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February  2020, 14(1): 89-96. doi: 10.3934/amc.2020007

New doubly even self-dual codes having minimum weight 20

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

Fund Project: This work was supported by JSPS KAKENHI Grant Number 15H03633

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.

Citation: Masaaki Harada. New doubly even self-dual codes having minimum weight 20. Advances in Mathematics of Communications, 2020, 14 (1) : 89-96. doi: 10.3934/amc.2020007
References:
[1]

W. BosmaJ. Cannon and C. Playoust, The Magma algebra system Ⅰ: The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[2]

R. A. Brualdi and V. S. Pless, Weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, 37 (1991), 1222-1225.  doi: 10.1109/18.86979.  Google Scholar

[3]

J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333.  doi: 10.1109/18.59931.  Google Scholar

[4]

P. Gaborit, Table of Type Ⅱ Codes, Online available at http://www.unilim.fr/pages_perso/philippe.gaborit/SD/GF2/GF2II.htm, Accessed on October 6, 2017. Google Scholar

[5]

P. GaboritC.-S. Nedeloaia and A. Wassermann, On the weight enumerators of duadic and quadratic residue codes, IEEE Trans. Inform. Theory, 51 (2005), 402-407.  doi: 10.1109/TIT.2004.839522.  Google Scholar

[6]

A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identities, Actes du Congrés International des Mathématiciens (Nice, 1970), Tome 3, Gauthier-Villars, Paris, (1971), 211-215.  Google Scholar

[7]

M. Harada, An extremal doubly even self-dual code of length 112, Electron. J. Combin., 15 (2008), Note 33, 5 pp.  Google Scholar

[8]

M. HaradaW. HolzmannH. Kharaghani and M. Khorvash, Extremal ternary self-dual codes constructed from negacirculant matrices, Graphs Combin., 23 (2007), 401-417.  doi: 10.1007/s00373-007-0731-2.  Google Scholar

[9]

C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200.  doi: 10.1016/S0019-9958(73)90273-8.  Google Scholar

[10]

E. M. Rains and N. J. A. Sloane, Self-dual codes, Handbook of Coding Theory, V. S. Pless and W. C. Huffman (Editors), Elsevier, Amsterdam, (1998), 177-294.  Google Scholar

[11]

R. Yorgova and A. Wassermann, Binary self-dual codes with automorphisms of order 23, Des. Codes Cryptogr., 48 (2008), 155-164.  doi: 10.1007/s10623-007-9152-8.  Google Scholar

show all references

References:
[1]

W. BosmaJ. Cannon and C. Playoust, The Magma algebra system Ⅰ: The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[2]

R. A. Brualdi and V. S. Pless, Weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, 37 (1991), 1222-1225.  doi: 10.1109/18.86979.  Google Scholar

[3]

J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333.  doi: 10.1109/18.59931.  Google Scholar

[4]

P. Gaborit, Table of Type Ⅱ Codes, Online available at http://www.unilim.fr/pages_perso/philippe.gaborit/SD/GF2/GF2II.htm, Accessed on October 6, 2017. Google Scholar

[5]

P. GaboritC.-S. Nedeloaia and A. Wassermann, On the weight enumerators of duadic and quadratic residue codes, IEEE Trans. Inform. Theory, 51 (2005), 402-407.  doi: 10.1109/TIT.2004.839522.  Google Scholar

[6]

A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identities, Actes du Congrés International des Mathématiciens (Nice, 1970), Tome 3, Gauthier-Villars, Paris, (1971), 211-215.  Google Scholar

[7]

M. Harada, An extremal doubly even self-dual code of length 112, Electron. J. Combin., 15 (2008), Note 33, 5 pp.  Google Scholar

[8]

M. HaradaW. HolzmannH. Kharaghani and M. Khorvash, Extremal ternary self-dual codes constructed from negacirculant matrices, Graphs Combin., 23 (2007), 401-417.  doi: 10.1007/s00373-007-0731-2.  Google Scholar

[9]

C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200.  doi: 10.1016/S0019-9958(73)90273-8.  Google Scholar

[10]

E. M. Rains and N. J. A. Sloane, Self-dual codes, Handbook of Coding Theory, V. S. Pless and W. C. Huffman (Editors), Elsevier, Amsterdam, (1998), 177-294.  Google Scholar

[11]

R. Yorgova and A. Wassermann, Binary self-dual codes with automorphisms of order 23, Des. Codes Cryptogr., 48 (2008), 155-164.  doi: 10.1007/s10623-007-9152-8.  Google Scholar

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