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A connection between sumsets and covering codes of a module
On self-orthogonal designs and codes related to Held's simple group
1. | Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia |
2. | School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, University Road Private Bag X54001, Westville Campus, Durban 4041, South Africa |
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).
References:
[1] |
E. F. Assmus, Jr and J. D. Key, Designs and Their Codes, Cambridge: Cambridge University Press, 1992. Cambridge Tracts in Mathematics, Vol. 103 (Second printing with corrections, 1993).
doi: 10.1017/CBO9781316529836. |
[2] |
M. Behbahani and C. Lam,
Strongly regular graphs with non-trivial automorphisms, Discrete Math, 311 (2011), 132-144.
doi: 10.1016/j.disc.2010.10.005. |
[3] |
J. v. Bon, A. M. Cohen and H. Cuypers,
Graphs related to {H}eld's simple group, J. Algebra, 123 (1989), 6-26.
doi: 10.1016/0021-8693(89)90032-X. |
[4] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system I: The user language, J. Symb. Comp., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[5] |
A. E. Brouwer, Strongly regular graphs, C. J. Colbourn, J. H. Dinitz (Eds.), Handbook of Combinatorial Designs (second ed.), Chapman & Hall/CRC, Boca Raton (2007), 852-868. |
[6] |
A. E. Brouwer and C. J. van Eijl,
On the $p$-rank of the adjacency matrices of strongly regular graphs, J. Algebraic Combin., 1 (1992), 329-346.
doi: 10.1023/A:1022438616684. |
[7] |
G. Butler,
The maximal subgroups of the sporadic simple group of Held, J. Algebra, 69 (1981), 67-81.
doi: 10.1016/0021-8693(81)90127-7. |
[8] |
J. Cannon, A. Steel and G. White, Linear codes over finite fields, In J. Cannon and W. Bosma, editors, Handbook of Magma Functions, pages 3951-4023. Computational Algebra Group, Department of Mathematics, University of Sydney, 2006. V2. 13, http://magma.maths.usyd.edu.au/magma. |
[9] |
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, An Atlas of Finite Groups, Oxford: Oxford University Press, 1985. |
[10] |
D. Crnković and V. Mikulić,
Unitals, projective planes and other combinatorial structures constructed from the unitary groups ${U}_3(q)q = 3,4,5,7$, Ars Combin, 110 (2013), 3-13.
|
[11] |
D. Crnković, B. G. Rodrigues, L. Simčić and S. Rukavina,
Self-orthogonal codes from orbit matrices of 2-designs, Adv. Math. Commun., 7 (2013), 161-174.
doi: 10.3934/amc.2013.7.161. |
[12] |
W. Haemers, C. Parker, V. Pless and V. Tonchev,
A design and a code invariant under the simple group $\rm{Co}_3$, J. Combin. Theory, Ser. A, 62 (1993), 225-233.
doi: 10.1016/0097-3165(93)90045-A. |
[13] |
M. Harada and V. D. Tonchev,
Self-orthogonal codes from symmetric designs with fixed-point-free automorphisms, Discrete Math., 264 (2003), 81-90.
doi: 10.1016/S0012-365X(02)00553-8. |
[14] |
D. Held,
The simple groups related to ${\rm M}_{24}$, Journal of Algebra, 13 (1969), 253-296.
doi: 10.1016/0021-8693(69)90074-X. |
[15] |
D. Held,
The simple groups related to ${\rm M}_{24}$, Journal of Austral. Math. Soc., 16 (1973), 24-28.
doi: 10.1017/S1446788700013902. |
[16] |
J. Hrabě de Angelis,
A Presentation and a Representation of the Held Group, Acta Appl. Math., 52 (1998), 285-290.
doi: 10.1023/A:1005900217631. |
[17] |
C. Jansen,
The minimal degrees of faithful representations of the sporadic simple groups and their covering groups, LMS J. Comput. Math., 8 (2005), 122-144.
doi: 10.1112/S1461157000000930. |
[18] |
Decomposition Matrices, The Modular Atlas homepage, 2014. http://www.math.rwth-aachen.de/~MOC/decomposition/tex/He/Hemod2.pdf, 2014. |
[19] |
J. D. Key and J. Moori,
Designs, codes and graphs from the Janko groups ${J}_1$ and ${J}_2$, J. Combin. Math. and Combin. Comput., 40 (2002), 143-159.
|
[20] |
J. D. Key and J. Moori,
Some irreducible codes invariant under the Janko group, ${J}_1$ or ${J}_2$, J. Combin. Math. Combin. Comput., 81 (2012), 165-189.
|
[21] |
J. D. Key, J. Moori and B. G. Rodrigues,
On some designs and codes from primitive representations of some finite simple groups, J. Combin. Math. and Combin. Comput., 45 (2003), 3-19.
|
[22] |
J. D. Key and J. Moori, Correction to: "Codes, designs and graphs from the Janko groups J1 and J2", [J. Combin. Math. Combin. Comput., 40 (2002), 143-159], J. Combin. Math. Combin. Comput., 64 (2008), 153. |
[23] |
J. Moori and B. G. Rodrigues,
Some designs and codes invariant under the simple group ${\rm Co}_2$, J. Algebra, 316 (2007), 649-661.
doi: 10.1016/j.jalgebra.2007.02.004. |
[24] |
J. Moori and B. G. Rodrigues,
Some designs and binary codes preserved by the simple group ${\rm Ru}$ of Rudvalis, J. Algebra, 372 (2012), 702-710.
doi: 10.1016/j.jalgebra.2012.09.032. |
[25] |
C. Parker and V. D. Tonchev,
Linear Codes and Double Transitive Symmetric Design, Linear Algebra Appl., 226/228 (1995), 237-246.
doi: 10.1016/0024-3795(95)00104-Y. |
[26] |
C. E. Praeger and L. H. Soicher, Low Rank Representations and Graphs for Sporadic Groups, Cambridge: Cambridge University Press. 1997. Australian Mathematical Society Lecture Series, Vol. 8. |
[27] |
B. G. Rodrigues, Codes of Designs and Graphs from Finite Simple Groups, Ph. D. thesis, University of Natal, Pietermaritzburg, 2002. |
[28] |
C. M. Roney-Dougal,
The primitive permutation groups of degree less than 2500, J. Algebra, 292 (2005), 154-183.
doi: 10.1016/j.jalgebra.2005.04.017. |
[29] |
V. D. Tonchev,
Self-orthogonal designs and extremal doubly even codes, J. Combin. Theory, A, 52 (1989), 197-205.
doi: 10.1016/0097-3165(89)90030-7. |
[30] |
R. A. Wilson,
Maximal subgroups of automorphism groups of simple groups, J. London Math. Soc., 32 (1985), 406-466.
doi: 10.1112/jlms/s2-32.3.460. |
[31] |
R. A. Wilson, The Finite Simple Groups, London: Springer-Verlag London Ltd., 2009. Graduate Texts in Mathematics, Vol. 251.
doi: 10.1007/978-1-84800-988-2. |
[32] |
E. Witt,
Die 5-Fach transitiven Gruppen von Mathieu, Abh. Math. Sem. Univ. Hamburg, 12 (1937), 256-264.
doi: 10.1007/BF02948947. |
[33] |
E. Witt,
Uber Steinersche Systeme, Abh. Math. Sem. Univ. Hamburg, 12 (1937), 265-275.
doi: 10.1007/BF02948948. |
show all references
References:
[1] |
E. F. Assmus, Jr and J. D. Key, Designs and Their Codes, Cambridge: Cambridge University Press, 1992. Cambridge Tracts in Mathematics, Vol. 103 (Second printing with corrections, 1993).
doi: 10.1017/CBO9781316529836. |
[2] |
M. Behbahani and C. Lam,
Strongly regular graphs with non-trivial automorphisms, Discrete Math, 311 (2011), 132-144.
doi: 10.1016/j.disc.2010.10.005. |
[3] |
J. v. Bon, A. M. Cohen and H. Cuypers,
Graphs related to {H}eld's simple group, J. Algebra, 123 (1989), 6-26.
doi: 10.1016/0021-8693(89)90032-X. |
[4] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system I: The user language, J. Symb. Comp., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[5] |
A. E. Brouwer, Strongly regular graphs, C. J. Colbourn, J. H. Dinitz (Eds.), Handbook of Combinatorial Designs (second ed.), Chapman & Hall/CRC, Boca Raton (2007), 852-868. |
[6] |
A. E. Brouwer and C. J. van Eijl,
On the $p$-rank of the adjacency matrices of strongly regular graphs, J. Algebraic Combin., 1 (1992), 329-346.
doi: 10.1023/A:1022438616684. |
[7] |
G. Butler,
The maximal subgroups of the sporadic simple group of Held, J. Algebra, 69 (1981), 67-81.
doi: 10.1016/0021-8693(81)90127-7. |
[8] |
J. Cannon, A. Steel and G. White, Linear codes over finite fields, In J. Cannon and W. Bosma, editors, Handbook of Magma Functions, pages 3951-4023. Computational Algebra Group, Department of Mathematics, University of Sydney, 2006. V2. 13, http://magma.maths.usyd.edu.au/magma. |
[9] |
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, An Atlas of Finite Groups, Oxford: Oxford University Press, 1985. |
[10] |
D. Crnković and V. Mikulić,
Unitals, projective planes and other combinatorial structures constructed from the unitary groups ${U}_3(q)q = 3,4,5,7$, Ars Combin, 110 (2013), 3-13.
|
[11] |
D. Crnković, B. G. Rodrigues, L. Simčić and S. Rukavina,
Self-orthogonal codes from orbit matrices of 2-designs, Adv. Math. Commun., 7 (2013), 161-174.
doi: 10.3934/amc.2013.7.161. |
[12] |
W. Haemers, C. Parker, V. Pless and V. Tonchev,
A design and a code invariant under the simple group $\rm{Co}_3$, J. Combin. Theory, Ser. A, 62 (1993), 225-233.
doi: 10.1016/0097-3165(93)90045-A. |
[13] |
M. Harada and V. D. Tonchev,
Self-orthogonal codes from symmetric designs with fixed-point-free automorphisms, Discrete Math., 264 (2003), 81-90.
doi: 10.1016/S0012-365X(02)00553-8. |
[14] |
D. Held,
The simple groups related to ${\rm M}_{24}$, Journal of Algebra, 13 (1969), 253-296.
doi: 10.1016/0021-8693(69)90074-X. |
[15] |
D. Held,
The simple groups related to ${\rm M}_{24}$, Journal of Austral. Math. Soc., 16 (1973), 24-28.
doi: 10.1017/S1446788700013902. |
[16] |
J. Hrabě de Angelis,
A Presentation and a Representation of the Held Group, Acta Appl. Math., 52 (1998), 285-290.
doi: 10.1023/A:1005900217631. |
[17] |
C. Jansen,
The minimal degrees of faithful representations of the sporadic simple groups and their covering groups, LMS J. Comput. Math., 8 (2005), 122-144.
doi: 10.1112/S1461157000000930. |
[18] |
Decomposition Matrices, The Modular Atlas homepage, 2014. http://www.math.rwth-aachen.de/~MOC/decomposition/tex/He/Hemod2.pdf, 2014. |
[19] |
J. D. Key and J. Moori,
Designs, codes and graphs from the Janko groups ${J}_1$ and ${J}_2$, J. Combin. Math. and Combin. Comput., 40 (2002), 143-159.
|
[20] |
J. D. Key and J. Moori,
Some irreducible codes invariant under the Janko group, ${J}_1$ or ${J}_2$, J. Combin. Math. Combin. Comput., 81 (2012), 165-189.
|
[21] |
J. D. Key, J. Moori and B. G. Rodrigues,
On some designs and codes from primitive representations of some finite simple groups, J. Combin. Math. and Combin. Comput., 45 (2003), 3-19.
|
[22] |
J. D. Key and J. Moori, Correction to: "Codes, designs and graphs from the Janko groups J1 and J2", [J. Combin. Math. Combin. Comput., 40 (2002), 143-159], J. Combin. Math. Combin. Comput., 64 (2008), 153. |
[23] |
J. Moori and B. G. Rodrigues,
Some designs and codes invariant under the simple group ${\rm Co}_2$, J. Algebra, 316 (2007), 649-661.
doi: 10.1016/j.jalgebra.2007.02.004. |
[24] |
J. Moori and B. G. Rodrigues,
Some designs and binary codes preserved by the simple group ${\rm Ru}$ of Rudvalis, J. Algebra, 372 (2012), 702-710.
doi: 10.1016/j.jalgebra.2012.09.032. |
[25] |
C. Parker and V. D. Tonchev,
Linear Codes and Double Transitive Symmetric Design, Linear Algebra Appl., 226/228 (1995), 237-246.
doi: 10.1016/0024-3795(95)00104-Y. |
[26] |
C. E. Praeger and L. H. Soicher, Low Rank Representations and Graphs for Sporadic Groups, Cambridge: Cambridge University Press. 1997. Australian Mathematical Society Lecture Series, Vol. 8. |
[27] |
B. G. Rodrigues, Codes of Designs and Graphs from Finite Simple Groups, Ph. D. thesis, University of Natal, Pietermaritzburg, 2002. |
[28] |
C. M. Roney-Dougal,
The primitive permutation groups of degree less than 2500, J. Algebra, 292 (2005), 154-183.
doi: 10.1016/j.jalgebra.2005.04.017. |
[29] |
V. D. Tonchev,
Self-orthogonal designs and extremal doubly even codes, J. Combin. Theory, A, 52 (1989), 197-205.
doi: 10.1016/0097-3165(89)90030-7. |
[30] |
R. A. Wilson,
Maximal subgroups of automorphism groups of simple groups, J. London Math. Soc., 32 (1985), 406-466.
doi: 10.1112/jlms/s2-32.3.460. |
[31] |
R. A. Wilson, The Finite Simple Groups, London: Springer-Verlag London Ltd., 2009. Graduate Texts in Mathematics, Vol. 251.
doi: 10.1007/978-1-84800-988-2. |
[32] |
E. Witt,
Die 5-Fach transitiven Gruppen von Mathieu, Abh. Math. Sem. Univ. Hamburg, 12 (1937), 256-264.
doi: 10.1007/BF02948947. |
[33] |
E. Witt,
Uber Steinersche Systeme, Abh. Math. Sem. Univ. Hamburg, 12 (1937), 265-275.
doi: 10.1007/BF02948948. |
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 |
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 |
|
|||
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 |
|
|||
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 |
Orbits | Parameters | Full Automorphism Group |
Orbits | Parameters | Full Automorphism Group |
Orbits | Parameters | Full Automorphism Group |
Orbits | Parameters | Full Automorphism Group |
Design | Orbits | Parameters | Full Automorphism Group |
Design | Orbits | Parameters | Full Automorphism Group |
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1 | 2 | ||
1 | 2 | ||
2 | 1 | ||
2 | 3 | ||
1 | 1 | ||
3 | 2 | ||
1 | 2 | ||
1 | 2 | ||
1 | 10 |
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1 | 2 | ||
1 | 2 | ||
2 | 1 | ||
2 | 3 | ||
1 | 1 | ||
3 | 2 | ||
1 | 2 | ||
1 | 2 | ||
1 | 10 |
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SRG | Automorphism group | Code | |
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SRG | Automorphism group | Code | |
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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 |
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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