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Gong (github.com/gongaa/RM127, triply-even/): [[79,1,15]] quantum QR code (extended QR [80,40,16]) doubled onto [[401,1,13]]; the doubling theorem (Jain and Albert, arXiv:2408.12752, Sec. III) gives d >= min(d_sd, d_tri + 2) = 15; 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],[271,7],[272,7],[273,7],[274,7],[275,7],[276,7],[277,7],[278,7],[279,7],[280,7],[281,7],[282,7],[283,7],[284,7],[285,7],[286,7],[287,7],[288,7],[289,7],[290,7],[291,7],[292,7],[293,7],[294,7],[295,7],[296,7],[297,7],[298,7],[299,7],[300,7],[301,7],[302,7],[303,7],[304,7],[305,7],[306,7],[307,7],[308,7],[309,7],[310,7],[311,7],[312,7],[313,7],[314,7],[315,7],[316,7],[317,7],[318,7],[319,7],[320,7],[321,7],[322,7],[323,7],[324,7],[325,7],[326,7],[327,7],[328,7],[329,7],[330,7],[331,7],[332,7],[333,7],[334,7],[335,7],[336,7],[337,7],[338,7],[339,7],[340,7],[341,7],[342,7],[343,7],[344,7],[345,7],[346,7],[347,7],[348,7],[349,7],[350,7],[351,7],[352,7],[353,7],[354,7],[355,7],[356,7],[357,7],[358,7],[359,7],[360,7],[361,7],[362,7],[363,7],[364,7],[365,7],[366,7],[367,7],[368,7],[369,7],[370,7],[371,7],[372,7],[373,7],[374,7],[375,7],[376,7],[377,7],[378,7],[379,7],[380,7],[381,7],[382,7],[383,7],[384,7],[385,7],[386,7],[387,7],[388,7],[389,7],[390,7],[391,7],[392,7],[393,7],[394,7],[395,7],[396,7],[397,7],[398,7],[399,7],[400,7],[401,7],[402,7],[403,7],[404,7],[405,7],[406,7],[407,7],[408,7],[409,7],[410,7],[411,7],[412,7],[413,7],[414,7],[415,7],[416,7],[417,7],[418,7],[419,7],[420,7],[421,7],[422,7],[423,7],[424,7],[425,7],[426,7],[427,7],[428,7],[429,7],[430,7],[431,7],[432,7],[433,7],[434,7],[435,7],[436,7],[437,7],[438,7],[439,7],[440,7],[441,7],[442,7],[443,7],[444,7],[445,7],[446,7],[447,7],[448,7],[449,7],[450,7],[451,7],[452,7],[453,7],[454,7],[455,7],[456,7],[457,7],[458,7],[459,7],[460,7],[461,7],[462,7],[463,7],[464,7],[465,7],[466,7],[467,7],[468,7],[469,7],[470,7],[471,7],[472,7],[473,7],[474,7],[475,7],[476,7],[477,7],[478,7],[479,7],[480,7],[481,7],[482,7],[483,7],[484,7],[485,7],[486,7],[487,7],[488,7],[489,7],[490,7],[491,7],[492,7],[493,7],[494,7],[495,7],[496,7],[497,7],[498,7],[499,7],[500,7],[501,7],[502,7],[503,7],[504,7],[505,7],[506,7],[507,7],[508,7],[509,7],[510,7],[511,7],[512,7],[513,7],[514,7],[515,7],[516,7],[517,7],[518,7],[519,7],[520,7],[521,7],[522,7],[523,7],[524,7],[525,7],[526,7],[527,7],[528,7],[529,7],[530,7],[531,7],[532,7],[533,7],[534,7],[535,7],[536,7],[537,7],[538,7],[539,7],[540,7],[541,7],[542,7],[543,7],[544,7],[545,7],[546,7],[547,7],[548,7],[549,7],[550,7],[551,7],[552,7],[553,7],[554,7],[555,7],[556,7],[557,7],[558,7]]},"metrics":{"gamma":3.530691246119435,"gamma_t":3.530691246119435,"gamma_rho":3.530691246119435,"gamma_claim":2.3360532498918514,"gamma_t_claim":2.3360532498918514,"gamma_rho_claim":2.3360532498918514,"v_ex":1,"rate":0.0017889087656529517,"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-559","N":144,"d":6,"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":"[[79,1,15]] quantum QR code (extended QR [80,40,16]) doubled onto [[401,1,13]] (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"}]}