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76,1155,1234,1303,1372,1419,1466,1511,1556,1579,1602,1619,1636,1643,1650],"A_d":null,"certified":{"d":23,"is_exact":false,"met":true,"source":"Jain and Albert (arXiv:2408.12752), Table I: [[167,1,23]] quantum QR code (extended QR [168,84,24]) doubled onto [[1317,1,21]]; the doubling theorem of their Sec. III gives d >= min(d_sd, d_tri + 2) = 23, 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":[[0,3],[218,2],[219,2],[220,2],[221,2],[225,2],[227,2],[229,2],[233,2],[234,2],[238,2],[243,2],[244,2],[245,2],[250,2],[252,2],[253,2],[254,2],[255,2],[259,2],[262,2],[265,2],[267,2],[318,2],[319,2],[320,2],[321,2],[322,2],[323,2],[324,2],[325,2],[326,2],[327,2],[328,2],[329,2],[330,2],[331,2],[332,2],[333,2],[334,2],[335,2],[336,2],[337,2],[338,2],[339,2],[340,2],[364,2],[367,2],[368,2],[371,2],[372,2],[374,2],[375,2],[378,2],[381,2],[383,2],[411,2],[413,2],[415,2],[417,1],[418,2]],"CZ":[[0,417],[218,415],[219,415],[220,415],[221,415],[225,415],[227,415],[229,415],[233,415],[234,415],[238,415],[243,415],[244,415],[245,415],[250,415],[252,415],[253,415],[254,415],[255,415],[259,415],[262,415],[265,415],[267,415],[318,413],[319,413],[320,413],[321,413],[322,413],[323,413],[324,413],[325,413],[326,413],[327,413],[328,413],[329,413],[330,413],[331,413],[332,413],[333,413],[334,413],[335,413],[336,413],[337,413],[338,413],[339,413],[340,413],[364,411],[367,411],[368,411],[371,411],[372,411],[374,411],[375,411],[378,411],[381,411],[383,411],[417,418]]},"rotation_powers":[[0,7],[1,7],[4,7],[6,7],[8,7],[10,7],[12,7],[13,7],[17,7],[19,7],[23,7],[24,7],[25,7],[26,7],[27,7],[29,7],[31,7],[32,7],[33,7],[35,7],[36,7],[40,7],[42,7],[45,7],[46,7],[47,7],[49,7],[50,7],[52,7],[53,7],[56,7],[59,7],[60,7],[62,7],[64,7],[67,7],[70,7],[71,7],[73,7],[76,7],[78,7],[81,7],[83,7],[169,3],[170,3],[172,3],[174,3],[176,3],[178,3],[181,3],[182,3],[183,3],[185,3],[187,3],[188,3],[189,3],[195,3],[197,3],[201,3],[204,3],[205,3],[206,3],[208,3],[210,3],[211,3],[215,3],[218,3],[221,3],[222,3],[224,3],[225,3],[228,3],[230,3],[232,3],[233,3],[235,3],[236,3],[239,3],[241,3],[242,3],[244,3],[246,3],[247,3],[249,3],[251,3],[252,3],[253,3],[254,3],[255,3],[256,3],[257,3],[258,3],[259,3],[260,3],[261,3],[262,3],[263,3],[264,3],[265,3],[266,3],[267,3],[268,3],[269,3],[270,3],[271,3],[272,3],[273,3],[274,3],[275,3],[276,3],[277,3],[278,3],[279,3],[280,3],[281,3],[282,3],[283,3],[284,3],[285,3],[286,3],[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],[334,7],[337,3],[339,3],[341,3],[343,3],[345,3],[346,3],[350,3],[352,3],[356,3],[357,3],[358,3],[359,3],[360,3],[362,3],[364,3],[365,3],[366,3],[368,3],[369,3],[373,3],[375,3],[378,3],[379,3],[380,3],[382,3],[383,3],[385,3],[386,3],[389,3],[392,3],[393,3],[395,3],[397,3],[400,3],[403,3],[404,3],[406,3],[409,3],[411,3],[414,3],[416,3],[500,3],[501,7],[504,3],[506,3],[508,3],[510,3],[512,3],[513,3],[517,3],[519,3],[523,3],[524,3],[525,3],[526,3],[527,3],[529,3],[531,3],[532,3],[533,3],[535,3],[536,3],[540,3],[542,3],[545,3],[546,3],[547,3],[549,3],[550,3],[552,3],[553,3],[556,3],[559,3],[560,3],[562,3],[564,3],[567,3],[570,3],[571,3],[573,3],[576,3],[578,3],[581,3],[583,3],[667,3],[874,3],[877,3],[880,3],[881,3],[882,3],[883,3],[884,3],[885,3],[888,3],[890,3],[893,3],[894,3],[898,3],[901,3],[905,3],[906,3],[907,3],[911,3],[914,3],[915,3],[917,3],[918,3],[920,3],[921,3],[922,3],[925,3],[975,3],[978,3],[981,3],[982,3],[983,3],[984,3],[985,3],[986,3],[989,3],[991,3],[994,3],[995,3],[999,3],[1002,3],[1006,3],[1007,3],[1008,3],[1012,3],[1015,3],[1016,3],[1018,3],[1019,3],[1021,3],[1022,3],[1023,3],[1026,3],[1236,3],[1237,3],[1238,3],[1239,3],[1240,3],[1241,3],[1245,3],[1247,3],[1248,3],[1251,3],[1252,3],[1257,3],[1259,3],[1261,3],[1262,3],[1263,3],[1267,3],[1268,3],[1305,3],[1306,3],[1307,3],[1308,3],[1309,3],[1310,3],[1314,3],[1316,3],[1317,3],[1320,3],[1321,3],[1326,3],[1328,3],[1330,3],[1331,3],[1332,3],[1336,3],[1337,3],[1467,3],[1470,3],[1471,3],[1477,3],[1478,3],[1479,3],[1480,3],[1481,3],[1482,3],[1484,3],[1485,3],[1486,3],[1489,3],[1510,3],[1511,3],[1513,3],[1514,3],[1517,3],[1518,3],[1519,3],[1520,3],[1521,3],[1528,3],[1532,3],[1533,3],[1535,3],[1536,3],[1537,3],[1538,3],[1539,3],[1540,3],[1541,3],[1542,3],[1543,3],[1544,3],[1545,3],[1546,3],[1547,3],[1548,3],[1549,3],[1550,3],[1551,3],[1552,3],[1553,3],[1554,3],[1556,3],[1560,3],[1561,3],[1562,3],[1564,3],[1566,3],[1578,3],[1579,3],[1583,3],[1584,3],[1585,3],[1587,3],[1589,3],[1601,3],[1603,3],[1604,3],[1606,3],[1612,3],[1618,3],[1619,3],[1622,3],[1624,3],[1625,3],[1626,3],[1627,3],[1628,3],[1630,3],[1631,3],[1632,3],[1633,3],[1634,3]]},"metrics":{"gamma":6.744086626686573,"gamma_t":6.744086626686573,"gamma_rho":6.744086626686573,"gamma_claim":2.3629883947097503,"gamma_t_claim":2.3629883947097503,"gamma_rho_claim":2.3629883947097503,"v_ex":1,"rate":0.0006056935190793458,"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-1651","N":419,"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":"[[167,1,23]] quantum QR code (extended QR [168,84,24]) doubled onto [[1317,1,21]] (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"}]}