1 Introduction
2 Preliminaries
2.1 Self-dual codes
2.2 Alphabets
2.3 Special matrices
2.4 Group rings and composite matrices
3 Composite matrix constructions
4 Results
\({\mathbb {F}}_2+u{\mathbb {F}}_2\) | \({\mathbb {F}}_4\) | Symbol |
---|---|---|
0 | 0 | 0 |
1 | 1 | 1 |
u | w | 2 |
\(1+u\) | \(1+w\) | 3 |
4.1 New self-dual codes of length 80
-
\(\beta =0\) and \(\alpha \in \{-z:z=65,80,120,125,130,135,140,145,150,155,165,175,190,195,205,210,215,230,235,250,270,275,280,360\}\);
-
\(\beta =10\) and \(\alpha \in \{-2z:z=130,150,160,185\}\).
\({\mathcal {C}}_{80,i}\) | \(\mathbf{v} \) | \(\alpha \) | \(\beta \) | \(|{{\,\mathrm{Aut}\,}}({\mathcal {C}}_{80,i})|\) |
---|---|---|---|---|
1 | (31223333300320201200) | \(-275\) | 0 | \(2^{2}\cdot 5\) |
2 | (13111130203000233223) | \(-270\) | 0 | \(2^{2}\cdot 5\) |
3 | (00302012331122313103) | \(-250\) | 0 | \(2^{2}\cdot 5\) |
4 | (01332030221111113310) | \(-235\) | 0 | \(2^{2}\cdot 5\) |
5 | (23320213130330103221) | \(-230\) | 0 | \(2^{3}\cdot 5\) |
6 | (22011233231033013100) | \(-210\) | 0 | \(2^{2}\cdot 5\) |
7 | (11333122033223331212) | \(-205\) | 0 | \(2^{2}\cdot 5\) |
8 | (02222010112332220213) | \(-195\) | 0 | \(2^{2}\cdot 5\) |
9 | (30333313000233100021) | \(-190\) | 0 | \(2^{2}\cdot 5\) |
10 | (00111023231213313321) | \(-175\) | 0 | \(2^{2}\cdot 5\) |
11 | (22310231030332003032) | \(-165\) | 0 | \(2^{2}\cdot 5\) |
12 | (02002030232203221313) | \(-155\) | 0 | \(2^{2}\cdot 5\) |
13 | (03121003232002123332) | \(-150\) | 0 | \(2^{2}\cdot 5\) |
14 | (23200233120101002302) | \(-145\) | 0 | \(2^{2}\cdot 5\) |
15 | (22133232333121133232) | \(-140\) | 0 | \(2^{3}\cdot 5\) |
16 | (31031330230000203122) | \(-135\) | 0 | \(2^{2}\cdot 5\) |
17 | (01103022122003122122) | \(-130\) | 0 | \(2^{2}\cdot 5\) |
18 | (22010203131000112213) | \(-65\) | 0 | \(2^{2}\cdot 5\) |
19 | (02330020210322001303) | \(-260\) | 10 | \(2^{3}\cdot 5\) |
\({\mathcal {C}}_{80,i}\) | \(\mathbf{v} \) | \(\alpha \) | \(\beta \) | \(|{{\,\mathrm{Aut}\,}}({\mathcal {C}}_{80,i})|\) |
---|---|---|---|---|
20 | (12222331200322021203) | \(-280\) | 0 | \(2^{3}\cdot 5\) |
21 | (23330310032021331010) | \(-120\) | 0 | \(2^{3}\cdot 5\) |
22 | (30320122023203322322) | \(-80\) | 0 | \(2^{3}\cdot 5\) |
23 | (21222311321120112303) | \(-320\) | 10 | \(2^{4}\cdot 5\) |
\({\mathcal {C}}_{80,i}\) | \(\mathbf{v} \) | \(\alpha \) | \(\beta \) | \(|{{\,\mathrm{Aut}\,}}({\mathcal {C}}_{80,i})|\) |
---|---|---|---|---|
24 | (31211223330300232332) | \(-360\) | 0 | \(2^{2}\cdot 5\) |
25 | (10201301032322330300) | \(-215\) | 0 | \(2^{2}\cdot 5\) |
26 | (01021003132222203113) | \(-125\) | 0 | \(2^{2}\cdot 5\) |
27 | (31003000101110232322) | \(-370\) | 10 | \(2^{2}\cdot 5\) |
28 | (11210213102203230313) | \(-300\) | 10 | \(2^{2}\cdot 5\) |
4.2 New self-dual codes of length 84
-
\(\beta =0\) and \(\alpha \in \{6z:z=336,350,358,365,372,386,392,393,399,400,406,407,413,414,420,421,427,428,434,435,441,442,448,449,455,456,462,463,469,470,476,477,483,484,490,491,497,498,504,505,511,512,518,519,525,526,532,533,539,540,546,553,554,560,567\}\);
-
\(\beta =21\) and \(\alpha \in \{6z:z=413,434,435,441,442,449,455,456,462,463,469,470,476,477,483,484,490,491,497,498,504,505,511,512,518,519,525,526,532,533,539,540,546,547,553,560,568,575,595\}\);
-
\(\beta =42\) and \(\alpha \in \{6z:z=490,512,518,525,526,539,540,547,553,560,568\}\);
-
\(\beta =63\) and \(\alpha \in \{6z:z=574,575\}\).
\({\mathcal {C}}_{84,i}\) | \(\mathbf{v} \) | \(W_{84,j}\) | \(\alpha \) | \(\beta \) | \(|{{\,\mathrm{Aut}\,}}({\mathcal {C}}_{84,i})|\) |
---|---|---|---|---|---|
1 | (110001110100101111010000011100010000011111) | 3 | 2988 | 0 | \(2\cdot 3\cdot 7\) |
2 | (111111011111011000011010010000101000001001) | 3 | 3024 | 0 | \(2\cdot 3\cdot 7\) |
3 | (001001101100110111101011010000011100011010) | 3 | 3030 | 0 | \(2\cdot 3\cdot 7\) |
4 | (101111111010011001101100101011000001001000) | 3 | 3066 | 0 | \(2\cdot 3\cdot 7\) |
5 | (101001011101011110110100111111001011010100) | 3 | 3072 | 0 | \(2\cdot 3\cdot 7\) |
6 | (111100010111001011001010011100110100001001) | 3 | 3108 | 0 | \(2\cdot 3\cdot 7\) |
7 | (110101110100001100100000110101010010101111) | 3 | 3114 | 0 | \(2\cdot 3\cdot 7\) |
8 | (000000000110000110110010101101100110111000) | 3 | 3150 | 0 | \(2\cdot 3\cdot 7\) |
9 | (101010111001111011101001100100110100100000) | 3 | 3156 | 0 | \(2\cdot 3\cdot 7\) |
10 | (101100110111001110010100000010110101111000) | 3 | 3192 | 0 | \(2\cdot 3\cdot 7\) |
\({\mathcal {C}}_{84,i}\) | \(\mathbf{v} \) | \(W_{84,j}\) | \(\alpha \) | \(\beta \) | \(|{{\,\mathrm{Aut}\,}}({\mathcal {C}}_{84,i})|\) |
---|---|---|---|---|---|
56 | (011001100101000010101000000000011110111100) | 3 | 2016 | 0 | \(2^{2}\cdot 3\cdot 7\) |
57 | (100101010001111110100110011011000001011001) | 3 | 2100 | 0 | \(2\cdot 3\cdot 7\) |
58 | (010110001001010100011100001111000111011011) | 3 | 2148 | 0 | \(2\cdot 3\cdot 7\) |
59 | (010101100111000010011001000001000000000001) | 3 | 2190 | 0 | \(2\cdot 3\cdot 7\) |
60 | (101110110000001010001011111001000000000101) | 3 | 2232 | 0 | \(2\cdot 3\cdot 7\) |
61 | (001000100010110011001101111011001001111100) | 3 | 2316 | 0 | \(2\cdot 3\cdot 7\) |
62 | (010010101101010100100111001011011001110001) | 3 | 2352 | 0 | \(2\cdot 3\cdot 7\) |
63 | (001101000100110000001101011011011011110011) | 3 | 2358 | 0 | \(2\cdot 3\cdot 7\) |
64 | (000011000001100101110100001010111101110111) | 3 | 2394 | 0 | \(2\cdot 3\cdot 7\) |
65 | (011101100100110011000111001110111101000000) | 3 | 2400 | 0 | \(2\cdot 3\cdot 7\) |
4.3 New self-dual codes of length 96
\({\mathcal {C}}_{96,i}^{\text {I}}\) | \(\mathbf{v} \) | \(W_{96,j}^{\text {I}}\) | \(\alpha \) | \(\beta \) | \(\gamma \) | \(|{{\,\mathrm{Aut}\,}}({\mathcal {C}}_{96,i}^{\text {I}})|\) |
---|---|---|---|---|---|---|
1 | (021111013112231302031321) | 2 | 15336 | \(-240\) | 0 | \(2^{4}\cdot 3\) |
2 | (332030221021223333303031) | 2 | 14664 | \(-144\) | 0 | \(2^{4}\cdot 3\) |
3 | (310201300213103023131203) | 2 | 12456 | \(-120\) | 0 | \(2^{4}\cdot 3\) |
4 | (110330330331133112022003) | 2 | 16608 | \(-432\) | 12 | \(2^{6}\cdot 3\) |
5 | (301201202300231031203031) | 2 | 14712 | \(-336\) | 12 | \(2^{5}\cdot 3\) |
\({\mathcal {C}}_{96,i}^{\text {I}}\) | \(\mathbf{v} \) | \(W_{96,j}^{\text {I}}\) | \(\alpha \) | \(\beta \) | \(\gamma \) | \(|{{\,\mathrm{Aut}\,}}({\mathcal {C}}_{96,i}^{\text {I}})|\) |
---|---|---|---|---|---|---|
6 | (301220102333222223210331) | 2 | 14448 | \(-208\) | 0 | \(2^{4}\cdot 3\) |
7 | (111322103200321233201211) | 2 | 13776 | \(-184\) | 0 | \(2^{4}\cdot 3\) |
8 | (333110012302102113330110) | 2 | 11652 | \(-136\) | 0 | \(2^{4}\cdot 3\) |
9 | (321212110001220122211301) | 2 | 12624 | \(-124\) | 0 | \(2^{3}\cdot 3\) |
10 | (000232332103103311032121) | 2 | 11364 | \(-112\) | 0 | \(2^{3}\cdot 3\) |
11 | (231232002131031220200120) | 2 | 12036 | \(-108\) | 0 | \(2^{3}\cdot 3\) |
12 | (021301113010112220211130) | 2 | 11580 | \(-100\) | 0 | \(2^{3}\cdot 3\) |
13 | (000332130323021220110022) | 2 | 11880 | \(-96\) | 0 | \(2^{3}\cdot 3\) |
14 | (001022122300133130333310) | 2 | 11424 | \(-88\) | 0 | \(2^{3}\cdot 3\) |
15 | (213121322231133130230323) | 2 | 11256 | \(-84\) | 0 | \(2^{3}\cdot 3\) |
-
\(\gamma =0\) and \((\alpha ,\beta )\in \{(12z_1,-4z_2):(z_1,z_2)=(850,0),(896,0),(904,0),(805,1),(854,3),(808,4),(837,6),(926,6),(822,7),(865,9),(860,10),(860,12),(897,12),(900,12),(929,12),(1014,12),(877,13),(910,15),(877,16),(933,18),(908,19),(938,21),(952,22),(957,24),(990,24),(965,25),(1003,27),(947,28),(1038,30),(1052,31),(971,34),(1045,36),(1222,36),(1148,46),(1244,48),(1260,48),(1204,52),(1278,60)\}\);
-
\(\gamma =6\) and \((\alpha ,\beta )\in \{(12z_1,-4z_2):(z_1,z_2)=(909,30),(913,31),(922,33),(901,34),(902,36),(918,37),(944,39),(948,40),(932,42),(995,43),(949,45),(980,46),(1034,48),(1018,49),(969,51),(978,52),(1120,64)\}\);
-
\(\gamma =12\) and \((\alpha ,\beta )\in \{(12z_1,-4z_2):(z_1,z_2)=(928,60),(988,60),(992,60),(1048,60),(1056,60),(1076,60),(1096,60),(1104,60),(1120,60),(1148,60),(1160,60),(1168,60),(1176,60),(1208,60),(1216,60),(1232,60),(1240,60),(1264,60),(1280,60),(1288,60),(1320,60),(1336,60),(1520,60),(982,61),(975,63),(984,64),(997,66),(1133,66),(1148,66),(1236,66),(977,67),(1075,69),(1042,70),(1080,72),(1112,72),(1120,72),(1137,72),(1272,72),(1544,72),(1036,73),(1046,76),(1098,78),(1121,78),(1072,79),(1226,84),(1352,84),(1528,84),(1224,85),(1332,100),(1384,108)\}\).
\({\mathcal {C}}_{96,i}^{\text {I}}\) | \(\mathbf{v} \) | \(W_{96,j}^{\text {I}}\) | \(\alpha \) | \(\beta \) | \(\gamma \) | \(|{{\,\mathrm{Aut}\,}}({\mathcal {C}}_{96,i}^{\text {I}})|\) |
---|---|---|---|---|---|---|
62 | (222222222220220133213123) | 2 | 14928 | \(-192\) | 0 | \(2^{6}\cdot 3\) |
63 | (222222222220220133211121) | 2 | 15120 | \(-192\) | 0 | \(2^{6}\cdot 3\) |
64 | (222220222011020210021113) | 2 | 12540 | \(-144\) | 0 | \(2^{4}\cdot 3\) |
65 | (222222222011021013011303) | 2 | 11484 | \(-96\) | 0 | \(2^{4}\cdot 3\) |
66 | (222222222011202110211131) | 2 | 10764 | \(-48\) | 0 | \(2^{4}\cdot 3\) |
67 | (222222222101200131212230) | 2 | 10800 | \(-48\) | 0 | \(2^{5}\cdot 3\) |
68 | (222222222011202110213111) | 2 | 11148 | \(-48\) | 0 | \(2^{4}\cdot 3\) |
69 | (222222220103021223012121) | 2 | 12168 | \(-48\) | 0 | \(2^{4}\cdot 3\) |
70 | (222222222101200131221023) | 2 | 10752 | 0 | 0 | \(2^{5}\cdot 3\) |
71 | (222222202121200111221203) | 2 | 10848 | 0 | 0 | \(2^{5}\cdot 3\) |
\({\mathcal {C}}_{96,i}^{\text {I}}\) | \(\mathbf{v} \) | \(W_{96,j}^{\text {I}}\) | \(\alpha \) | \(\beta \) | \(\gamma \) | \(|{{\,\mathrm{Aut}\,}}({\mathcal {C}}_{96,i}^{\text {I}})|\) |
---|---|---|---|---|---|---|
91 | (222220222111201001210311) | 2 | 11112 | \(-24\) | 0 | \(2^{4}\cdot 3\) |
92 | (222222202111001223010313) | 2 | 16224 | \(-336\) | 12 | \(2^{6}\cdot 3\) |
93 | (222222222011101333122333) | 2 | 18336 | \(-336\) | 12 | \(2^{5}\cdot 3\) |
94 | (222222222101220113212010) | 2 | 15264 | \(-288\) | 12 | \(2^{5}\cdot 3\) |
95 | (222220222111221003212111) | 2 | 18528 | \(-288\) | 12 | \(2^{6}\cdot 3\) |
96 | (222220220101211331210113) | 2 | 14832 | \(-264\) | 12 | \(2^{4}\cdot 3\) |
97 | (222220200103211331212113) | 2 | 13776 | \(-240\) | 12 | \(2^{4}\cdot 3\) |
98 | (222220222111221003212313) | 2 | 13920 | \(-240\) | 12 | \(2^{5}\cdot 3\) |
99 | (222222220021202121211101) | 2 | 14496 | \(-240\) | 12 | \(2^{6}\cdot 3\) |
100 | (222222222113212131201003) | 2 | 14592 | \(-240\) | 12 | \(2^{5}\cdot 3\) |
101 | (222222222011212313201101) | 2 | 14784 | \(-240\) | 12 | \(2^{5}\cdot 3\) |
102 | (222222220021020101011121) | 2 | 14880 | \(-240\) | 12 | \(2^{5}\cdot 3\) |
103 | (222222222113212333201003) | 2 | 15360 | \(-240\) | 12 | \(2^{5}\cdot 3\) |
104 | (222222222011011111002013) | 2 | 15456 | \(-240\) | 12 | \(2^{5}\cdot 3\) |
105 | (222222222211020101021321) | 2 | 16032 | \(-240\) | 12 | \(2^{5}\cdot 3\) |
\({\mathcal {C}}_{96,i}^{\text {II}}\) | \(\mathbf{v} \) | \(\alpha \) | \(|{{\,\mathrm{Aut}\,}}({\mathcal {C}}_{96,i}^{\text {II}})|\) |
---|---|---|---|
1 | (320210300223213323022021) | 8514 | \(2^{4}\cdot 3\) |
2 | (122313111112022110302021) | 8754 | \(2^{4}\cdot 3\) |
3 | (122123010133300221011031) | 8994 | \(2^{4}\cdot 3\) |
4 | (001212011312020203212003) | 9042 | \(2^{4}\cdot 3\) |
5 | (122000032021320000301313) | 9138 | \(2^{4}\cdot 3\) |
6 | (010220032021103212312322) | 9234 | \(2^{4}\cdot 3\) |
7 | (210231130330223123221020) | 9282 | \(2^{4}\cdot 3\) |
8 | (032311303332300120032321) | 9378 | \(2^{4}\cdot 3\) |
9 | (213201111011203112303130) | 9474 | \(2^{4}\cdot 3\) |
10 | (110230310113303323101232) | 9618 | \(2^{4}\cdot 3\) |
\({\mathcal {C}}_{96,i}^{\text {II}}\) | \(\mathbf{v} \) | \(\alpha \) | \(|{{\,\mathrm{Aut}\,}}({\mathcal {C}}_{96,i}^{\text {II}})|\) |
---|---|---|---|
89 | (332010230212013330233103) | 8274 | \(2^{3}\cdot 3\) |
90 | (121001211131002223313030) | 8418 | \(2^{3}\cdot 3\) |
91 | (330222312102031223221213) | 8658 | \(2^{3}\cdot 3\) |
92 | (331001322120111003113202) | 8838 | \(2^{3}\cdot 3\) |
93 | (322112032202123203331221) | 11478 | \(2^{3}\cdot 3\) |
94 | (333003302201123232100313) | 11526 | \(2^{3}\cdot 3\) |
95 | (201120113100000113122122) | 11742 | \(2^{3}\cdot 3\) |
96 | (000232210010130121123202) | 13194 | \(2^{5}\cdot 3\) |
\({\mathcal {C}}_{96,i}^{\text {II}}\) | \(\mathbf{v} \) | \(\alpha \) | \(|{{\,\mathrm{Aut}\,}}({\mathcal {C}}_{96,i}^{\text {II}})|\) |
---|---|---|---|
97 | (222222220103200133210030) | 10002 | \(2^{4}\cdot 3\) |
98 | (222222220103021003012303) | 10098 | \(2^{4}\cdot 3\) |
99 | (222222220103200133212032) | 10578 | \(2^{4}\cdot 3\) |
100 | (222222220103021003010123) | 10818 | \(2^{4}\cdot 3\) |
101 | (222222220103221203210101) | 10866 | \(2^{4}\cdot 3\) |
102 | (222220202013221331211111) | 12138 | \(2^{5}\cdot 3\) |
103 | (222222222101122211113131) | 12234 | \(2^{6}\cdot 3\) |
104 | (222222222101020131001221) | 12522 | \(2^{6}\cdot 3\) |
105 | (222222220103200113212230) | 12546 | \(2^{4}\cdot 3\) |
106 | (222222222101201021210101) | 12810 | \(2^{6}\cdot 3\) |
107 | (222220222211020212001111) | 13290 | \(2^{6}\cdot 3\) |
108 | (222222202013220110213131) | 13578 | \(2^{5}\cdot 3\) |
109 | (222222222220222111213123) | 28506 | \(2^{8}\cdot 3\cdot 5\) |
\({\mathcal {C}}_{96,i}^{\text {II}}\) | \(\mathbf{v} \) | \(\alpha \) | \(|{{\,\mathrm{Aut}\,}}({\mathcal {C}}_{96,i}^{\text {II}})|\) |
---|---|---|---|
110 | (222220200103011331010113) | 12186 | \(2^{4}\cdot 3\) |
111 | (222222222011202121201123) | 12426 | \(2^{5}\cdot 3\) |
112 | (222222222011211111220213) | 12714 | \(2^{5}\cdot 3\) |
113 | (222220220101011331012113) | 12762 | \(2^{4}\cdot 3\) |
114 | (222222222011212313221303) | 13002 | \(2^{5}\cdot 3\) |
115 | (222020200101011331012133) | 13050 | \(2^{4}\cdot 3\) |
116 | (222220220101011331012133) | 13338 | \(2^{4}\cdot 3\) |
117 | (222222220211121333100113) | 13866 | \(2^{6}\cdot 3\) |
118 | (222222220211121333100131) | 14826 | \(2^{6}\cdot 3\) |
119 | (222222222011211311220213) | 15978 | \(2^{6}\cdot 3\) |
120 | (222222222011121333100333) | 16170 | \(2^{5}\cdot 3\) |
121 | (222222222011121333100311) | 16554 | \(2^{5}\cdot 3\) |
-
\(\alpha \in \{6z:z=1379,1403,1419,1443,1459,1473,1499,1507,1523,1539,1547,1563,\)\(1579,1603,1619,1627,1643,1659,1667,1683,1699,1707,1723,1747,1759,1763,1779,\)\(1787,1795,1803,1811,1819,1827,1835,1843,1851,1859,1867,1875,1879,1883,1891,\)\(1899,1903,1907,1913,1915,1921,1923,1931,1939,1947,1957,1963,1971,1975,1979,\)\(1987,1995,2003,2007,2011,2015,2019,2023,2027,2031,2039,2043,2055,2059,2067,\)\(2071,2079,2083,2087,2091,2095,2103,2107,2119,2127,2135,2143,2147,2151,2163,\)\(2167,2175,2195,2199,2203,2207,2211,2215,2223,2231,2247,2255,2259,2263,2279,\)\(2283,2295,2311,2359,2379,2407,2423,2471,2483,2503,2519,2567,2599,2663,2695,\)\(2711,2759,2887,4751\}\).