,3],[344,3],[347,3],[348,3],[350,3],[351,3],[353,3],[354,3],[355,3],[358,3],[568,3],[569,3],[570,3],[571,3],[572,3],[573,3],[577,3],[579,3],[580,3],[583,3],[584,3],[589,3],[591,3],[593,3],[594,3],[595,3],[599,3],[600,3],[637,3],[638,3],[639,3],[640,3],[641,3],[642,3],[646,3],[648,3],[649,3],[652,3],[653,3],[658,3],[660,3],[662,3],[663,3],[664,3],[668,3],[669,3],[799,3],[802,3],[803,3],[809,3],[810,3],[811,3],[812,3],[813,3],[814,3],[816,3],[817,3],[818,3],[821,3],[842,3],[843,3],[845,3],[846,3],[849,3],[850,3],[851,3],[852,3],[853,3],[860,3],[864,3],[865,3],[867,3],[868,3],[869,3],[870,3],[871,3],[872,3],[873,3],[874,3],[875,3],[876,3],[877,3],[878,3],[879,3],[880,3],[881,3],[882,3],[883,3],[884,3],[885,3],[886,3],[888,3],[892,3],[893,3],[894,3],[896,3],[898,3],[910,3],[911,3],[915,3],[916,3],[917,3],[919,3],[921,3],[933,3],[935,3],[936,3],[938,3],[944,3],[950,3],[951,3],[954,3],[956,3],[957,3],[958,3],[959,3],[960,3],[962,3],[963,3],[964,3],[965,3],[966,3]]},"metrics":{"gamma":3.8457221733650737,"gamma_t":3.8457221733650737,"gamma_rho":3.8457221733650737,"gamma_claim":2.3402112147346563,"gamma_t_claim":2.3402112147346563,"gamma_rho_claim":2.3402112147346563,"v_ex":1,"rate":0.001017293997965412,"t_count":1,"poly_degree":1,"effective_width":1},"provenance":{"regimes":["Jain-Albert doubling: triorthogonal family"],"discovery":"pre-existing","strongest_claim":"a code of the triorthogonal family of Jain and Albert (arXiv:2408.12752, Table I; transversal_t_codes/), self-dual CSS codes doubled onto [[95,1,7]], rebuilt here from the paper's construction; a verified witness, not a maximum","sources":[{"regime":"Jain-Albert doubling: triorthogonal family","file":"transversal_t_codes/factories.json","label":"JA-983","N":251,"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 [[777,1,17]] (Table I)","notes":"its [[69,1,13]] input is the self-dual [70,35,12] code found by transversal_t_codes/search_sd70.py, whose words through the punctured coordinate all weigh at least 14, in place of the paper's [70,35,14] code, which is only formally self-dual"}]},"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"}]}