Article Contents
Article Contents

# New doubly even self-dual codes having minimum weight 20

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.

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

 Citation:

• 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

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

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

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

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

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

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

Tables(6)