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Gong (github.com/gongaa/RM127, triply-even/): [[47,1,11]] quantum QR code (extended QR [48,24,12]) doubled onto [[177,1,9]]; the doubling theorem (Jain and Albert, arXiv:2408.12752, Sec. III) gives d >= min(d_sd, d_tri + 2) = 11; the distances of Gong's self-dual [[41,1,9]] and [[65,1,13]] codes are proved here by enumerating every word of their stabiliser spaces"}},"clifford":{"kind":"powers","correction":{"S":[[0,1]],"CZ":[]},"rotation_powers":[[0,7],[1,7],[2,7],[3,7],[4,7],[5,7],[6,7],[7,7],[8,7],[9,7],[10,7],[11,7],[12,7],[13,7],[14,7],[15,7],[16,7],[17,7],[18,7],[19,7],[20,7],[21,7],[22,7],[23,7],[24,7],[25,7],[26,7],[27,7],[28,7],[29,7],[30,7],[31,7],[32,7],[33,7],[34,7],[35,7],[36,7],[37,7],[38,7],[39,7],[40,7],[41,7],[42,7],[43,7],[44,7],[45,7],[46,7],[47,7],[48,7],[49,7],[50,7],[51,7],[52,7],[53,7],[54,7],[55,7],[56,7],[57,7],[58,7],[59,7],[60,7],[61,7],[62,7],[63,7],[64,7],[65,7],[66,7],[67,7],[68,7],[69,7],[70,7],[71,7],[72,7],[73,7],[74,7],[75,7],[76,7],[77,7],[78,7],[79,7],[80,7],[81,7],[82,7],[83,7],[84,7],[85,7],[86,7],[87,7],[88,7],[89,7],[90,7],[91,7],[92,7],[93,7],[94,7],[95,7],[96,7],[97,7],[98,7],[99,7],[100,7],[101,7],[102,7],[103,7],[104,7],[105,7],[106,7],[107,7],[108,7],[109,7],[110,7],[111,7],[112,7],[113,7],[114,7],[115,7],[116,7],[117,7],[118,7],[119,7],[120,7],[121,7],[122,7],[123,7],[124,7],[125,7],[126,7],[127,7],[128,7],[129,7],[130,7],[131,7],[132,7],[133,7],[134,7],[135,7],[136,7],[137,7],[138,7],[139,7],[140,7],[141,7],[142,7],[143,7],[144,7],[145,7],[146,7],[147,7],[148,7],[149,7],[150,7],[151,7],[152,7],[153,7],[154,7],[155,7],[156,7],[157,7],[158,7],[159,7],[160,7],[161,7],[162,7],[163,7],[164,7],[165,7],[166,7],[167,7],[168,7],[169,7],[170,7],[171,7],[172,7],[173,7],[174,7],[175,7],[176,7],[177,7],[178,7],[179,7],[180,7],[181,7],[182,7],[183,7],[184,7],[185,7],[186,7],[187,7],[188,7],[189,7],[190,7],[191,7],[192,7],[193,7],[194,7],[195,7],[196,7],[197,7],[198,7],[199,7],[200,7],[201,7],[202,7],[203,7],[204,7],[205,7],[206,7],[207,7],[208,7],[209,7],[210,7],[211,7],[212,7],[213,7],[214,7],[215,7],[216,7],[217,7],[218,7],[219,7],[220,7],[221,7],[222,7],[223,7],[224,7],[225,7],[226,7],[227,7],[228,7],[229,7],[230,7],[231,7],[232,7],[233,7],[234,7],[235,7],[236,7],[237,7],[238,7],[239,7],[240,7],[241,7],[242,7],[243,7],[244,7],[245,7],[246,7],[247,7],[248,7],[249,7],[250,7],[251,7],[252,7],[253,7],[254,7],[255,7],[256,7],[257,7],[258,7],[259,7],[260,7],[261,7],[262,7],[263,7],[264,7],[265,7],[266,7],[267,7],[268,7],[269,7],[270,7]]},"metrics":{"gamma":2.8789195758084607,"gamma_t":2.8789195758084607,"gamma_rho":2.8789195758084607,"gamma_claim":2.3362650089142405,"gamma_t_claim":2.3362650089142405,"gamma_rho_claim":2.3362650089142405,"v_ex":1,"rate":0.0036900369003690036,"t_count":1,"poly_degree":1,"effective_width":1},"provenance":{"regimes":["Gong doubling: triply even family"],"discovery":"pre-existing","strongest_claim":"a code of the triply even family of A. Gong (note in github.com/gongaa/RM127, triply-even/; transversal_t_codes/): her doubly even self-dual [[41,1,9]] and [[65,1,13]] codes and quantum QR codes doubled onto [[95,1,7]], rebuilt here from her self-dual codes, where T on every qubit is a logical T or T-dagger; a verified witness, not a maximum","sources":[{"regime":"Gong doubling: triply even family","file":"transversal_t_codes/factories.json","label":"G-271","N":71,"d":7,"d_is_exact":false,"origin":"A. Gong, \"On some K=1 codes admitting strongly transversal T\", note in github.com/gongaa/RM127, triply-even/ (2025)","provenance":"[[47,1,11]] quantum QR code (extended QR [48,24,12]) doubled onto [[177,1,9]] (Gong's note)"}]},"references":[{"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"}]}