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here by the doubling of Jain and Albert (arXiv:2408.12752) from A. Gong's [[559,1,15]] (github.com/gongaa/RM127, triply-even/): [[199,1,31]] quantum QR code (extended QR [200,100,32]) doubled onto [[2797,1,29]]; their doubling theorem (Sec. III) gives d >= min(d_sd, d_tri + 2) = 31, 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],[606,2],[607,2],[608,2],[609,2],[613,2],[615,2],[617,2],[621,2],[622,2],[626,2],[631,2],[632,2],[633,2],[638,2],[640,2],[641,2],[642,2],[643,2],[647,2],[650,2],[653,2],[655,2],[799,2],[801,1],[802,2],[803,3],[805,1],[806,2]],"CZ":[[0,801],[0,803],[0,805],[606,799],[607,799],[608,799],[609,799],[613,799],[615,799],[617,799],[621,799],[622,799],[626,799],[631,799],[632,799],[633,799],[638,799],[640,799],[641,799],[642,799],[643,799],[647,799],[650,799],[653,799],[655,799],[801,802],[801,803],[801,804],[801,805],[801,806],[803,804],[803,805],[803,806],[805,806]]},"rotation_powers":[[0,7],[1,7],[3,7],[5,7],[6,7],[8,7],[10,7],[12,7],[14,7],[17,7],[21,7],[22,7],[23,7],[27,7],[29,7],[30,7],[32,7],[33,7],[35,7],[37,7],[40,7],[41,7],[42,7],[43,7],[45,7],[48,7],[49,7],[52,7],[54,7],[55,7],[56,7],[57,7],[59,7],[66,7],[67,7],[69,7],[71,7],[75,7],[76,7],[77,7],[79,7],[83,7],[87,7],[88,7],[89,7],[90,7],[91,7],[94,7],[96,7],[97,7],[99,7],[201,3],[203,3],[206,3],[208,3],[210,3],[212,3],[214,3],[215,3],[217,3],[218,3],[219,3],[223,3],[224,3],[225,3],[227,3],[230,3],[233,3],[235,3],[237,3],[238,3],[243,3],[245,3],[246,3],[249,3],[250,3],[252,3],[257,3],[259,3],[260,3],[261,3],[262,3],[263,3],[264,3],[267,3],[269,3],[271,3],[272,3],[273,3],[277,3],[279,3],[280,3],[281,3],[283,3],[284,3],[285,3],[291,3],[292,3],[294,3],[297,3],[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],[335,3],[336,3],[337,3],[338,3],[339,3],[340,3],[341,3],[342,3],[343,3],[344,3],[345,3],[346,3],[347,3],[348,3],[349,3],[350,3],[351,3],[352,3],[353,3],[354,3],[355,3],[356,3],[357,3],[358,3],[359,3],[360,3],[361,3],[362,3],[363,3],[364,3],[365,3],[366,3],[367,3],[368,3],[369,3],[370,3],[371,3],[372,3],[373,3],[374,3],[375,3],[376,3],[377,3],[378,3],[379,3],[380,3],[381,3],[382,3],[383,3],[384,3],[385,3],[386,3],[387,3],[388,3],[389,3],[390,3],[391,3],[392,3],[393,3],[394,3],[395,3],[396,3],[397,3],[398,7],[400,3],[402,3],[403,3],[405,3],[407,3],[409,3],[411,3],[414,3],[418,3],[419,3],[420,3],[424,3],[426,3],[427,3],[429,3],[430,3],[432,3],[434,3],[437,3],[438,3],[439,3],[440,3],[442,3],[445,3],[446,3],[449,3],[451,3],[452,3],[453,3],[454,3],[456,3],[463,3],[464,3],[466,3],[468,3],[472,3],[473,3],[474,3],[476,3],[480,3],[484,3],[485,3],[486,3],[487,3],[488,3],[491,3],[493,3],[494,3],[496,3],[596,3],[597,7],[599,3],[601,3],[602,3],[604,3],[606,3],[608,3],[610,3],[613,3],[617,3],[618,3],[619,3],[623,3],[625,3],[626,3],[628,3],[629,3],[631,3],[633,3],[636,3],[637,3],[638,3],[639,3],[641,3],[644,3],[645,3],[648,3],[650,3],[651,3],[652,3],[653,3],[655,3],[662,3],[663,3],[665,3],[667,3],[671,3],[672,3],[673,3],[675,3],[679,3],[683,3],[684,3],[685,3],[686,3],[687,3],[690,3],[692,3],[693,3],[695,3],[795,3],[1178,7],[1179,3],[1180,3],[1181,3],[1182,3],[1183,3],[1188,3],[1189,3],[1190,3],[1193,3],[1194,3],[1196,3],[1197,3],[1199,3],[1202,3],[1204,3],[1208,3],[1209,3],[1210,3],[1214,3],[1217,3],[1219,3],[1221,3],[1222,3],[1224,3],[1225,3],[1226,3],[1227,3],[1231,3],[1232,3],[1234,3],[1236,3],[1237,3],[1238,3],[1241,3],[1242,3],[1243,3],[1245,3],[1246,3],[1249,3],[1253,3],[1254,3],[1256,3],[1261,3],[1265,3],[1272,3],[1368,3],[1369,5],[1375,3],[1376,3],[1377,3],[1378,3],[1382,3],[1383,3],[1386,3],[1389,3],[1391,3],[1392,3],[1394,3],[1396,3],[1397,3],[1398,3],[1402,3],[1403,3],[1404,3],[1406,3],[1407,3],[1409,3],[1411,3],[1414,3],[1419,3],[1420,3],[1421,3],[1424,3],[1426,3],[1430,3],[1431,3],[1435,3],[1438,3],[1439,3],[1441,3],[1442,3],[1443,3],[1446,3],[1448,3],[1449,3],[1450,3],[1451,3],[1453,3],[1454,3],[1455,3],[1457,3],[1458,3],[1459,3],[1460,3],[1461,3],[1462,3],[1464,3],[1465,3],[1466,3],[1467,3],[1468,3],[1469,3],[1470,3],[1471,3],[1472,3],[1473,3],[1474,3],[1475,3],[1476,3],[1477,3],[1478,3],[1479,3],[1480,3],[1481,3],[1482,3],[1483,3],[1484,3],[1485,3],[1486,3],[1487,3],[1488,3],[1489,3],[1490,3],[1491,3],[1492,3],[1493,3],[1494,3],[1495,3],[1496,3],[1497,3],[1498,3],[1499,3],[1500,3],[1501,3],[1502,3],[1503,3],[1504,3],[1505,3],[1506,3],[1507,3],[1508,3],[1509,3],[1510,3],[1511,3],[1512,3],[1513,3],[1514,3],[1515,3],[1516,3],[1517,3],[1518,3],[1519,3],[1520,3],[1521,3],[1522,3],[1523,3],[1524,3],[1525,3],[1526,3],[1527,3],[1528,3],[1529,3],[1530,3],[1531,3],[1532,3],[1533,3],[1534,3],[1535,3],[1536,3],[1537,3],[1538,3],[1539,3],[1540,3],[1541,3],[1542,3],[1543,3],[1544,3],[1545,3],[1546,3],[1547,3],[1548,3],[1549,3],[1550,3],[1551,3],[1552,3],[1553,3],[1554,3],[1555,3],[1556,3],[1557,3],[1558,3],[1560,3],[1563,3],[1565,3],[1567,3],[1569,3],[1571,3],[1572,3],[1576,3],[1578,3],[1582,3],[1583,3],[1584,3],[1585,3],[1586,3],[1588,3],[1590,3],[1591,3],[1592,3],[1594,3],[1595,3],[1599,3],[1601,3],[1604,3],[1605,3],[1606,3],[1608,3],[1609,3],[1611,3],[1612,3],[1615,3],[1618,3],[1619,3],[1621,3],[1623,3],[1626,3],[1629,3],[1630,3],[1632,3],[1635,3],[1637,3],[1640,3],[1642,3],[1726,3],[1727,3],[1730,3],[1732,3],[1734,3],[1736,3],[1738,3],[1739,3],[1743,3],[1745,3],[1749,3],[1750,3],[1751,3],[1752,3],[1753,3],[1755,3],[1757,3],[1758,3],[1759,3],[1761,3],[1762,3],[1766,3],[1768,3],[1771,3],[1772,3],[1773,3],[1775,3],[1776,3],[1778,3],[1779,3],[1782,3],[1785,3],[1786,3],[1788,3],[1790,3],[1793,3],[1796,3],[1797,3],[1799,3],[1802,3],[1804,3],[1807,3],[1809,3],[1893,3],[1894,5],[2061,5],[2228,3],[2230,3],[2235,3],[2236,3],[2239,3],[2240,3],[2241,3],[2242,3],[2244,3],[2245,3],[2246,3],[2247,3],[2253,3],[2255,3],[2256,3],[2257,3],[2258,3],[2259,3],[2262,3],[2264,3],[2269,3],[2270,3],[2271,3],[2275,3],[2276,3],[2278,3],[2330,3],[2331,3],[2333,3],[2338,3],[2339,3],[2342,3],[2343,3],[2344,3],[2345,3],[2347,3],[2348,3],[2349,3],[2350,3],[2356,3],[2358,3],[2359,3],[2360,3],[2361,3],[2362,3],[2365,3],[2367,3],[2372,3],[2373,3],[2374,3],[2378,3],[2379,3],[2381,3],[2433,3],[2434,3],[2436,3],[2437,3],[2439,3],[2441,3],[2444,3],[2445,3],[2447,3],[2453,3],[2455,3],[2459,3],[2460,3],[2461,3],[2462,3],[2463,3],[2464,3],[2465,3],[2470,3],[2471,3],[2472,3],[2475,3],[2476,3],[2477,3],[2478,3],[2482,3],[2485,3],[2531,3],[2535,3],[2537,3],[2538,3],[2540,3],[2542,3],[2545,3],[2546,3],[2548,3],[2554,3],[2556,3],[2560,3],[2561,3],[2562,3],[2563,3],[2564,3],[2565,3],[2566,3],[2571,3],[2572,3],[2573,3],[2576,3],[2577,3],[2578,3],[2579,3],[2583,3],[2586,3],[2632,3],[2636,3],[2637,3],[2639,3],[2640,3],[2646,3],[2648,3],[2649,3],[2651,3],[2653,3],[2654,3],[2655,3],[2656,3],[2659,3],[2660,3],[2661,3],[2662,3],[2664,3],[2665,3],[2666,3],[2670,3],[2671,3],[2674,3],[2714,3],[2717,3],[2720,3],[2721,3],[2722,3],[2723,3],[2724,3],[2726,3],[2729,3],[2731,3],[2736,3],[2737,3],[2742,3],[2746,3],[2747,3],[2748,3],[2751,3],[2752,3],[2754,3],[2755,3],[2756,3],[2757,3],[2758,3],[2759,3],[2760,3],[2761,3],[2762,3],[2763,3],[2764,3],[2765,3],[2766,3],[2767,3],[2768,3],[2769,3],[2770,3],[2771,3],[2772,3],[2773,3],[2774,3],[2775,3],[2776,3],[2777,3],[2778,3],[2779,3],[2780,3],[2781,3],[2782,3],[2783,3],[2784,3],[2785,3],[2786,3],[2787,3],[2788,3],[2789,3],[2790,3],[2791,3],[2792,3]]},"metrics":{"gamma":7.345031955353797,"gamma_t":7.345031955353797,"gamma_rho":7.345031955353797,"gamma_claim":2.3498463699201415,"gamma_t_claim":2.3498463699201415,"gamma_rho_claim":2.3498463699201415,"v_ex":1,"rate":0.00031298904538341156,"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-3195","N":807,"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":"[[199,1,31]] quantum QR code (extended QR [200,100,32]) doubled onto [[2797,1,29]]: 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"}]}