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0,971,1162,1329,1496,1663,1830,1933,2036,2137,2238,2317,2396,2465,2534,2581,2628,2673,2718,2741,2764,2781,2798,2805,2812],"A_d":null,"certified":{"d":29,"is_exact":false,"met":true,"source":"Jain and Albert (arXiv:2408.12752), Table I: [[199,1,31]] quantum QR code (extended QR [200,100,32]) doubled onto [[2415,1,27]]; the doubling theorem of their Sec. III gives d >= min(d_sd, d_tri + 2) = 29, 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":[[0,2],[507,2],[508,2],[509,2],[510,2],[514,2],[516,2],[518,2],[522,2],[523,2],[527,2],[532,2],[533,2],[534,2],[539,2],[541,2],[542,2],[543,2],[544,2],[548,2],[551,2],[554,2],[556,2],[607,2],[608,2],[609,2],[610,2],[611,2],[612,2],[613,2],[614,2],[615,2],[616,2],[617,2],[618,2],[619,2],[620,2],[621,2],[622,2],[623,2],[624,2],[625,2],[626,2],[627,2],[628,2],[629,2],[653,2],[656,2],[657,2],[660,2],[661,2],[663,2],[664,2],[667,2],[670,2],[672,2],[700,2],[702,2],[704,2],[706,1],[707,2],[708,3],[710,1]],"CZ":[[0,706],[0,708],[0,710],[507,704],[508,704],[509,704],[510,704],[514,704],[516,704],[518,704],[522,704],[523,704],[527,704],[532,704],[533,704],[534,704],[539,704],[541,704],[542,704],[543,704],[544,704],[548,704],[551,704],[554,704],[556,704],[607,702],[608,702],[609,702],[610,702],[611,702],[612,702],[613,702],[614,702],[615,702],[616,702],[617,702],[618,702],[619,702],[620,702],[621,702],[622,702],[623,702],[624,702],[625,702],[626,702],[627,702],[628,702],[629,702],[653,700],[656,700],[657,700],[660,700],[661,700],[663,700],[664,700],[667,700],[670,700],[672,700],[706,707],[706,708],[706,709],[706,710],[708,709],[708,710]]},"rotation_powers":[[0,7],[1,3],[3,3],[5,3],[6,3],[8,3],[10,3],[12,3],[14,3],[17,3],[21,3],[22,3],[23,3],[27,3],[29,3],[30,3],[32,3],[33,3],[35,3],[37,3],[40,3],[41,3],[42,3],[43,3],[45,3],[48,3],[49,3],[52,3],[54,3],[55,3],[56,3],[57,3],[59,3],[66,3],[67,3],[69,3],[71,3],[75,3],[76,3],[77,3],[79,3],[83,3],[87,3],[88,3],[89,3],[90,3],[91,3],[94,3],[96,3],[97,3],[99,3],[199,5],[201,3],[203,3],[206,3],[208,3],[210,3],[212,3],[214,3],[215,3],[217,3],[218,3],[219,3],[223,3],[224,3],[225,3],[227,3],[230,3],[233,3],[235,3],[237,3],[238,3],[243,3],[245,3],[246,3],[249,3],[250,3],[252,3],[257,3],[259,3],[260,3],[261,3],[262,3],[263,3],[264,3],[267,3],[269,3],[271,3],[272,3],[273,3],[277,3],[279,3],[280,3],[281,3],[283,3],[284,3],[285,3],[291,3],[292,3],[294,3],[297,3],[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],[382,3],[383,3],[384,3],[385,3],[386,3],[387,3],[388,3],[389,3],[390,3],[391,3],[392,3],[393,3],[394,3],[395,3],[396,3],[397,3],[398,3],[399,3],[400,3],[401,3],[402,3],[403,3],[408,3],[409,3],[410,3],[413,3],[414,3],[416,3],[417,3],[419,3],[422,3],[424,3],[428,3],[429,3],[430,3],[434,3],[437,3],[439,3],[441,3],[442,3],[444,3],[445,3],[446,3],[447,3],[451,3],[452,3],[454,3],[456,3],[457,3],[458,3],[461,3],[462,3],[463,3],[465,3],[466,3],[469,3],[473,3],[474,3],[476,3],[481,3],[485,3],[492,3],[588,3],[595,3],[596,3],[597,3],[598,3],[602,3],[603,3],[606,3],[609,3],[611,3],[612,3],[614,3],[616,3],[617,3],[618,3],[622,3],[623,3],[624,3],[626,3],[627,3],[629,3],[631,3],[634,3],[639,3],[640,3],[641,3],[644,3],[646,3],[650,3],[651,3],[655,3],[658,3],[659,3],[661,3],[662,3],[663,3],[666,3],[668,3],[669,3],[670,3],[671,3],[673,3],[674,3],[675,3],[677,3],[678,3],[679,3],[680,3],[681,3],[682,3],[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],[763,3],[764,3],[765,3],[766,3],[767,3],[768,3],[769,3],[770,3],[771,3],[772,3],[773,3],[774,3],[775,3],[776,3],[777,3],[778,3],[780,7],[781,3],[782,3],[783,3],[784,3],[785,3],[790,3],[791,3],[792,3],[795,3],[796,3],[798,3],[799,3],[801,3],[804,3],[806,3],[810,3],[811,3],[812,3],[816,3],[819,3],[821,3],[823,3],[824,3],[826,3],[827,3],[828,3],[829,3],[833,3],[834,3],[836,3],[838,3],[839,3],[840,3],[843,3],[844,3],[845,3],[847,3],[848,3],[851,3],[855,3],[856,3],[858,3],[863,3],[867,3],[874,3],[970,3],[971,5],[977,3],[978,3],[979,3],[980,3],[984,3],[985,3],[988,3],[991,3],[993,3],[994,3],[996,3],[998,3],[999,3],[1000,3],[1004,3],[1005,3],[1006,3],[1008,3],[1009,3],[1011,3],[1013,3],[1016,3],[1021,3],[1022,3],[1023,3],[1026,3],[1028,3],[1032,3],[1033,3],[1037,3],[1040,3],[1041,3],[1043,3],[1044,3],[1045,3],[1048,3],[1050,3],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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-2813","N":711,"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":"[[199,1,31]] quantum QR code (extended QR [200,100,32]) doubled onto [[2415,1,27]] (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"}]}