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and Albert (arXiv:2408.12752), Table I: [[69,1,13]] code of the self-dual [70,35,12] code SD70 doubled onto [[279,1,11]]; the doubling theorem of their Sec. III gives d >= min(d_sd, d_tri + 2) = 13, 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":[[12,2],[13,2],[14,2],[15,2],[16,2],[17,2],[18,2],[19,2],[20,2],[21,2],[22,2],[23,2],[24,2],[25,2],[26,2],[27,2],[28,2],[29,2],[30,2],[31,2],[32,2],[33,2],[34,2],[58,2],[61,2],[62,2],[65,2],[66,2],[68,2],[69,2],[72,2],[75,2],[77,2],[105,2],[107,2]],"CZ":[[12,107],[13,107],[14,107],[15,107],[16,107],[17,107],[18,107],[19,107],[20,107],[21,107],[22,107],[23,107],[24,107],[25,107],[26,107],[27,107],[28,107],[29,107],[30,107],[31,107],[32,107],[33,107],[34,107],[58,105],[61,105],[62,105],[65,105],[66,105],[68,105],[69,105],[72,105],[75,105],[77,105]]},"rotation_powers":[[2,3],[3,3],[4,3],[5,3],[6,3],[7,3],[11,3],[13,3],[14,3],[17,3],[18,3],[23,3],[25,3],[27,3],[28,3],[29,3],[33,3],[34,3],[71,3],[72,3],[73,3],[74,3],[75,3],[76,3],[80,3],[82,3],[83,3],[86,3],[87,3],[92,3],[94,3],[96,3],[97,3],[98,3],[102,3],[103,3],[233,3],[236,3],[237,3],[243,3],[244,3],[245,3],[246,3],[247,3],[248,3],[250,3],[251,3],[252,3],[255,3],[276,3],[277,3],[279,3],[280,3],[283,3],[284,3],[285,3],[286,3],[287,3],[294,3],[298,3],[299,3],[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],[322,3],[326,3],[327,3],[328,3],[330,3],[332,3],[344,3],[345,3],[349,3],[350,3],[351,3],[353,3],[355,3],[367,3],[369,3],[370,3],[372,3],[378,3],[384,3],[385,3],[388,3],[390,3],[391,3],[392,3],[393,3],[394,3],[396,3],[397,3],[398,3],[399,3],[400,3]]},"metrics":{"gamma":3.3671295312858263,"gamma_t":3.3671295312858263,"gamma_rho":3.3671295312858263,"gamma_claim":2.352126838001039,"gamma_t_claim":2.352126838001039,"gamma_rho_claim":2.352126838001039,"v_ex":1,"rate":0.002398081534772182,"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-417","N":108,"d":6,"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":"[[69,1,13]] code of the self-dual [70,35,12] code SD70 doubled onto [[279,1,11]] (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"}]}