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507,508,509,510,511,512,513,514],[0,502,503,504,505,506,507,508,509,510,511,512,513,514],[0,503,504,505,506,507,508,509,510,511,512,513,514]]},"distance":{"d":3,"is_exact":false,"upper":25,"witness":[0,191,382,549,716,883,1050,1153,1256,1357,1458,1537,1616,1685,1754,1801,1848,1893,1938,1961,1984,2001,2018,2025,2032],"A_d":null,"certified":{"d":25,"is_exact":false,"met":true,"source":"Jain and Albert (arXiv:2408.12752), Table I: [[191,1,27]] quantum QR code (extended QR [192,96,28]) doubled onto [[1651,1,23]]; the doubling theorem of their Sec. III gives d >= min(d_sd, d_tri + 2) = 25, 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":[[313,2],[314,2],[315,2],[316,2],[320,2],[322,2],[324,2],[328,2],[329,2],[333,2],[338,2],[339,2],[340,2],[345,2],[347,2],[348,2],[349,2],[350,2],[354,2],[357,2],[360,2],[362,2],[413,2],[414,2],[415,2],[416,2],[417,2],[418,2],[419,2],[420,2],[421,2],[422,2],[423,2],[424,2],[425,2],[426,2],[427,2],[428,2],[429,2],[430,2],[431,2],[432,2],[433,2],[434,2],[435,2],[459,2],[462,2],[463,2],[466,2],[467,2],[469,2],[470,2],[473,2],[476,2],[478,2],[506,2],[508,2],[510,2],[512,1],[513,2],[514,3]],"CZ":[[0,512],[0,514],[313,510],[314,510],[315,510],[316,510],[320,510],[322,510],[324,510],[328,510],[329,510],[333,510],[338,510],[339,510],[340,510],[345,510],[347,510],[348,510],[349,510],[350,510],[354,510],[357,510],[360,510],[362,510],[413,508],[414,508],[415,508],[416,508],[417,508],[418,508],[419,508],[420,508],[421,508],[422,508],[423,508],[424,508],[425,508],[426,508],[427,508],[428,508],[429,508],[430,508],[431,508],[432,508],[433,508],[434,508],[435,508],[459,506],[462,506],[463,506],[466,506],[467,506],[469,506],[470,506],[473,506],[476,506],[478,506],[512,513],[512,514]]},"rotation_powers":[[0,7],[1,3],[2,3],[3,3],[4,3],[5,3],[6,3],[11,3],[12,3],[13,3],[16,3],[17,3],[19,3],[20,3],[22,3],[25,3],[27,3],[31,3],[32,3],[33,3],[37,3],[40,3],[42,3],[44,3],[45,3],[47,3],[48,3],[49,3],[50,3],[54,3],[55,3],[57,3],[59,3],[60,3],[61,3],[64,3],[65,3],[66,3],[68,3],[69,3],[72,3],[76,3],[77,3],[79,3],[84,3],[88,3],[95,3],[191,5],[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],[382,3],[385,3],[387,3],[389,3],[391,3],[393,3],[394,3],[398,3],[400,3],[404,3],[405,3],[406,3],[407,3],[408,3],[410,3],[412,3],[413,3],[414,3],[416,3],[417,3],[421,3],[423,3],[426,3],[427,3],[428,3],[430,3],[431,3],[433,3],[434,3],[437,3],[440,3],[441,3],[443,3],[445,3],[448,3],[451,3],[452,3],[454,3],[457,3],[459,3],[462,3],[464,3],[548,3],[549,3],[552,3],[554,3],[556,3],[558,3],[560,3],[561,3],[565,3],[567,3],[571,3],[572,3],[573,3],[574,3],[575,3],[577,3],[579,3],[580,3],[581,3],[583,3],[584,3],[588,3],[590,3],[593,3],[594,3],[595,3],[597,3],[598,3],[600,3],[601,3],[604,3],[607,3],[608,3],[610,3],[612,3],[615,3],[618,3],[619,3],[621,3],[624,3],[626,3],[629,3],[631,3],[715,3],[716,5],[883,5],[1050,3],[1052,3],[1057,3],[1058,3],[1061,3],[1062,3],[1063,3],[1064,3],[1066,3],[1067,3],[1068,3],[1069,3],[1075,3],[1077,3],[1078,3],[1079,3],[1080,3],[1081,3],[1084,3],[1086,3],[1091,3],[1092,3],[1093,3],[1097,3],[1098,3],[1100,3],[1152,3],[1153,3],[1155,3],[1160,3],[1161,3],[1164,3],[1165,3],[1166,3],[1167,3],[1169,3],[1170,3],[1171,3],[1172,3],[1178,3],[1180,3],[1181,3],[1182,3],[1183,3],[1184,3],[1187,3],[1189,3],[1194,3],[1195,3],[1196,3],[1200,3],[1201,3],[1203,3],[1255,3],[1256,3],[1258,3],[1259,3],[1261,3],[1263,3],[1266,3],[1267,3],[1269,3],[1275,3],[1277,3],[1281,3],[1282,3],[1283,3],[1284,3],[1285,3],[1286,3],[1287,3],[1292,3],[1293,3],[1294,3],[1297,3],[1298,3],[1299,3],[1300,3],[1304,3],[1307,3],[1353,3],[1357,3],[1359,3],[1360,3],[1362,3],[1364,3],[1367,3],[1368,3],[1370,3],[1376,3],[1378,3],[1382,3],[1383,3],[1384,3],[1385,3],[1386,3],[1387,3],[1388,3],[1393,3],[1394,3],[1395,3],[1398,3],[1399,3],[1400,3],[1401,3],[1405,3],[1408,3],[1454,3],[1458,3],[1459,3],[1461,3],[1462,3],[1468,3],[1470,3],[1471,3],[1473,3],[1475,3],[1476,3],[1477,3],[1478,3],[1481,3],[1482,3],[1483,3],[1484,3],[1486,3],[1487,3],[1488,3],[1492,3],[1493,3],[1496,3],[1536,3],[1539,3],[1542,3],[1543,3],[1544,3],[1545,3],[1546,3],[1548,3],[1551,3],[1553,3],[1558,3],[1559,3],[1564,3],[1568,3],[1569,3],[1570,3],[1573,3],[1574,3],[1576,3],[1577,3],[1578,3],[1579,3],[1580,3],[1581,3],[1582,3],[1583,3],[1584,3],[1585,3],[1586,3],[1587,3],[1588,3],[1589,3],[1590,3],[1591,3],[1592,3],[1593,3],[1594,3],[1595,3],[1596,3],[1597,3],[1598,3],[1599,3],[1600,3],[1601,3],[1602,3],[1603,3],[1604,3],[1605,3],[1606,3],[1607,3],[1608,3],[1609,3],[1610,3],[161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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-2033","N":515,"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 [[1651,1,23]] (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"}]}