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1586,1603,1620,1627,1634],"A_d":null,"certified":{"d":23,"is_exact":false,"met":true,"source":"built here by the doubling of Jain and Albert (arXiv:2408.12752) from A. Gong's [[559,1,15]] (github.com/gongaa/RM127, triply-even/): [[167,1,23]] quantum QR code (extended QR [168,84,24]) doubled onto [[1301,1,21]]; their doubling theorem (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) and the distances of Gong's self-dual [[41,1,9]] and [[65,1,13]] codes proved here by enumerating every word of their stabiliser spaces"}},"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],[411,2],[413,1],[414,2]],"CZ":[[0,413],[218,411],[219,411],[220,411],[221,411],[225,411],[227,411],[229,411],[233,411],[234,411],[238,411],[243,411],[244,411],[245,411],[250,411],[252,411],[253,411],[254,411],[255,411],[259,411],[262,411],[265,411],[267,411],[413,414]]},"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]]},"metrics":{"gamma":6.735222388875634,"gamma_t":6.735222388875634,"gamma_rho":6.735222388875634,"gamma_claim":2.3598825492135918,"gamma_t_claim":2.3598825492135918,"gamma_rho_claim":2.3598825492135918,"v_ex":1,"rate":0.0006116207951070336,"t_count":1,"poly_degree":1,"effective_width":1},"provenance":{"regimes":["Jain-Albert doubling of Gong's codes"],"discovery":"pre-existing","strongest_claim":"a code built here by the doubling method of Jain and Albert (arXiv:2408.12752) from the codes in A. Gong's repository (github.com/gongaa/RM127, triply-even/): her [[559,1,15]] doubled with the self-dual CSS codes of the paper's Table I from distance 17 on (transversal_t_codes/); a verified witness, not a maximum","sources":[{"regime":"Jain-Albert doubling of Gong's codes","file":"transversal_t_codes/factories.json","label":"GJ-1635","N":415,"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; A. Gong, \"On some K=1 codes admitting strongly transversal T\", note in github.com/gongaa/RM127, triply-even/ (2025)","provenance":"[[167,1,23]] quantum QR code (extended QR [168,84,24]) doubled onto [[1301,1,21]]: the doubling method of Jain and Albert (arXiv:2408.12752) applied to the codes in A. Gong's repository (github.com/gongaa/RM127)"}]},"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"},{"key":"gong2025triply","short":"Gong (2025)","full":"A. Gong, \"On some K=1 codes admitting strongly transversal T,\" note in the RM127 repository, triply-even/ (2025).","url":"https://github.com/gongaa/RM127/tree/main/triply-even","date":"2025-04-10"}]}