New doubly even self-dual codes having minimum weight 20

Masaaki Harada

doi:10.3934/amc.2020007

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2020, Volume 14, Issue 1: 89-96. Doi: 10.3934/amc.2020007

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New doubly even self-dual codes having minimum weight 20

Masaaki Harada

Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan

Received: August 2018

Revised: February 2019

Published: August 2019

This work was supported by JSPS KAKENHI Grant Number 15H03633

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Abstract

In this note, we construct new doubly even self-dual codes having minimum weight 20 for lengths 112,120 and 128. This implies that there are at least three inequivalent extremal doubly even self-dual codes of length 112.

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Mathematics Subject Classification: Primary: 94B05; Secondary: 94B25.

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Table 1. Weight distribution of $ C_{112} $

$ i $	$ A_i $	$ i $	$ A_i $
$ 0,112 $	1	$ 38, 74 $	31676520067584
$ 18, 94 $	8512	$ 40, 72 $	109690203298312
$ 20, 92 $	186060	$ 42, 70 $	325630986391040
$ 22, 90 $	3239936	$ 44, 68 $	831288282918576
$ 24, 88 $	47551798	$ 46, 66 $	1829637194737408
$ 26, 86 $	561437184	$ 48, 64 $	3479230392288469
$ 28, 84 $	5424089452	$ 50, 62 $	5725819388994432
$ 30, 82 $	43459872064	$ 52, 60 $	8165553897114152
$ 32, 80 $	291008417322	$ 54, 58 $	10099951175046656
$ 34, 78 $	1639219687168	$ 56 $	10841051388476292
$ 36, 76 $	7813559379696

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Table 2. $ m(H_{112}),m(D_{112}) $ and $ m(E_{112}) $

$ m(H_{112}) $
10613, 10649, 10661, 10703, 10709, 10715, 10721, 10727, 10733, 10739, 10745,
10769, 10775, 10781, 10787, 10799, 10805, 10811, 10823, 10829, 10835, 10841,
10847, 10853, 10859, 10865, 10871, 10883, 10895, 10901, 10907, 10913, 10919,
10925, 10931, 10937, 10943, 10949, 10967, 10973, 10985, 10991, 10997, 11009,
11021, 11033, 11045, 11057, 11063, 11069, 11093, 11099, 11117, 63525
$ m(D_{112}) $
10618, 10663, 10672, 10702, 10708, 10717, 10735, 10750, 10765, 10768, 10771,
10777, 10783, 10786, 10789, 10801, 10810, 10819, 10831, 10834, 10837, 10840,
10843, 10846, 10849, 10852, 10858, 10861, 10864, 10867, 10873, 10882, 10885,
10900, 10903, 10906, 10909, 10912, 10918, 10921, 10924, 10927, 10930, 10936,
10945, 10954, 10957, 10978, 10984, 10987, 11002, 11011, 11023, 11041, 11044,
11056, 11065, 11080, 11086, 11098, 11110, 63525
$ m(E_{112}) $
10581, 10620, 10641, 10653, 10659, 10668, 10674, 10689, 10698, 10701, 10704,
10707, 10719, 10728, 10734, 10749, 10758, 10761, 10764, 10770, 10776, 10779,
10782, 10785, 10791, 10794, 10797, 10806, 10809, 10812, 10815, 10818, 10821,
10824, 10827, 10830, 10833, 10842, 10848, 10851, 10854, 10860, 10863, 10866,
10872, 10875, 10878, 10881, 10884, 10890, 10893, 10896, 10899, 10902, 10905,
10911, 10914, 10917, 10923, 10926, 10929, 10932, 10935, 10938, 10941, 10950,
10953, 10959, 10965, 10968, 10971, 10977, 10980, 10989, 10992, 10995, 11001,
11013, 11016, 11025, 11028, 11040, 11046, 11049, 11052, 11055, 11073, 11085,
11103, 11151, 63525

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Table 3. Weight enumerators for length 120

Numbers of codewords of weight 20
93180, 93936, 94512, 95136, 95202, 95376, 95496, 95532, 95826, 95946, 95952, 96012, 96096, 96126,
96156, 96216, 96240, 96312, 96336, 96360, 96366, 96372, 96486, 96540, 96576, 96666, 96690, 96720,
96762, 96780, 96816, 96840, 96846, 96876, 96906, 96912, 96936, 96996, 97026, 97056, 97092, 97116,
97176, 97230, 97260, 97266, 97272, 97296, 97326, 97356, 97422, 97446, 97452, 97476, 97566, 97572,
97590, 97596, 97626, 97632, 97656, 97716, 97746, 97770, 97776, 97782, 97836, 97842, 97866, 97890,
97896, 97926, 97950, 97962, 97986, 98016, 98040, 98076, 98130, 98136, 98166, 98196, 98220, 98226,
98250, 98256, 98262, 98286, 98292, 98316, 98346, 98412, 98466, 98496, 98502, 98526, 98532, 98556,
98562, 98580, 98586, 98610, 98616, 98622, 98640, 98646, 98670, 98676, 98682, 98700, 98706, 98712,
98730, 98742, 98772, 98796, 98802, 98826, 98832, 98856, 98886, 98910, 98916, 98940, 98952, 98976,
99000, 99036, 99066, 99090, 99096, 99120, 99126, 99156, 99162, 99180, 99186, 99210, 99216, 99222,
99240, 99246, 99252, 99270, 99282, 99306, 99312, 99330, 99336, 99342, 99372, 99390, 99396, 99402,
99432, 99450, 99456, 99486, 99516, 99540, 99546, 99576, 99612, 99666, 99672, 99690, 99696, 99702,
99720, 99726, 99750, 99756, 99786, 99792, 99810, 99816, 99846, 99876, 99906, 99936, 99942, 99966,
99972, 99996, 100026, 100032, 100062, 100086, 100110, 100116, 100122, 100140, 100146, 100170,
100176, 100182, 100200, 100206, 100212, 100236, 100242, 100260, 100266, 100290, 100296, 100350,
100356, 100380, 100446, 100452, 100476, 100482, 100500, 100506, 100512, 100536, 100542, 100560,
100566, 100590, 100596, 100626, 100650, 100656, 100662, 100680, 100686, 100716, 100722, 100746,
100752, 100770, 100776, 100782, 100800, 100806, 100842, 100860, 100872, 100896, 100902, 100920,
100926, 100956, 100980, 100986, 100992, 101046, 101052, 101070, 101076, 101082, 101106, 101112,
101130, 101136, 101142, 101160, 101166, 101196, 101202, 101226, 101232, 101250, 101256, 101280,
101286, 101316, 101376, 101382, 101400, 101406, 101412, 101436, 101442, 101472, 101496, 101526,
101532, 101550, 101556, 101586, 101616, 101622, 101640, 101646, 101652, 101670, 101676, 101700,
101706, 101730, 101736, 101760, 101766, 101772, 101790, 101796, 101802, 101820, 101826, 101850,
101856, 101862, 101880, 101892, 101910, 101916, 101940, 101946, 101952, 101970, 101976, 101982,
102000, 102006, 102030, 102036, 102042, 102066, 102072, 102096, 102120, 102126, 102150, 102156,
102180, 102186, 102210, 102216, 102240, 102246, 102252, 102270, 102312, 102336, 102342, 102360,
102366, 102372, 102402, 102420, 102426, 102456, 102480, 102486, 102492, 102516, 102540, 102546,
102570, 102576, 102582, 102606, 102636, 102660, 102666, 102672, 102690, 102696, 102702, 102726,
102732, 102750, 102756, 102780, 102786, 102792, 102816, 102840, 102846, 102870, 102876, 102906,
102930, 102936, 102942, 102966, 102972, 102996, 103002, 103020, 103026, 103032, 103050, 103056,
103080, 103086, 103092, 103116, 103140, 103146, 103176, 103182, 103206, 103236, 103266, 103272,
103296, 103320, 103326, 103332, 103356, 103380, 103386, 103410, 103416, 103422, 103452, 103500,
103506, 103530, 103560, 103566, 103590, 103596, 103632, 103650, 103656, 103686, 103692, 103710,
103716, 103722, 103740, 103746, 103752, 103770, 103776, 103800, 103806, 103830, 103836, 103860,
103896, 103932, 103962, 103986, 104022, 104046, 104076, 104106, 104166, 104220, 104226, 104232,
104256, 104286, 104316, 104346, 104436, 104442, 104496, 104502, 104532, 104556, 104580, 104592,
104616, 104622, 104646, 104652, 104676, 104736, 104772, 104796, 104820, 104880, 104886, 104892,
104910, 104916, 104970, 104982, 105066, 105096, 105156, 105336, 105396, 105426, 105456, 105510,
105546, 105576, 105636, 105666, 105696, 105762, 105966, 106152, 106236, 106266, 106290, 106386,
106626, 106662, 106812, 106836, 107220, 107406, 108486, 108600

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Table 4. Doubly even self-dual codes of length $ 120 $ and minimum weight $ 20 $

$ r_A $	$ r_B $
$ (100000111110000011111010001101) $	$ (010000000010010010110101001111) $
$ (100001100101001111000100110010) $	$ (100101100010011100001110011100) $
$ (100000011001010111001110101001) $	$ (111101110101111111001100111010) $
$ (100001001111010001100000000011) $	$ (010110111001100010000111011101) $
$ (100001011011001010010110000010) $	$ (001010001100000100000010001000) $
$ (100000110010110111100100111110) $	$ (100111000011110011101010001010) $
$ (100000000000111000011000111101) $	$ (111000101010011000111000111011) $
$ (100001001111110101101101110111) $	$ (011001000000100100101101110100) $
$ (100000001100001100111011101110) $	$ (111111010001001110011000000000) $
$ (100000101111001000010100100110) $	$ (001101011010111010101111011110) $

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Table 5. Weight enumerators for length $ 128 $

Numbers of codewords of weight $ 20 $
21376, 21824, 22016, 22400, 22464, 22880, 22944, 23008, 23104, 23136, 23232,
23296, 23328, 23360, 23392, 23520, 23552, 23616, 23648, 23680, 23808, 23936,
24000, 24032, 24064, 24096, 24128, 24160, 24192, 24224, 24256, 24288, 24320,
24352, 24384, 24416, 24448, 24480, 24512, 24544, 24576, 24640, 24672, 24704,
24736, 24768, 24800, 24832, 24864, 24896, 24928, 24960, 24992, 25024, 25056,
25088, 25120, 25152, 25184, 25216, 25248, 25280, 25312, 25344, 25376, 25408,
25440, 25472, 25504, 25536, 25568, 25600, 25632, 25664, 25696, 25728, 25760,
25824, 25856, 25888, 25920, 25952, 25984, 26016, 26048, 26080, 26112, 26144,
26176, 26208, 26240, 26272, 26304, 26336, 26368, 26400, 26432, 26464, 26496,
26528, 26560, 26592, 26624, 26656, 26688, 26720, 26752, 26784, 26816, 26848,
26880, 26912, 26944, 26976, 27008, 27040, 27072, 27104, 27136, 27168, 27200,
27232, 27264, 27296, 27328, 27360, 27392, 27424, 27456, 27488, 27520, 27584,
27616, 27648, 27680, 27712, 27744, 27776, 27808, 27840, 27872, 27904, 27936,
27968, 28000, 28032, 28064, 28096, 28128, 28160, 28192, 28224, 28256, 28288,
28320, 28352, 28384, 28416, 28448, 28480, 28512, 28544, 28576, 28608, 28640,
28736, 28768, 28800, 28832, 28864, 28896, 28928, 28992, 29024, 29056, 29088,
29120, 29152, 29216, 29248, 29312, 29344, 29376, 29536, 29600, 29632, 29696,
29760, 29792, 29824, 29856, 29888, 30048, 30144, 30176, 30208, 30240, 30304,
30368, 31584

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Table 6. Doubly even self-dual codes of length $ 128 $ and minimum weight $ 20 $

$ r_A $	$ r_B $
$ (10000111111110100010000110111100) $	$ (00101010111100011101101011110100) $
$ (10000111010101110101100011100110) $	$ (10001000101110011011111011100110) $
$ (10000010011000000001101010010001) $	$ (01010001111001100001001011000010) $
$ (10000111000111101111100111100111) $	$ (11010000000111100110110101111111) $
$ (10000010001100010100011010000111) $	$ (01110100011000110110100110000110) $
$ (10000111010101110010100010001010) $	$ (01011110001110011011111011100111) $
$ (10000100011111000010100010100010) $	$ (10001101000000111010111010010101) $
$ (10000101110011001010100011011010) $	$ (00111111111010001111100110111000) $
$ (10000000110010100000011111111000) $	$ (11110000010011001110110110110101) $
$ (10000010010000110011101110110100) $	$ (10101000001000010101011011100001) $

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New doubly even self-dual codes having minimum weight 20

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