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,"source":"Jain and Albert (arXiv:2408.12752), Table I: [[167,1,23]] quantum QR code (extended QR [168,84,24]) doubled onto [[983,1,19]]; the doubling theorem of their Sec. III gives d >= min(d_sd, d_tri + 2) = 21, 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],[135,2],[136,2],[137,2],[138,2],[142,2],[144,2],[146,2],[150,2],[151,2],[155,2],[160,2],[161,2],[162,2],[167,2],[169,2],[170,2],[171,2],[172,2],[176,2],[179,2],[182,2],[184,2],[235,2],[236,2],[237,2],[238,2],[239,2],[240,2],[241,2],[242,2],[243,2],[244,2],[245,2],[246,2],[247,2],[248,2],[249,2],[250,2],[251,2],[252,2],[253,2],[254,2],[255,2],[256,2],[257,2],[281,2],[284,2],[285,2],[288,2],[289,2],[291,2],[292,2],[295,2],[298,2],[300,2],[328,2],[330,2],[332,2],[334,1]],"CZ":[[0,334],[135,332],[136,332],[137,332],[138,332],[142,332],[144,332],[146,332],[150,332],[151,332],[155,332],[160,332],[161,332],[162,332],[167,332],[169,332],[170,332],[171,332],[172,332],[176,332],[179,332],[182,332],[184,332],[235,330],[236,330],[237,330],[238,330],[239,330],[240,330],[241,330],[242,330],[243,330],[244,330],[245,330],[246,330],[247,330],[248,330],[249,330],[250,330],[251,330],[252,330],[253,330],[254,330],[255,330],[256,330],[257,330],[281,328],[284,328],[285,328],[288,328],[289,328],[291,328],[292,328],[295,328],[298,328],[300,328]]},"rotation_powers":[[0,7],[1,3],[4,3],[6,3],[8,3],[10,3],[12,3],[13,3],[17,3],[19,3],[23,3],[24,3],[25,3],[26,3],[27,3],[29,3],[31,3],[32,3],[33,3],[35,3],[36,3],[40,3],[42,3],[45,3],[46,3],[47,3],[49,3],[50,3],[52,3],[53,3],[56,3],[59,3],[60,3],[62,3],[64,3],[67,3],[70,3],[71,3],[73,3],[76,3],[78,3],[81,3],[83,3],[167,5],[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],[334,3],[336,3],[341,3],[342,3],[345,3],[346,3],[347,3],[348,3],[350,3],[351,3],[352,3],[353,3],[359,3],[361,3],[362,3],[363,3],[364,3],[365,3],[368,3],[370,3],[375,3],[376,3],[377,3],[381,3],[382,3],[384,3],[436,3],[437,3],[439,3],[444,3],[445,3],[448,3],[449,3],[450,3],[451,3],[453,3],[454,3],[455,3],[456,3],[462,3],[464,3],[465,3],[466,3],[467,3],[468,3],[471,3],[473,3],[478,3],[479,3],[480,3],[484,3],[485,3],[487,3],[539,3],[540,3],[542,3],[543,3],[545,3],[547,3],[550,3],[551,3],[553,3],[559,3],[561,3],[565,3],[566,3],[567,3],[568,3],[569,3],[570,3],[571,3],[576,3],[577,3],[578,3],[581,3],[582,3],[583,3],[584,3],[588,3],[591,3],[637,3],[641,3],[643,3],[644,3],[646,3],[648,3],[651,3],[652,3],[654,3],[660,3],[662,3],[666,3],[667,3],[668,3],[669,3],[670,3],[671,3],[672,3],[677,3],[678,3],[679,3],[682,3],[683,3],[684,3],[685,3],[689,3],[692,3],[738,3],[742,3],[743,3],[745,3],[746,3],[752,3],[754,3],[755,3],[757,3],[759,3],[760,3],[761,3],[762,3],[765,3],[766,3],[767,3],[768,3],[770,3],[771,3],[772,3],[776,3],[777,3],[780,3],[820,3],[823,3],[826,3],[827,3],[828,3],[829,3],[830,3],[832,3],[835,3],[837,3],[842,3],[843,3],[848,3],[852,3],[853,3],[854,3],[857,3],[858,3],[860,3],[861,3],[862,3],[863,3],[864,3],[865,3],[866,3],[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],[887,3],[888,3],[889,3],[890,3],[891,3],[892,3],[893,3],[894,3],[895,3],[896,3],[897,3],[898,3],[902,3],[903,3],[904,3],[905,3],[906,3],[907,3],[911,3],[913,3],[914,3],[917,3],[918,3],[923,3],[925,3],[927,3],[928,3],[929,3],[933,3],[934,3],[971,3],[972,3],[973,3],[974,3],[975,3],[976,3],[980,3],[982,3],[983,3],[986,3],[987,3],[992,3],[994,3],[996,3],[997,3],[998,3],[1002,3],[1003,3],[1133,3],[1136,3],[1137,3],[1143,3],[1144,3],[1145,3],[1146,3],[1147,3],[1148,3],[1150,3],[1151,3],[1152,3],[1155,3],[1176,3],[1177,3],[1179,3],[1180,3],[1183,3],[1184,3],[1185,3],[1186,3],[1187,3],[1194,3],[1198,3],[1199,3],[1201,3],[1202,3],[1203,3],[1204,3],[1205,3],[1206,3],[1207,3],[1208,3],[1209,3],[1210,3],[1211,3],[1212,3],[1213,3],[1214,3],[1215,3],[1216,3],[1217,3],[1218,3],[1219,3],[1220,3],[1222,3],[1226,3],[1227,3],[1228,3],[1230,3],[1232,3],[1244,3],[1245,3],[1249,3],[1250,3],[1251,3],[1253,3],[1255,3],[1267,3],[1269,3],[1270,3],[1272,3],[1278,3],[1284,3],[1285,3],[1288,3],[1290,3],[1291,3],[1292,3],[1293,3],[1294,3],[1296,3],[1297,3],[1298,3],[1299,3],[1300,3]]},"metrics":{"gamma":6.538350040168989,"gamma_t":6.538350040168989,"gamma_rho":6.538350040168989,"gamma_claim":2.35935580987688,"gamma_t_claim":2.35935580987688,"gamma_rho_claim":2.35935580987688,"v_ex":1,"rate":0.0007593014426727411,"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-1317","N":335,"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 [[983,1,19]] (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"}]}