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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

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

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.

Keywords: Self-dual code, doubly even code, four-circulant code, minimum weight, neighbor.
Mathematics Subject Classification: Primary: 94B05; Secondary: 94B25.
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. Bosma, J. 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.

[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.

[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.

[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.

[5]

P. Gaborit, C.-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.

[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.

[7]

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

[8]

M. Harada, W. Holzmann, H. 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.

[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.

[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.

[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.

show all references

References:
[1]

W. Bosma, J. 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.

[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.

[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.

[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.

[5]

P. Gaborit, C.-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.

[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.

[7]

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

[8]

M. Harada, W. Holzmann, H. 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.

[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.

[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.

[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.

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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$ 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
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$ 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
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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) $
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$ 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
Table Options

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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) $
Table Options

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$ 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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