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and Albert (arXiv:2408.12752), Table I: [[191,1,27]] quantum QR code (extended QR [192,96,28]) doubled onto [[2033,1,25]]; the doubling theorem of their Sec. III gives d >= min(d_sd, d_tri + 2) = 27, 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],[408,2],[409,2],[410,2],[411,2],[415,2],[417,2],[419,2],[423,2],[424,2],[428,2],[433,2],[434,2],[435,2],[440,2],[442,2],[443,2],[444,2],[445,2],[449,2],[452,2],[455,2],[457,2],[508,2],[509,2],[510,2],[511,2],[512,2],[513,2],[514,2],[515,2],[516,2],[517,2],[518,2],[519,2],[520,2],[521,2],[522,2],[523,2],[524,2],[525,2],[526,2],[527,2],[528,2],[529,2],[530,2],[554,2],[557,2],[558,2],[561,2],[562,2],[564,2],[565,2],[568,2],[571,2],[573,2],[601,2],[603,2],[605,2],[607,1],[608,2],[609,3]],"CZ":[[0,607],[0,609],[408,605],[409,605],[410,605],[411,605],[415,605],[417,605],[419,605],[423,605],[424,605],[428,605],[433,605],[434,605],[435,605],[440,605],[442,605],[443,605],[444,605],[445,605],[449,605],[452,605],[455,605],[457,605],[508,603],[509,603],[510,603],[511,603],[512,603],[513,603],[514,603],[515,603],[516,603],[517,603],[518,603],[519,603],[520,603],[521,603],[522,603],[523,603],[524,603],[525,603],[526,603],[527,603],[528,603],[529,603],[530,603],[554,601],[557,601],[558,601],[561,601],[562,601],[564,601],[565,601],[568,601],[571,601],[573,601],[607,608],[607,609],[607,610],[609,610]]},"rotation_powers":[[0,7],[1,7],[2,7],[3,7],[4,7],[5,7],[6,7],[11,7],[12,7],[13,7],[16,7],[17,7],[19,7],[20,7],[22,7],[25,7],[27,7],[31,7],[32,7],[33,7],[37,7],[40,7],[42,7],[44,7],[45,7],[47,7],[48,7],[49,7],[50,7],[54,7],[55,7],[57,7],[59,7],[60,7],[61,7],[64,7],[65,7],[66,7],[68,7],[69,7],[72,7],[76,7],[77,7],[79,7],[84,7],[88,7],[95,7],[198,3],[199,3],[200,3],[201,3],[205,3],[206,3],[209,3],[212,3],[214,3],[215,3],[217,3],[219,3],[220,3],[221,3],[225,3],[226,3],[227,3],[229,3],[230,3],[232,3],[234,3],[237,3],[242,3],[243,3],[244,3],[247,3],[249,3],[253,3],[254,3],[258,3],[261,3],[262,3],[264,3],[265,3],[266,3],[269,3],[271,3],[272,3],[273,3],[274,3],[276,3],[277,3],[278,3],[280,3],[281,3],[282,3],[283,3],[284,3],[285,3],[287,3],[288,3],[289,3],[290,3],[291,3],[292,3],[293,3],[294,3],[295,3],[296,3],[297,3],[298,3],[299,3],[300,3],[301,3],[302,3],[303,3],[304,3],[305,3],[306,3],[307,3],[308,3],[309,3],[310,3],[311,3],[312,3],[313,3],[314,3],[315,3],[316,3],[317,3],[318,3],[319,3],[320,3],[321,3],[322,3],[323,3],[324,3],[325,3],[326,3],[327,3],[328,3],[329,3],[330,3],[331,3],[332,3],[333,3],[334,3],[335,3],[336,3],[337,3],[338,3],[339,3],[340,3],[341,3],[342,3],[343,3],[344,3],[345,3],[346,3],[347,3],[348,3],[349,3],[350,3],[351,3],[352,3],[353,3],[354,3],[355,3],[356,3],[357,3],[358,3],[359,3],[360,3],[361,3],[362,3],[363,3],[364,3],[365,3],[366,3],[367,3],[368,3],[369,3],[370,3],[371,3],[372,3],[373,3],[374,3],[375,3],[376,3],[377,3],[378,3],[379,3],[380,3],[381,3],[382,7],[383,3],[384,3],[385,3],[386,3],[387,3],[392,3],[393,3],[394,3],[397,3],[398,3],[400,3],[401,3],[403,3],[406,3],[408,3],[412,3],[413,3],[414,3],[418,3],[421,3],[423,3],[425,3],[426,3],[428,3],[429,3],[430,3],[431,3],[435,3],[436,3],[438,3],[440,3],[441,3],[442,3],[445,3],[446,3],[447,3],[449,3],[450,3],[453,3],[457,3],[458,3],[460,3],[465,3],[469,3],[476,3],[572,3],[573,5],[579,3],[580,3],[581,3],[582,3],[586,3],[587,3],[590,3],[593,3],[595,3],[596,3],[598,3],[600,3],[601,3],[602,3],[606,3],[607,3],[608,3],[610,3],[611,3],[613,3],[615,3],[618,3],[623,3],[624,3],[625,3],[628,3],[630,3],[634,3],[635,3],[639,3],[642,3],[643,3],[645,3],[646,3],[647,3],[650,3],[652,3],[653,3],[654,3],[655,3],[657,3],[658,3],[659,3],[661,3],[662,3],[663,3],[664,3],[665,3],[666,3],[668,3],[669,3],[670,3],[671,3],[672,3],[673,3],[674,3],[675,3],[676,3],[677,3],[678,3],[679,3],[680,3],[681,3],[682,3],[683,3],[684,3],[685,3],[686,3],[687,3],[688,3],[689,3],[690,3],[691,3],[692,3],[693,3],[694,3],[695,3],[696,3],[697,3],[698,3],[699,3],[700,3],[701,3],[702,3],[703,3],[704,3],[705,3],[706,3],[707,3],[708,3],[709,3],[710,3],[711,3],[712,3],[713,3],[714,3],[715,3],[716,3],[717,3],[718,3],[719,3],[720,3],[721,3],[722,3],[723,3],[724,3],[725,3],[726,3],[727,3],[728,3],[729,3],[730,3],[731,3],[732,3],[733,3],[734,3],[735,3],[736,3],[737,3],[738,3],[739,3],[740,3],[741,3],[742,3],[743,3],[744,3],[745,3],[746,3],[747,3],[748,3],[749,3],[750,3],[751,3],[752,3],[753,3],[754,3],[755,3],[756,3],[757,3],[758,3],[759,3],[760,3],[761,3],[762,3],[764,3],[767,3],[769,3],[771,3],[773,3],[775,3],[776,3],[780,3],[782,3],[786,3],[787,3],[788,3],[789,3],[790,3],[792,3],[794,3],[795,3],[796,3],[798,3],[799,3],[803,3],[805,3],[808,3],[809,3],[810,3],[812,3],[813,3],[815,3],[816,3],[819,3],[822,3],[823,3],[825,3],[827,3],[830,3],[833,3],[834,3],[836,3],[839,3],[841,3],[844,3],[846,3],[930,3],[931,3],[934,3],[936,3],[938,3],[940,3],[942,3],[943,3],[947,3],[949,3],[953,3],[954,3],[955,3],[956,3],[957,3],[959,3],[961,3],[962,3],[963,3],[965,3],[966,3],[970,3],[972,3],[975,3],[976,3],[977,3],[979,3],[980,3],[982,3],[983,3],[986,3],[989,3],[990,3],[992,3],[994,3],[997,3],[1000,3],[1001,3],[1003,3],[1006,3],[1008,3],[1011,3],[1013,3],[1097,3],[1098,5],[1265,5],[1432,3],[1434,3],[1439,3],[1440,3],[1443,3],[1444,3],[1445,3],[1446,3],[1448,3],[1449,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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-2415","N":611,"d":3,"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":"[[191,1,27]] quantum QR code (extended QR [192,96,28]) doubled onto [[2033,1,25]] (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"}]}