American Institute of Mathematical Sciences

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August  2018, 12(3): 607-628. doi: 10.3934/amc.2018036

On self-orthogonal designs and codes related to Held's simple group

Crnković Dean 1, , Vedrana Mikulić Crnković 1, and  Bernardo G. Rodrigues 2,

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

Received  November 2017 Published  July 2018

Fund Project: This work has been supported by Croatian Science Foundation under the project 1637
This work is based on the research supported by the National Research Foundation of South Africa (Grant Numbers 95725 and 106071).

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).

Keywords: Design, orbit matrix, linear code, sporadic simple group, Held group.
Mathematics Subject Classification: Primary: 05E18, 94B05; Secondary: 20D08, 05B05, 05E30.
Citation: Crnković Dean, Vedrana Mikulić Crnković, Bernardo G. Rodrigues. On self-orthogonal designs and codes related to Held's simple group. Advances in Mathematics of Communications, 2018, 12 (3) : 607-628. doi: 10.3934/amc.2018036
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.

Table 1.  Maximal subgroups of ${\rm He}$, up to conjugation.
No. Max. sub. Deg.
1 $S_4(4):2$ 2058
2 ${2^2}\, ^{\cdot} L_3(4){:} S_3$ 8330
3 ${2^6{:}3}\, ^{\cdot}\, S_6$ 29155
4 ${2^6{:}3}\, ^{\cdot} S_6$ 29155
5 $2^{1+6}_{+} {.} L_3(2)$ 187425
6 $7^2{:} L_2(7)$ 244800
7 $3 ^\cdot\, S_7$ 266560
8 $7^{1+2}_{+}{:}(3 \times S_3)$ 652800
9 $S_4 \times L_3(2)$ 999600
10 $7{:}3 \times L_3(2)$ 1142400
11 $5^2{:}4A_4$ 3358656
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No. Max. sub. Deg.
1 $S_4(4):2$ 2058
2 ${2^2}\, ^{\cdot} L_3(4){:} S_3$ 8330
3 ${2^6{:}3}\, ^{\cdot}\, S_6$ 29155
4 ${2^6{:}3}\, ^{\cdot} S_6$ 29155
5 $2^{1+6}_{+} {.} L_3(2)$ 187425
6 $7^2{:} L_2(7)$ 244800
7 $3 ^\cdot\, S_7$ 266560
8 $7^{1+2}_{+}{:}(3 \times S_3)$ 652800
9 $S_4 \times L_3(2)$ 999600
10 $7{:}3 \times L_3(2)$ 1142400
11 $5^2{:}4A_4$ 3358656
Table 2.  Block intersection numbers of $\mathcal D_{k}$ and $\bar{\mathcal D}_k = \mathcal D_{2058-k}$
$k$ $\mathcal D_{k}$ $i$ $l_i$
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
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$k$ $\mathcal D_{k}$ $i$ $l_i$
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 $1$-designs on 2058 points
Orbits Parameters Full Automorphism Group
$\Omega_1$, $\Omega_4$ $1$-$(2058,426,426)$ $\rm {He}{:}2$
$\Omega_1$, $\Omega_4$, $\Omega_2$ $1$-$(2058,562,562)$ $\rm He$
$\Omega_1$, $\Omega_4$, $\Omega_2$, $\Omega_3$ $1$-$(2058,698,698)$ $\rm {He}{:}2$
$\Omega_1$, $\Omega_4$, $\Omega_3$ $1$-$(2058,562,562)$ $\rm He$
$\Omega_1$, $\Omega_2$ $1$-$(2058,137,137)$ $\rm He$
$\Omega_1$, $\Omega_2$, $\Omega_3$ $1$-$(2058,273,273)$ $\rm {He}{:}2$
$\Omega_1$, $\Omega_3$ $1$-$(2058,137,137)$ $\rm He$
$\Omega_4$ $1$-$(2058,425,425)$ $\rm {He}{:}2$
$\Omega_4$, $\Omega_2$ $1$-$(2058,561,561)$ $\rm He$
$\Omega_4$, $\Omega_2$, $\Omega_3$ $1$-$(2058,697,697)$ $\rm {He}{:}2$
$\Omega_4$, $\Omega_3$ $1$-$(2058,561,561)$ $\rm He$
$\Omega_2$ $1$-$(2058,136,136)$ $\rm He$
$\Omega_2$, $\Omega_3$ $1$-$(2058,272,272)$ $\rm {He}{:}2$
$\Omega_3$ $1$-$(2058,136,136)$ $\rm He$
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Orbits Parameters Full Automorphism Group
$\Omega_1$, $\Omega_4$ $1$-$(2058,426,426)$ $\rm {He}{:}2$
$\Omega_1$, $\Omega_4$, $\Omega_2$ $1$-$(2058,562,562)$ $\rm He$
$\Omega_1$, $\Omega_4$, $\Omega_2$, $\Omega_3$ $1$-$(2058,698,698)$ $\rm {He}{:}2$
$\Omega_1$, $\Omega_4$, $\Omega_3$ $1$-$(2058,562,562)$ $\rm He$
$\Omega_1$, $\Omega_2$ $1$-$(2058,137,137)$ $\rm He$
$\Omega_1$, $\Omega_2$, $\Omega_3$ $1$-$(2058,273,273)$ $\rm {He}{:}2$
$\Omega_1$, $\Omega_3$ $1$-$(2058,137,137)$ $\rm He$
$\Omega_4$ $1$-$(2058,425,425)$ $\rm {He}{:}2$
$\Omega_4$, $\Omega_2$ $1$-$(2058,561,561)$ $\rm He$
$\Omega_4$, $\Omega_2$, $\Omega_3$ $1$-$(2058,697,697)$ $\rm {He}{:}2$
$\Omega_4$, $\Omega_3$ $1$-$(2058,561,561)$ $\rm He$
$\Omega_2$ $1$-$(2058,136,136)$ $\rm He$
$\Omega_2$, $\Omega_3$ $1$-$(2058,272,272)$ $\rm {He}{:}2$
$\Omega_3$ $1$-$(2058,136,136)$ $\rm He$
Table 4.  Symmetric $1$-designs on 8330 points
Orbits Parameters Full Automorphism Group
$\Delta_1$, $\Delta_2$ $1$-$(8330,106,106)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_3$ $1$-$(8330, 1450, 1450)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_4$ $1$-$(8330, 2290, 2290)$ $\rm {He}$
$\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 3010, 3010)$ $\rm {He}$
$\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3850, 3850)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 3130, 3130)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_5$ $1$-$(8330, 2170, 2170)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3010, 3010)$ $\rm {He}$
$\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_6$ $1$-$(8330, 2290, 2290)$ $\rm {He}$
$\Delta_1$, $\Delta_2$, $\Delta_4$ $1$-$(8330,946,946)$ $\rm {He}$
$\Delta_1$, $\Delta_2$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 1666, 1666)$ $\rm {He}$
$\Delta_1$, $\Delta_2$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2506, 2506)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 1786, 1786)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_5$ $1$-$(8330,826,826)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 1666, 1666)$ $\rm {He}$
$\Delta_1$, $\Delta_2$, $\Delta_6$ $1$-$(8330,946,946)$ $\rm {He}$
$\Delta_1$, $\Delta_3$ $1$-$(8330, 1345, 1345)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_3$, $\Delta_4$ $1$-$(8330, 2185, 2185)$ $\rm {He}$
$\Delta_1$, $\Delta_3$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 2905, 2905)$ $\rm {He}$
$\Delta_1$, $\Delta_3$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3745, 3745)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_3$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 3025, 3025)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_3$, $\Delta_5$ $1$-$(8330, 2065, 2065)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_3$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2905, 2905)$ $\rm {He}$
$\Delta_1$, $\Delta_3$, $\Delta_6$ $1$-$(8330, 2185, 2185)$ $\rm {He}$
$\Delta_1$, $\Delta_4$ $1$-$(8330,841,841)$ $\rm {He}$
$\Delta_1$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 1561, 1561)$ $\rm {He}$
$\Delta_1$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2401, 2401)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 1681, 1681)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_5$ $1$-$(8330,721,721)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 1561, 1561)$ $\rm {He}$
$\Delta_1$, $\Delta_6$ $1$-$(8330,841,841)$ $\rm {He}$
$\Delta_2$ $1$-$(8330,105,105)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_3$ $1$-$(8330, 1449, 1449)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_3$, $\Delta_4$ $1$-$(8330, 2289, 2289)$ $\rm {He}$
$\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 3009, 3009)$ $\rm {He}$
$\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3849, 3849)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 3129, 3129)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_3$, $\Delta_5$ $1$-$(8330, 2169, 2169)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_3$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3009, 3009)$ $\rm {He}$
$\Delta_2$, $\Delta_3$, $\Delta_6$ $1$-$(8330, 2289, 2289)$ $\rm {He}$
$\Delta_2$, $\Delta_4$ $1$-$(8330,945,945)$ $\rm {He}$
$\Delta_2$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 1665, 1665)$ $\rm {He}$
$\Delta_2$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2505, 2505)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 1785, 1785)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_5$ $1$-$(8330,825,825)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 1665, 1665)$ $\rm {He}$
$\Delta_2$, $\Delta_6$ $1$-$(8330,945,945)$ $\rm {He}$
$\Delta_3$ $1$-$(8330, 1344, 1344)$ $\rm {He}{:}2$
$\Delta_3$, $\Delta_4$ $1$-$(8330, 2184, 2184)$ $\rm {He}$
$\Delta_3$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 2904, 2904)$ $\rm {He}$
$\Delta_3$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3744, 3744)$ $\rm {He}{:}2$
$\Delta_3$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 3024, 3024)$ $\rm {He}{:}2$
$\Delta_3$, $\Delta_5$ $1$-$(8330, 2064, 2064)$ $\rm {He}{:}2$
$\Delta_3$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2904, 2904)$ $\rm {He}$
$\Delta_3$, $\Delta_6$ $1$-$(8330, 2184, 2184)$ $\rm {He}$
$\Delta_4$ $1$-$(8330,840,840)$ $\rm {He}$
$\Delta_4$, $\Delta_5$ $1$-$(8330, 1560, 1560)$ $\rm {He}$
$\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2400, 2400)$ $\rm {He}{:}2$
$\Delta_4$, $\Delta_6$ $1$-$(8330, 1680, 1680)$ $\rm {He}{:}2$
$\Delta_5$ $1$-$(8330,720,720)$ $\rm {He}{:}2$
$\Delta_5$, $\Delta_6$ $1$-$(8330, 1560, 1560)$ $\rm {He}$
$\Delta_6$ $1$-$(8330,840,840)$ $\rm {He}$
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Orbits Parameters Full Automorphism Group
$\Delta_1$, $\Delta_2$ $1$-$(8330,106,106)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_3$ $1$-$(8330, 1450, 1450)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_4$ $1$-$(8330, 2290, 2290)$ $\rm {He}$
$\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 3010, 3010)$ $\rm {He}$
$\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3850, 3850)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 3130, 3130)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_5$ $1$-$(8330, 2170, 2170)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3010, 3010)$ $\rm {He}$
$\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_6$ $1$-$(8330, 2290, 2290)$ $\rm {He}$
$\Delta_1$, $\Delta_2$, $\Delta_4$ $1$-$(8330,946,946)$ $\rm {He}$
$\Delta_1$, $\Delta_2$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 1666, 1666)$ $\rm {He}$
$\Delta_1$, $\Delta_2$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2506, 2506)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 1786, 1786)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_5$ $1$-$(8330,826,826)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_2$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 1666, 1666)$ $\rm {He}$
$\Delta_1$, $\Delta_2$, $\Delta_6$ $1$-$(8330,946,946)$ $\rm {He}$
$\Delta_1$, $\Delta_3$ $1$-$(8330, 1345, 1345)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_3$, $\Delta_4$ $1$-$(8330, 2185, 2185)$ $\rm {He}$
$\Delta_1$, $\Delta_3$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 2905, 2905)$ $\rm {He}$
$\Delta_1$, $\Delta_3$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3745, 3745)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_3$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 3025, 3025)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_3$, $\Delta_5$ $1$-$(8330, 2065, 2065)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_3$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2905, 2905)$ $\rm {He}$
$\Delta_1$, $\Delta_3$, $\Delta_6$ $1$-$(8330, 2185, 2185)$ $\rm {He}$
$\Delta_1$, $\Delta_4$ $1$-$(8330,841,841)$ $\rm {He}$
$\Delta_1$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 1561, 1561)$ $\rm {He}$
$\Delta_1$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2401, 2401)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 1681, 1681)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_5$ $1$-$(8330,721,721)$ $\rm {He}{:}2$
$\Delta_1$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 1561, 1561)$ $\rm {He}$
$\Delta_1$, $\Delta_6$ $1$-$(8330,841,841)$ $\rm {He}$
$\Delta_2$ $1$-$(8330,105,105)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_3$ $1$-$(8330, 1449, 1449)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_3$, $\Delta_4$ $1$-$(8330, 2289, 2289)$ $\rm {He}$
$\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 3009, 3009)$ $\rm {He}$
$\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3849, 3849)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 3129, 3129)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_3$, $\Delta_5$ $1$-$(8330, 2169, 2169)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_3$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3009, 3009)$ $\rm {He}$
$\Delta_2$, $\Delta_3$, $\Delta_6$ $1$-$(8330, 2289, 2289)$ $\rm {He}$
$\Delta_2$, $\Delta_4$ $1$-$(8330,945,945)$ $\rm {He}$
$\Delta_2$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 1665, 1665)$ $\rm {He}$
$\Delta_2$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2505, 2505)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 1785, 1785)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_5$ $1$-$(8330,825,825)$ $\rm {He}{:}2$
$\Delta_2$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 1665, 1665)$ $\rm {He}$
$\Delta_2$, $\Delta_6$ $1$-$(8330,945,945)$ $\rm {He}$
$\Delta_3$ $1$-$(8330, 1344, 1344)$ $\rm {He}{:}2$
$\Delta_3$, $\Delta_4$ $1$-$(8330, 2184, 2184)$ $\rm {He}$
$\Delta_3$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 2904, 2904)$ $\rm {He}$
$\Delta_3$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3744, 3744)$ $\rm {He}{:}2$
$\Delta_3$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 3024, 3024)$ $\rm {He}{:}2$
$\Delta_3$, $\Delta_5$ $1$-$(8330, 2064, 2064)$ $\rm {He}{:}2$
$\Delta_3$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2904, 2904)$ $\rm {He}$
$\Delta_3$, $\Delta_6$ $1$-$(8330, 2184, 2184)$ $\rm {He}$
$\Delta_4$ $1$-$(8330,840,840)$ $\rm {He}$
$\Delta_4$, $\Delta_5$ $1$-$(8330, 1560, 1560)$ $\rm {He}$
$\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2400, 2400)$ $\rm {He}{:}2$
$\Delta_4$, $\Delta_6$ $1$-$(8330, 1680, 1680)$ $\rm {He}{:}2$
$\Delta_5$ $1$-$(8330,720,720)$ $\rm {He}{:}2$
$\Delta_5$, $\Delta_6$ $1$-$(8330, 1560, 1560)$ $\rm {He}$
$\Delta_6$ $1$-$(8330,840,840)$ $\rm {He}$
Table 5.  $1$-designs on 2058 points and 8330 blocks
Design Orbits Parameters Full Automorphism Group
$\mathcal D_{1}''$ $\Delta_5$ $1$-$(2058,840, 3400)$ $\rm He$
$\mathcal D_{2}''$ $\Delta_5$, $\Delta_2$ $1$-$(2058,861, 3485)$ $\rm He$
$\mathcal D_{3}''$ $\Delta_5$, $\Delta_2$, $\Delta_1$ $1$-$(2058,882, 3570)$ $\rm He$
$\mathcal D_{4}''$ $\Delta_5$, $\Delta_1$ $1$-$(2058,861, 3485)$ $\rm He$
$\mathcal D_{5}''$ $\Delta_4$ $1$-$(2058,840, 3400)$ $\rm He$
$\mathcal D_{6}''$ $\Delta_4$, $\Delta_2$ $1$-$(2058,861, 3485)$ $\rm He$
$\mathcal D_{7}''$ $\Delta_4$, $\Delta_2$, $\Delta_1$ $1$-$(2058,882, 3570)$ $\rm He$
$\mathcal D_{8}''$ $\Delta_4$, $\Delta_1$ $1$-$(2058,861, 3485)$ $\rm He$
$\mathcal D_{9}''$ $\Delta_3$ $1$-$(2058,336, 1360)$ $\rm {He}{:}2$
$\mathcal D_{10}''$ $\Delta_3$, $\Delta_2$ $1$-$(2058,357, 1445)$ $\rm He$
$\mathcal D_{11}''$ $\Delta_3$, $\Delta_2$, $\Delta_1$ $1$-$(2058,378, 1530)$ $\rm {He}{:}2$
$\mathcal D_{12}''$ $\Delta_3$, $\Delta_1$ $1$-$(2058,357, 1445)$ $\rm He$
$\mathcal D_{13}''$ $\Delta_2$ $1$-$(2058, 21, 85)$ $\rm He$
$\mathcal D_{14}''$ $\Delta_2$, $\Delta_1$ $1$-$(2058, 42,170)$ $\rm {He}{:}2$
$\mathcal D_{15}''$ $\Delta_1$ $1$-$(2058, 21, 85)$ $\rm He$
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Design Orbits Parameters Full Automorphism Group
$\mathcal D_{1}''$ $\Delta_5$ $1$-$(2058,840, 3400)$ $\rm He$
$\mathcal D_{2}''$ $\Delta_5$, $\Delta_2$ $1$-$(2058,861, 3485)$ $\rm He$
$\mathcal D_{3}''$ $\Delta_5$, $\Delta_2$, $\Delta_1$ $1$-$(2058,882, 3570)$ $\rm He$
$\mathcal D_{4}''$ $\Delta_5$, $\Delta_1$ $1$-$(2058,861, 3485)$ $\rm He$
$\mathcal D_{5}''$ $\Delta_4$ $1$-$(2058,840, 3400)$ $\rm He$
$\mathcal D_{6}''$ $\Delta_4$, $\Delta_2$ $1$-$(2058,861, 3485)$ $\rm He$
$\mathcal D_{7}''$ $\Delta_4$, $\Delta_2$, $\Delta_1$ $1$-$(2058,882, 3570)$ $\rm He$
$\mathcal D_{8}''$ $\Delta_4$, $\Delta_1$ $1$-$(2058,861, 3485)$ $\rm He$
$\mathcal D_{9}''$ $\Delta_3$ $1$-$(2058,336, 1360)$ $\rm {He}{:}2$
$\mathcal D_{10}''$ $\Delta_3$, $\Delta_2$ $1$-$(2058,357, 1445)$ $\rm He$
$\mathcal D_{11}''$ $\Delta_3$, $\Delta_2$, $\Delta_1$ $1$-$(2058,378, 1530)$ $\rm {He}{:}2$
$\mathcal D_{12}''$ $\Delta_3$, $\Delta_1$ $1$-$(2058,357, 1445)$ $\rm He$
$\mathcal D_{13}''$ $\Delta_2$ $1$-$(2058, 21, 85)$ $\rm He$
$\mathcal D_{14}''$ $\Delta_2$, $\Delta_1$ $1$-$(2058, 42,170)$ $\rm {He}{:}2$
$\mathcal D_{15}''$ $\Delta_1$ $1$-$(2058, 21, 85)$ $\rm He$
Table 6.  Non-trivial binary codes of the pairwise non-isomorphic symmetric $1$-designs on 2058 points
$k$ $C_{k}$ ${\rm Aut}(C_k)$ $\bar{C}_{k}$ ${\rm Aut}(\bar{C}_{k})$ $E_k$
$426$ $[2058,783]$ ${\rm He}\textrm{:}2$ $[2058,782]$ ${\rm He}\textrm{:}2$ $[2058,782]$
$562$ $[2058, 52]$ ${\rm He}$ $[2058, 51]$ ${\rm He}$ $[2058, 51]$
$698$ $[2058,681]$ ${\rm He}\textrm{:}2$ $[2058,680]$ ${\rm He}\textrm{:}2$ $[2058,680]$
$136$ $[2058,731]$ ${\rm He}$ $[2058,732]$ ${\rm He}$ $[2058,731]$
$272$ $[2058,102]$ ${\rm He}\textrm{:}2$ $[2058,103]$ ${\rm He}\textrm{:}2$ $[2058,102]$
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$k$ $C_{k}$ ${\rm Aut}(C_k)$ $\bar{C}_{k}$ ${\rm Aut}(\bar{C}_{k})$ $E_k$
$426$ $[2058,783]$ ${\rm He}\textrm{:}2$ $[2058,782]$ ${\rm He}\textrm{:}2$ $[2058,782]$
$562$ $[2058, 52]$ ${\rm He}$ $[2058, 51]$ ${\rm He}$ $[2058, 51]$
$698$ $[2058,681]$ ${\rm He}\textrm{:}2$ $[2058,680]$ ${\rm He}\textrm{:}2$ $[2058,680]$
$136$ $[2058,731]$ ${\rm He}$ $[2058,732]$ ${\rm He}$ $[2058,731]$
$272$ $[2058,102]$ ${\rm He}\textrm{:}2$ $[2058,103]$ ${\rm He}\textrm{:}2$ $[2058,102]$
Table 7.  Number of submodules of length 2058 invariant under ${\rm He}$
$k'$ $\#$ $k'$ $\#$
$0$ 1 $731$ 2
$1$ 1 $732$ 2
$51$ 2 $782$ 1
$52$ 2 $783$ 3
$102$ 1 $784$ 1
$103$ 3 $977$ 2
$104$ 1 $978$ 2
$680$ 1 $1028$ 2
$681$ 1 $1029$ 10
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$k'$ $\#$ $k'$ $\#$
$0$ 1 $731$ 2
$1$ 1 $732$ 2
$51$ 2 $782$ 1
$52$ 2 $783$ 3
$102$ 1 $784$ 1
$103$ 3 $977$ 2
$104$ 1 $978$ 2
$680$ 1 $1028$ 2
$681$ 1 $1029$ 10
Table 8.  Non-trivial binary codes of the pairwise non-isomorphic symmetric $1$-designs on 8330 points
$k$ $C_{k}'$ $\bar{C}_{k}'$ $E_k'$
$106$ $[8330, 7055]$ $[8330, 7054]$ $[8330, 7054]$
$1450$ $[8330,783]$ $[8330,782]$ $[8330,782]$
$2290$ $[8330, 1972]$* $[8330, 1971]$* $[8330, 1971]$
$3010$ $[8330, 7004]$* $[8330, 7003]$* $[8330, 7003]$
$3850$ $[8330, 4353]$ $[8330, 4352]$ $[8330, 4352]$
$3130$ $[8330,681]$ $[8330,680]$ $[8330,680]$
$2170$ $[8330, 4455]$ $[8330, 4454]$ $[8330, 4454]$
$946$ $[8330, 4404]$* $[8330, 4403]$* $[8330, 4403]$
$1666$ $[8330,732]$* $[8330,731]$* $[8330,731]$
$2506$ $[8330, 1921]$ $[8330, 1920]$ $[8330, 1920]$
$1786$ $[8330, 6953]$ $[8330, 6952]$ $[8330, 6952]$
$826$ $[8330, 2023]$ $[8330, 2022]$ $[8330, 2022]$
$1345$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$
$2185$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$
$3745$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$
$3025$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$
$2065$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$
$841$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$
$1561$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$
$2401$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$
$721$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$
$105$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$
$2289$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$
$3009$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$
$3849$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$
$2169$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$
$945$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$
$2505$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$
$1785$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$
$825$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$
$1344$ $[8330, 6272]$ $[8330, 6273]$ $[8330, 6272]$
$2184$ $[8330, 5083]$* $[8330, 5084]$* $[8330, 5083]$
$2904$ $[8330, 51]$* $[8330, 52]$* $[8330, 51]$
$3744$ $[8330, 2702]$ $[8330, 2703]$ $[8330, 2702]$
$3024$ $[8330, 6374]$ $[8330, 6375]$ $[8330, 6374]$
$2064$ $[8330, 2600]$ $[8330, 2601]$ $[8330, 2600]$
$840$ $[8330, 2651]$* $[8330, 2652]$* $[8330, 2651]$
$1560$ $[8330, 6323]$* $[8330, 6324]$* $[8330, 6323]$
$2400$ $[8330, 5134]$ $[8330, 5135]$ $[8330, 5134]$
$1680$ $[8330,102]$ $[8330,103]$ $[8330,102]$
$720$ $[8330, 5032]$ $[8330, 5033]$ $[8330, 5032]$
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$k$ $C_{k}'$ $\bar{C}_{k}'$ $E_k'$
$106$ $[8330, 7055]$ $[8330, 7054]$ $[8330, 7054]$
$1450$ $[8330,783]$ $[8330,782]$ $[8330,782]$
$2290$ $[8330, 1972]$* $[8330, 1971]$* $[8330, 1971]$
$3010$ $[8330, 7004]$* $[8330, 7003]$* $[8330, 7003]$
$3850$ $[8330, 4353]$ $[8330, 4352]$ $[8330, 4352]$
$3130$ $[8330,681]$ $[8330,680]$ $[8330,680]$
$2170$ $[8330, 4455]$ $[8330, 4454]$ $[8330, 4454]$
$946$ $[8330, 4404]$* $[8330, 4403]$* $[8330, 4403]$
$1666$ $[8330,732]$* $[8330,731]$* $[8330,731]$
$2506$ $[8330, 1921]$ $[8330, 1920]$ $[8330, 1920]$
$1786$ $[8330, 6953]$ $[8330, 6952]$ $[8330, 6952]$
$826$ $[8330, 2023]$ $[8330, 2022]$ $[8330, 2022]$
$1345$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$
$2185$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$
$3745$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$
$3025$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$
$2065$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$
$841$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$
$1561$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$
$2401$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$
$721$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$
$105$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$
$2289$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$
$3009$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$
$3849$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$
$2169$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$
$945$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$
$2505$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$
$1785$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$
$825$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$
$1344$ $[8330, 6272]$ $[8330, 6273]$ $[8330, 6272]$
$2184$ $[8330, 5083]$* $[8330, 5084]$* $[8330, 5083]$
$2904$ $[8330, 51]$* $[8330, 52]$* $[8330, 51]$
$3744$ $[8330, 2702]$ $[8330, 2703]$ $[8330, 2702]$
$3024$ $[8330, 6374]$ $[8330, 6375]$ $[8330, 6374]$
$2064$ $[8330, 2600]$ $[8330, 2601]$ $[8330, 2600]$
$840$ $[8330, 2651]$* $[8330, 2652]$* $[8330, 2651]$
$1560$ $[8330, 6323]$* $[8330, 6324]$* $[8330, 6323]$
$2400$ $[8330, 5134]$ $[8330, 5135]$ $[8330, 5134]$
$1680$ $[8330,102]$ $[8330,103]$ $[8330,102]$
$720$ $[8330, 5032]$ $[8330, 5033]$ $[8330, 5032]$
Table 9.  Non-trivial binary codes of the pairwise non-isomorphic $1$-designs on 2058 points and 8330 blocks
$k$ $C_{k}''$ ${\rm Aut}(C_k'')$ $\bar{C}_{k}''$ ${\rm Aut}(\bar{C}_{k}'')$ $E_{k}''$
$840$ $[2058,731]$ ${\rm He}$ $[2058,732]$ ${\rm He}$ $[2058,731]$
$882$ $[2058, 52]$ ${\rm He}$ $[2058, 51]$ ${\rm He}$ $[2058, 51]$
$336$ $[2058,680]$ ${\rm He}\textrm{:}2$ $[2058,681]$ ${\rm He}\textrm{:}2$ $[2058,680]$
$378$ $[2058,103]$ ${\rm He}\textrm{:}2$ $[2058,102]$ ${\rm He}\textrm{:}2$ $[2058,102]$
$42$ $[2058,783]$ ${\rm He}\textrm{:}2$ $[2058,782]$ ${\rm He}:2$ $[2058,782]$
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$k$ $C_{k}''$ ${\rm Aut}(C_k'')$ $\bar{C}_{k}''$ ${\rm Aut}(\bar{C}_{k}'')$ $E_{k}''$
$840$ $[2058,731]$ ${\rm He}$ $[2058,732]$ ${\rm He}$ $[2058,731]$
$882$ $[2058, 52]$ ${\rm He}$ $[2058, 51]$ ${\rm He}$ $[2058, 51]$
$336$ $[2058,680]$ ${\rm He}\textrm{:}2$ $[2058,681]$ ${\rm He}\textrm{:}2$ $[2058,680]$
$378$ $[2058,103]$ ${\rm He}\textrm{:}2$ $[2058,102]$ ${\rm He}\textrm{:}2$ $[2058,102]$
$42$ $[2058,783]$ ${\rm He}\textrm{:}2$ $[2058,782]$ ${\rm He}:2$ $[2058,782]$
Table 10.  Non-trivial binary codes of the pairwise non-isomorphic $1$-designs on 8330 points and 2058 blocks
$k$ ${C_k''}^t$ ${\rm Aut}({C_k''}^t)$ ${\bar{C}_k}^{''t}$ ${\rm Aut}({\bar{C}_k}^{''t})$
$840$ $[8330,731]$ ${\rm He}$ $[8330,732]$ ${\rm He}$
$882$ $[8330, 52]$ ${\rm He}$ $[8330, 51]$ ${\rm He}$
$336$ $[8330,680]$ ${\rm He}\textrm{:}2$ $[8330,681]$ ${\rm He}\textrm{:}2$
$378$ $[8330,103]$ ${\rm He}\textrm{:}2$ $[8330,102]$ ${\rm He}\textrm{:}2$
$42$ $[8330,783]$ ${\rm He}\textrm{:}2$ $[8330,782]$ ${\rm He}\textrm{:}2$
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$k$ ${C_k''}^t$ ${\rm Aut}({C_k''}^t)$ ${\bar{C}_k}^{''t}$ ${\rm Aut}({\bar{C}_k}^{''t})$
$840$ $[8330,731]$ ${\rm He}$ $[8330,732]$ ${\rm He}$
$882$ $[8330, 52]$ ${\rm He}$ $[8330, 51]$ ${\rm He}$
$336$ $[8330,680]$ ${\rm He}\textrm{:}2$ $[8330,681]$ ${\rm He}\textrm{:}2$
$378$ $[8330,103]$ ${\rm He}\textrm{:}2$ $[8330,102]$ ${\rm He}\textrm{:}2$
$42$ $[8330,783]$ ${\rm He}\textrm{:}2$ $[8330,782]$ ${\rm He}\textrm{:}2$
Table 11.  Non-trivial binary pairwise non-equivalent codes of the orbit matrices of the symmetric self-orthogonal $1$-designs on 2058 points
$[294,111]$ $[294,110]$ $[686,261]$ $[686,260]$
$[294, 10,126]$ $[294, 6,120]$ $[686, 18,224]$ $[686, 17,224]$
$[294, 93]$ $[294, 98]$ $[686,227]$ $[686,226]$
$[294,101]$ $[294,105]$ $[686,243]$ $[686,244]$
$[294, 18, 96]$ $[294, 13,112]$ $[686, 34,192]$ $[686, 35,126]$
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$[294,111]$ $[294,110]$ $[686,261]$ $[686,260]$
$[294, 10,126]$ $[294, 6,120]$ $[686, 18,224]$ $[686, 17,224]$
$[294, 93]$ $[294, 98]$ $[686,227]$ $[686,226]$
$[294,101]$ $[294,105]$ $[686,243]$ $[686,244]$
$[294, 18, 96]$ $[294, 13,112]$ $[686, 34,192]$ $[686, 35,126]$
Table 12.  Non-trivial binary pairwise non-equivalent codes of the orbit matrices of the self-orthogonal $1$-designs on 2058 points and 8330 blocks
$[294,104]$ $[294,105]$ $[686,243]$ $[686,244]$
$[294, 7,120]$ $[294, 6,120]$ $[686, 18,224]$ $[686, 17,224]$
$[294, 98]$ $[294, 99]$ $[686,226]$ $[686,227]$
$[294, 13,112]$ $[294, 12,112]$ $[686, 35,126]$ $[686, 34,192]$
$[294,111]$ $[294,110]$ $[686,261]$ $[686,260]$
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$[294,104]$ $[294,105]$ $[686,243]$ $[686,244]$
$[294, 7,120]$ $[294, 6,120]$ $[686, 18,224]$ $[686, 17,224]$
$[294, 98]$ $[294, 99]$ $[686,226]$ $[686,227]$
$[294, 13,112]$ $[294, 12,112]$ $[686, 35,126]$ $[686, 34,192]$
$[294,111]$ $[294,110]$ $[686,261]$ $[686,260]$
Table 13.  Non-trivial binary pairwise non-equivalent codes of the orbit matrices of the self-orthogonal $1$-designs on 8330 points and 2058 blocks
$[490, 43]$ $[490, 44]$ $[1190,101]$ $[1190,102]$
$[490, 4,210]$ $[490, 3,240]$ $[1190, 10,554]$ $[1190, 9,564]$
$[490, 40]$ $[490, 41]$ $[1190, 92]$ $[1190, 93]$
$[490, 7, 90]$ $[490, 6,224]$ $[1190, 19,360]$ $[1190, 18,360]$
$[490, 47]$ $[490, 46]$ $[1190,111]$ $[1190,110]$
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$[490, 43]$ $[490, 44]$ $[1190,101]$ $[1190,102]$
$[490, 4,210]$ $[490, 3,240]$ $[1190, 10,554]$ $[1190, 9,564]$
$[490, 40]$ $[490, 41]$ $[1190, 92]$ $[1190, 93]$
$[490, 7, 90]$ $[490, 6,224]$ $[1190, 19,360]$ $[1190, 18,360]$
$[490, 47]$ $[490, 46]$ $[1190,111]$ $[1190,110]$
Table 14.  Non-trivial binary pairwise non-equivalent codes of the orbit matrices of the self-orthogonal symmetric $1$-designs on 8330 points
$[490, 47]$ $[490, 46]$ $[1190,111]$ $[1190,110]$
$[490, 41]$ $[490, 40]$ $[1190, 99]$ $[1190, 98]$
$[490, 44]$ $[490, 43]$ $[1190,105]$ $[1190,104]$
$[490, 3,240]$ $[490, 4,210]$ $[1190, 6,568]$ $[1190, 7,568]$
$[490, 6,224]$ $[490, 7, 90]$ $[1190, 12,512]$ $[1190, 13,512]$
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$[490, 47]$ $[490, 46]$ $[1190,111]$ $[1190,110]$
$[490, 41]$ $[490, 40]$ $[1190, 99]$ $[1190, 98]$
$[490, 44]$ $[490, 43]$ $[1190,105]$ $[1190,104]$
$[490, 3,240]$ $[490, 4,210]$ $[1190, 6,568]$ $[1190, 7,568]$
$[490, 6,224]$ $[490, 7, 90]$ $[1190, 12,512]$ $[1190, 13,512]$
Table 15.  Parameters of non-trivial binary codes of the orbit matrices of the self-orthogonal symmetric $1$-designs with 2058 points (Theorem 5, item $1$, fixed points and blocks)
$[154, 75, 8]$ $[154, 74, 8]$ $[42, 15, 6]$ $[42, 14, 8]$
$[154, 12, 42]$ $[154, 11, 48]$ $[42, 4, 18]$ $[42, 3, 24]$
$[154, 57, 12]$ $[154, 56, 12]$ $[42, 9, 12]$ $[42, 8, 12]$
$[154, 65, 8]$ $[154, 66, 8]$ $[42, 11, 8]$ $[42, 12, 8]$
$[154, 20, 32]$ $[154, 21, 32]$ $[42, 6, 16]$ $[42, 7, 16]$
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$[154, 75, 8]$ $[154, 74, 8]$ $[42, 15, 6]$ $[42, 14, 8]$
$[154, 12, 42]$ $[154, 11, 48]$ $[42, 4, 18]$ $[42, 3, 24]$
$[154, 57, 12]$ $[154, 56, 12]$ $[42, 9, 12]$ $[42, 8, 12]$
$[154, 65, 8]$ $[154, 66, 8]$ $[42, 11, 8]$ $[42, 12, 8]$
$[154, 20, 32]$ $[154, 21, 32]$ $[42, 6, 16]$ $[42, 7, 16]$
Table 16.  Parameters of non-trivial binary codes of the orbit matrices of the self-orthogonal symmetric $1$-designs with 2058 points (Theorem 5, item $2$, non-fixed part of orbit matrices)
$[952,352]$ $[1008,384]$
$[952, 20,368]$ $[1008, 24,336]$
$[952,312]$ $[1008,336]$
$[952,332]$ $[1008,360]$
$[952, 40,224]$ $[1008, 48]$
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$[952,352]$ $[1008,384]$
$[952, 20,368]$ $[1008, 24,336]$
$[952,312]$ $[1008,336]$
$[952,332]$ $[1008,360]$
$[952, 40,224]$ $[1008, 48]$
Table 17.  Parameters of non-trivial binary codes of the orbit matrices of the self-orthogonal symmetric $1$-designs with 8330 points (Theorem 5, item $1$, fixed points and blocks)
$[346, 75, 26]$ $[346, 74, 32]$ $[42, 15, 6]$ $[42, 14, 8]$
$[346, 57, 32]$ $[346, 56, 32]$ $[42, 9, 12]$ $[42, 8, 12]$
$[346, 66, 26]$ $[346, 65, 32]$ $[42, 12, 8]$ $[42, 11, 8]$
$[346, 11,152]$ $[346, 12,106]$ $[42, 3, 24]$ $[42, 4, 18]$
$[346, 20, 96]$ $[346, 21, 96]$ $[42, 6, 16]$ $[42, 7, 16]$
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$[346, 75, 26]$ $[346, 74, 32]$ $[42, 15, 6]$ $[42, 14, 8]$
$[346, 57, 32]$ $[346, 56, 32]$ $[42, 9, 12]$ $[42, 8, 12]$
$[346, 66, 26]$ $[346, 65, 32]$ $[42, 12, 8]$ $[42, 11, 8]$
$[346, 11,152]$ $[346, 12,106]$ $[42, 3, 24]$ $[42, 4, 18]$
$[346, 20, 96]$ $[346, 21, 96]$ $[42, 6, 16]$ $[42, 7, 16]$
Table 18.  Parameters of non-trivial binary codes of the orbit matrices of the self-orthogonal symmetric $1$-designs with 8330 points (Theorem 5, item $2$, non-fixed part of orbit matrices)
$[3992,352]$ $[4144,384]$
$[3992,312]$ $[4144,336]$
$[3992,332]$ $[4144,360]$
$[3992, 20, 1776]$ $[4144, 24, 1776]$
$[3992, 40]$ $[4144, 48]$
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$[3992,352]$ $[4144,384]$
$[3992,312]$ $[4144,336]$
$[3992,332]$ $[4144,360]$
$[3992, 20, 1776]$ $[4144, 24, 1776]$
$[3992, 40]$ $[4144, 48]$
Table 19.  Strongly regular graphs constructed from the support designs of the codes of small dimension
SRG Automorphism group Code $w$
$(21, 10, 3, 6)$ $S_7$ $[42, 11]$ $8 $
$[42, 12]$ $8 $
$[42, 12]$ $34 $
$[1190, 12]$ $512 $
$[1190, 13]$ $512 $
$[1190, 13]$ $678 $
$(28, 12, 6, 4)$ $S_8$ $[42, 8]$ $12 $
$[42, 9]$ $12 $
$[42, 9]$ $30 $
$[1008, 24]$ $660 $
$(49, 12, 5, 2)$ $(S_7\times S_7){:}2$ $[294, 10]$ $138 $
$[294, 10]$ $156 $
$[1190, 9]$ $636 $
$[1190, 10]$ $554 $
$[1190, 10]$ $636 $
$[1190, 18]$ $360 $
$[1190, 19]$ $360 $
$[1190, 19]$ $830 $
$(49, 18, 7, 6)$ $((7{:}3\times 7{:}3){:}2){:}2$ $[294, 10]$ $138 $
$[294, 10]$ $156 $
$[294, 18]$ $96 $
$[1190, 9]$ $636$
$[1190, 10]$ $554 $
$[1190, 10]$ $636 $
$[1190, 18]$ $360 $
$[1190, 19]$ $360 $
$[1190, 19]$ $830 $
$(56, 10, 0, 2)$ $(L_3(4){:}2){:}2$ $[154, 20]$ $32 $
$[154, 21]$ $32 $
$[154, 21]$ $122 $
$(63, 30, 13, 15)$ $2^6{:}L_3(2)$ $[1008, 24]$ $336 $
$[4144, 24]$ $2512 $
$(105, 32, 4, 12)$ $((L_3(4){:}3){:}2){:}2$ $[346, 20]$ $96 $
$[346, 21]$ $96 $
$[346, 21]$ $250 $
$(112, 30, 2, 10)$ $(U_4(3){:}4){:}2$ $[952, 20]$ $416$
$[3992, 20]$ $ 2080$
$(120, 42, 8, 18)$ $(L_3(4){:}2){:}3$ $[346, 20]$ $112 $
$[346, 21]$ $112 $
$[346, 21]$ $234 $
$[952, 20]$ $588 $
$[3992, 20]$ $2284 $
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SRG Automorphism group Code $w$
$(21, 10, 3, 6)$ $S_7$ $[42, 11]$ $8 $
$[42, 12]$ $8 $
$[42, 12]$ $34 $
$[1190, 12]$ $512 $
$[1190, 13]$ $512 $
$[1190, 13]$ $678 $
$(28, 12, 6, 4)$ $S_8$ $[42, 8]$ $12 $
$[42, 9]$ $12 $
$[42, 9]$ $30 $
$[1008, 24]$ $660 $
$(49, 12, 5, 2)$ $(S_7\times S_7){:}2$ $[294, 10]$ $138 $
$[294, 10]$ $156 $
$[1190, 9]$ $636 $
$[1190, 10]$ $554 $
$[1190, 10]$ $636 $
$[1190, 18]$ $360 $
$[1190, 19]$ $360 $
$[1190, 19]$ $830 $
$(49, 18, 7, 6)$ $((7{:}3\times 7{:}3){:}2){:}2$ $[294, 10]$ $138 $
$[294, 10]$ $156 $
$[294, 18]$ $96 $
$[1190, 9]$ $636$
$[1190, 10]$ $554 $
$[1190, 10]$ $636 $
$[1190, 18]$ $360 $
$[1190, 19]$ $360 $
$[1190, 19]$ $830 $
$(56, 10, 0, 2)$ $(L_3(4){:}2){:}2$ $[154, 20]$ $32 $
$[154, 21]$ $32 $
$[154, 21]$ $122 $
$(63, 30, 13, 15)$ $2^6{:}L_3(2)$ $[1008, 24]$ $336 $
$[4144, 24]$ $2512 $
$(105, 32, 4, 12)$ $((L_3(4){:}3){:}2){:}2$ $[346, 20]$ $96 $
$[346, 21]$ $96 $
$[346, 21]$ $250 $
$(112, 30, 2, 10)$ $(U_4(3){:}4){:}2$ $[952, 20]$ $416$
$[3992, 20]$ $ 2080$
$(120, 42, 8, 18)$ $(L_3(4){:}2){:}3$ $[346, 20]$ $112 $
$[346, 21]$ $112 $
$[346, 21]$ $234 $
$[952, 20]$ $588 $
$[3992, 20]$ $2284 $
Table 20.  The weight distribution of $C_{562}$ ($\overline{m} = 2058-m$)
$m, \, \overline{m}$ $\;\, A_m$ $m, \, \overline{m}$ $\;\, A_m$ $m, \, \overline{m}$ $\;\, A_m$ $m, \, \overline{m}$ $\;\, A_m$
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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$m, \, \overline{m}$ $\;\, A_m$ $m, \, \overline{m}$ $\;\, A_m$ $m, \, \overline{m}$ $\;\, A_m$ $m, \, \overline{m}$ $\;\, A_m$
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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