No. | Max. sub. | Deg. |
1 | 2058 | |
2 | 8330 | |
3 | 29155 | |
4 | 29155 | |
5 | 187425 | |
6 | 244800 | |
7 | 266560 | |
8 | 652800 | |
9 | 999600 | |
10 | 1142400 | |
11 | 3358656 |
A construction of designs acted on by simple primitive groups is used to find some 1-designs and associated self-orthogonal, decomposable and irreducible codes that admit the simple group ${\rm He}$ of Held as an automorphism group. The properties of the codes are given and links with modular representation theory are established. Further, we introduce a method of constructing self-orthogonal binary codes from orbit matrices of weakly self-orthogonal designs. Furthermore, from the support designs of the obtained self-orthogonal codes we construct strongly regular graphs with parameters (21, 10, 3, 6), (28, 12, 6, 4), (49, 12, 5, 2), (49, 18, 7, 6), (56, 10, 0, 2), (63, 30, 13, 15), (105, 32, 4, 12), (112, 30, 2, 10) and (120, 42, 8, 18).
Citation: |
Table 1.
Maximal subgroups of
No. | Max. sub. | Deg. |
1 | 2058 | |
2 | 8330 | |
3 | 29155 | |
4 | 29155 | |
5 | 187425 | |
6 | 244800 | |
7 | 266560 | |
8 | 652800 | |
9 | 999600 | |
10 | 1142400 | |
11 | 3358656 |
Table 2.
Block intersection numbers of
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136 | 1-(2058,136,136) | ||
0 | 272 | ||
6 | 1360 | ||
24 | 425 | ||
272 | 1-(2058,272,272) | ||
32 | 1360 | ||
36 | 272 | ||
48 | 425 | ||
426 | 1-(2058,426,426) | ||
50 | 272 | ||
80 | 1360 | ||
138 | 425 | ||
562 | 1-(2058,562,562) | ||
126 | 272 | ||
146 | 1360 | ||
194 | 425 | ||
698 | 1-(2058,698,698) | ||
232 | 1360 | ||
238 | 272 | ||
250 | 425 | ||
1786 | 272 | ||
1792 | 1360 | ||
1810 | 425 | ||
1922 | 1-(2058, 1922, 1922) | ||
1786 | 1-(2058, 1786, 1786) | ||
1546 | 1360 | ||
1550 | 272 | ||
1562 | 425 | ||
1632 | 1-(2058, 1632, 1632) | ||
1256 | 272 | ||
1286 | 1360 | ||
1344 | 425 | ||
1496 | 1-(2058, 1496, 1496) | ||
1060 | 272 | ||
1080 | 1360 | ||
1128 | 425 | ||
1360 | 1-(2058, 1360, 1360) | ||
894 | 1360 | ||
900 | 272 | ||
912 | 425 |
Table 3.
Symmetric
Orbits | Parameters | Full Automorphism Group |
Table 4.
Symmetric
Orbits | Parameters | Full Automorphism Group |
Table 5.
Design | Orbits | Parameters | Full Automorphism Group |
Table 6.
Non-trivial binary codes of the pairwise non-isomorphic symmetric
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Table 7.
Number of submodules of length 2058 invariant under
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1 | 2 | ||
1 | 2 | ||
2 | 1 | ||
2 | 3 | ||
1 | 1 | ||
3 | 2 | ||
1 | 2 | ||
1 | 2 | ||
1 | 10 |
Table 8.
Non-trivial binary codes of the pairwise non-isomorphic symmetric
Table 9.
Non-trivial binary codes of the pairwise non-isomorphic
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Table 10.
Non-trivial binary codes of the pairwise non-isomorphic
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Table 11.
Non-trivial binary pairwise non-equivalent codes of the orbit matrices of the symmetric self-orthogonal
Table 12.
Non-trivial binary pairwise non-equivalent codes of the orbit matrices of the self-orthogonal
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Table 13.
Non-trivial binary pairwise non-equivalent codes of the orbit matrices of the self-orthogonal
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Table 14.
Non-trivial binary pairwise non-equivalent codes of the orbit matrices of the self-orthogonal symmetric
Table 15.
Parameters of non-trivial binary codes of the orbit matrices of the self-orthogonal symmetric
Table 16.
Parameters of non-trivial binary codes of the orbit matrices of the self-orthogonal symmetric
Table 17.
Parameters of non-trivial binary codes of the orbit matrices of the self-orthogonal symmetric
Table 18.
Parameters of non-trivial binary codes of the orbit matrices of the self-orthogonal symmetric
Table 19. Strongly regular graphs constructed from the support designs of the codes of small dimension
SRG | Automorphism group | Code | |
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Table 20.
The weight distribution of
0 | 1 | 898 | 627648840 | 942 | 124891623360 | 986 | 26006753124240 |
562 | 2058 | 900 | 363854400 | 944 | 198708584760 | 988 | 30513137860800 |
672 | 29155 | 902 | 1175529600 | 946 | 268972368000 | 990 | 36606837005760 |
736 | 437325 | 904 | 650739600 | 948 | 319520140800 | 992 | 42855107097600 |
738 | 291550 | 906 | 853658400 | 950 | 375609696000 | 994 | 48857456996880 |
822 | 9329600 | 908 | 1679328000 | 952 | 489407258760 | 996 | 56162611565760 |
824 | 1399440 | 910 | 2183126400 | 954 | 671122443600 | 998 | 63280991874240 |
828 | 9329600 | 912 | 3778721240 | 956 | 952794729600 | 1000 | 70628591358360 |
832 | 1399440 | 914 | 1081067400 | 958 | 1214389249920 | 1002 | 80190083930880 |
834 | 2826320 | 916 | 4450219200 | 960 | 1453185172495 | 1004 | 87981147058560 |
840 | 8496600 | 918 | 5911234560 | 962 | 1857007899600 | 1006 | 95494650854400 |
850 | 16793280 | 920 | 6976208400 | 964 | 2316778517760 | 1008 | 103483540186600 |
862 | 27988800 | 922 | 10498598880 | 966 | 3120234340160 | 1010 | 112284766020840 |
864 | 1049580 | 924 | 11283484800 | 968 | 4115758287900 | 1012 | 119212449538560 |
870 | 111955200 | 926 | 22872447360 | 970 | 5020380794100 | 1014 | 127053086428800 |
872 | 201799248 | 928 | 12748898400 | 972 | 6322656858560 | 1016 | 135073871830800 |
880 | 247001160 | 930 | 21121199806 | 974 | 7876759235520 | 1018 | 140267712163320 |
882 | 170654330 | 932 | 26869248000 | 976 | 9861465335400 | 1020 | 146287549564800 |
888 | 1272090960 | 934 | 45576961920 | 978 | 12248848983140 | 1022 | 149737964846400 |
890 | 13994400 | 936 | 59537775360 | 980 | 15215752863360 | 1024 | 153560682747360 |
894 | 574703360 | 938 | 70051968000 | 982 | 18335071036800 | 1026 | 154420810292000 |
896 | 636745200 | 940 | 90672516480 | 984 | 22005046459200 | 1028 | 157071376707840 |
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