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

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).

Mathematics Subject Classification: Primary: 05E18, 94B05; Secondary: 20D08, 05B05, 05E30.

 Citation:

• 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

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

Table 3.  Symmetric $1$-designs on 2058 points

 Orbits Parameters Full Automorphism Group $\Omega_1$, $\Omega_4$ $1$-$(2058,426,426)$ $\rm {He}{:}2$ $\Omega_1$, $\Omega_4$, $\Omega_2$ $1$-$(2058,562,562)$ $\rm He$ $\Omega_1$, $\Omega_4$, $\Omega_2$, $\Omega_3$ $1$-$(2058,698,698)$ $\rm {He}{:}2$ $\Omega_1$, $\Omega_4$, $\Omega_3$ $1$-$(2058,562,562)$ $\rm He$ $\Omega_1$, $\Omega_2$ $1$-$(2058,137,137)$ $\rm He$ $\Omega_1$, $\Omega_2$, $\Omega_3$ $1$-$(2058,273,273)$ $\rm {He}{:}2$ $\Omega_1$, $\Omega_3$ $1$-$(2058,137,137)$ $\rm He$ $\Omega_4$ $1$-$(2058,425,425)$ $\rm {He}{:}2$ $\Omega_4$, $\Omega_2$ $1$-$(2058,561,561)$ $\rm He$ $\Omega_4$, $\Omega_2$, $\Omega_3$ $1$-$(2058,697,697)$ $\rm {He}{:}2$ $\Omega_4$, $\Omega_3$ $1$-$(2058,561,561)$ $\rm He$ $\Omega_2$ $1$-$(2058,136,136)$ $\rm He$ $\Omega_2$, $\Omega_3$ $1$-$(2058,272,272)$ $\rm {He}{:}2$ $\Omega_3$ $1$-$(2058,136,136)$ $\rm He$

Table 4.  Symmetric $1$-designs on 8330 points

 Orbits Parameters Full Automorphism Group $\Delta_1$, $\Delta_2$ $1$-$(8330,106,106)$ $\rm {He}{:}2$ $\Delta_1$, $\Delta_2$, $\Delta_3$ $1$-$(8330, 1450, 1450)$ $\rm {He}{:}2$ $\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_4$ $1$-$(8330, 2290, 2290)$ $\rm {He}$ $\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 3010, 3010)$ $\rm {He}$ $\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3850, 3850)$ $\rm {He}{:}2$ $\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 3130, 3130)$ $\rm {He}{:}2$ $\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_5$ $1$-$(8330, 2170, 2170)$ $\rm {He}{:}2$ $\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3010, 3010)$ $\rm {He}$ $\Delta_1$, $\Delta_2$, $\Delta_3$, $\Delta_6$ $1$-$(8330, 2290, 2290)$ $\rm {He}$ $\Delta_1$, $\Delta_2$, $\Delta_4$ $1$-$(8330,946,946)$ $\rm {He}$ $\Delta_1$, $\Delta_2$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 1666, 1666)$ $\rm {He}$ $\Delta_1$, $\Delta_2$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2506, 2506)$ $\rm {He}{:}2$ $\Delta_1$, $\Delta_2$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 1786, 1786)$ $\rm {He}{:}2$ $\Delta_1$, $\Delta_2$, $\Delta_5$ $1$-$(8330,826,826)$ $\rm {He}{:}2$ $\Delta_1$, $\Delta_2$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 1666, 1666)$ $\rm {He}$ $\Delta_1$, $\Delta_2$, $\Delta_6$ $1$-$(8330,946,946)$ $\rm {He}$ $\Delta_1$, $\Delta_3$ $1$-$(8330, 1345, 1345)$ $\rm {He}{:}2$ $\Delta_1$, $\Delta_3$, $\Delta_4$ $1$-$(8330, 2185, 2185)$ $\rm {He}$ $\Delta_1$, $\Delta_3$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 2905, 2905)$ $\rm {He}$ $\Delta_1$, $\Delta_3$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3745, 3745)$ $\rm {He}{:}2$ $\Delta_1$, $\Delta_3$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 3025, 3025)$ $\rm {He}{:}2$ $\Delta_1$, $\Delta_3$, $\Delta_5$ $1$-$(8330, 2065, 2065)$ $\rm {He}{:}2$ $\Delta_1$, $\Delta_3$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2905, 2905)$ $\rm {He}$ $\Delta_1$, $\Delta_3$, $\Delta_6$ $1$-$(8330, 2185, 2185)$ $\rm {He}$ $\Delta_1$, $\Delta_4$ $1$-$(8330,841,841)$ $\rm {He}$ $\Delta_1$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 1561, 1561)$ $\rm {He}$ $\Delta_1$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2401, 2401)$ $\rm {He}{:}2$ $\Delta_1$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 1681, 1681)$ $\rm {He}{:}2$ $\Delta_1$, $\Delta_5$ $1$-$(8330,721,721)$ $\rm {He}{:}2$ $\Delta_1$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 1561, 1561)$ $\rm {He}$ $\Delta_1$, $\Delta_6$ $1$-$(8330,841,841)$ $\rm {He}$ $\Delta_2$ $1$-$(8330,105,105)$ $\rm {He}{:}2$ $\Delta_2$, $\Delta_3$ $1$-$(8330, 1449, 1449)$ $\rm {He}{:}2$ $\Delta_2$, $\Delta_3$, $\Delta_4$ $1$-$(8330, 2289, 2289)$ $\rm {He}$ $\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 3009, 3009)$ $\rm {He}$ $\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3849, 3849)$ $\rm {He}{:}2$ $\Delta_2$, $\Delta_3$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 3129, 3129)$ $\rm {He}{:}2$ $\Delta_2$, $\Delta_3$, $\Delta_5$ $1$-$(8330, 2169, 2169)$ $\rm {He}{:}2$ $\Delta_2$, $\Delta_3$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3009, 3009)$ $\rm {He}$ $\Delta_2$, $\Delta_3$, $\Delta_6$ $1$-$(8330, 2289, 2289)$ $\rm {He}$ $\Delta_2$, $\Delta_4$ $1$-$(8330,945,945)$ $\rm {He}$ $\Delta_2$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 1665, 1665)$ $\rm {He}$ $\Delta_2$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2505, 2505)$ $\rm {He}{:}2$ $\Delta_2$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 1785, 1785)$ $\rm {He}{:}2$ $\Delta_2$, $\Delta_5$ $1$-$(8330,825,825)$ $\rm {He}{:}2$ $\Delta_2$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 1665, 1665)$ $\rm {He}$ $\Delta_2$, $\Delta_6$ $1$-$(8330,945,945)$ $\rm {He}$ $\Delta_3$ $1$-$(8330, 1344, 1344)$ $\rm {He}{:}2$ $\Delta_3$, $\Delta_4$ $1$-$(8330, 2184, 2184)$ $\rm {He}$ $\Delta_3$, $\Delta_4$, $\Delta_5$ $1$-$(8330, 2904, 2904)$ $\rm {He}$ $\Delta_3$, $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 3744, 3744)$ $\rm {He}{:}2$ $\Delta_3$, $\Delta_4$, $\Delta_6$ $1$-$(8330, 3024, 3024)$ $\rm {He}{:}2$ $\Delta_3$, $\Delta_5$ $1$-$(8330, 2064, 2064)$ $\rm {He}{:}2$ $\Delta_3$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2904, 2904)$ $\rm {He}$ $\Delta_3$, $\Delta_6$ $1$-$(8330, 2184, 2184)$ $\rm {He}$ $\Delta_4$ $1$-$(8330,840,840)$ $\rm {He}$ $\Delta_4$, $\Delta_5$ $1$-$(8330, 1560, 1560)$ $\rm {He}$ $\Delta_4$, $\Delta_5$, $\Delta_6$ $1$-$(8330, 2400, 2400)$ $\rm {He}{:}2$ $\Delta_4$, $\Delta_6$ $1$-$(8330, 1680, 1680)$ $\rm {He}{:}2$ $\Delta_5$ $1$-$(8330,720,720)$ $\rm {He}{:}2$ $\Delta_5$, $\Delta_6$ $1$-$(8330, 1560, 1560)$ $\rm {He}$ $\Delta_6$ $1$-$(8330,840,840)$ $\rm {He}$

Table 5.  $1$-designs on 2058 points and 8330 blocks

 Design Orbits Parameters Full Automorphism Group $\mathcal D_{1}''$ $\Delta_5$ $1$-$(2058,840, 3400)$ $\rm He$ $\mathcal D_{2}''$ $\Delta_5$, $\Delta_2$ $1$-$(2058,861, 3485)$ $\rm He$ $\mathcal D_{3}''$ $\Delta_5$, $\Delta_2$, $\Delta_1$ $1$-$(2058,882, 3570)$ $\rm He$ $\mathcal D_{4}''$ $\Delta_5$, $\Delta_1$ $1$-$(2058,861, 3485)$ $\rm He$ $\mathcal D_{5}''$ $\Delta_4$ $1$-$(2058,840, 3400)$ $\rm He$ $\mathcal D_{6}''$ $\Delta_4$, $\Delta_2$ $1$-$(2058,861, 3485)$ $\rm He$ $\mathcal D_{7}''$ $\Delta_4$, $\Delta_2$, $\Delta_1$ $1$-$(2058,882, 3570)$ $\rm He$ $\mathcal D_{8}''$ $\Delta_4$, $\Delta_1$ $1$-$(2058,861, 3485)$ $\rm He$ $\mathcal D_{9}''$ $\Delta_3$ $1$-$(2058,336, 1360)$ $\rm {He}{:}2$ $\mathcal D_{10}''$ $\Delta_3$, $\Delta_2$ $1$-$(2058,357, 1445)$ $\rm He$ $\mathcal D_{11}''$ $\Delta_3$, $\Delta_2$, $\Delta_1$ $1$-$(2058,378, 1530)$ $\rm {He}{:}2$ $\mathcal D_{12}''$ $\Delta_3$, $\Delta_1$ $1$-$(2058,357, 1445)$ $\rm He$ $\mathcal D_{13}''$ $\Delta_2$ $1$-$(2058, 21, 85)$ $\rm He$ $\mathcal D_{14}''$ $\Delta_2$, $\Delta_1$ $1$-$(2058, 42,170)$ $\rm {He}{:}2$ $\mathcal D_{15}''$ $\Delta_1$ $1$-$(2058, 21, 85)$ $\rm He$

Table 6.  Non-trivial binary codes of the pairwise non-isomorphic symmetric $1$-designs on 2058 points

 $k$ $C_{k}$ ${\rm Aut}(C_k)$ $\bar{C}_{k}$ ${\rm Aut}(\bar{C}_{k})$ $E_k$ $426$ $[2058,783]$ ${\rm He}\textrm{:}2$ $[2058,782]$ ${\rm He}\textrm{:}2$ $[2058,782]$ $562$ $[2058, 52]$ ${\rm He}$ $[2058, 51]$ ${\rm He}$ $[2058, 51]$ $698$ $[2058,681]$ ${\rm He}\textrm{:}2$ $[2058,680]$ ${\rm He}\textrm{:}2$ $[2058,680]$ $136$ $[2058,731]$ ${\rm He}$ $[2058,732]$ ${\rm He}$ $[2058,731]$ $272$ $[2058,102]$ ${\rm He}\textrm{:}2$ $[2058,103]$ ${\rm He}\textrm{:}2$ $[2058,102]$

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

Table 8.  Non-trivial binary codes of the pairwise non-isomorphic symmetric $1$-designs on 8330 points

 $k$ $C_{k}'$ $\bar{C}_{k}'$ $E_k'$ $106$ $[8330, 7055]$ $[8330, 7054]$ $[8330, 7054]$ $1450$ $[8330,783]$ $[8330,782]$ $[8330,782]$ $2290$ $[8330, 1972]$* $[8330, 1971]$* $[8330, 1971]$ $3010$ $[8330, 7004]$* $[8330, 7003]$* $[8330, 7003]$ $3850$ $[8330, 4353]$ $[8330, 4352]$ $[8330, 4352]$ $3130$ $[8330,681]$ $[8330,680]$ $[8330,680]$ $2170$ $[8330, 4455]$ $[8330, 4454]$ $[8330, 4454]$ $946$ $[8330, 4404]$* $[8330, 4403]$* $[8330, 4403]$ $1666$ $[8330,732]$* $[8330,731]$* $[8330,731]$ $2506$ $[8330, 1921]$ $[8330, 1920]$ $[8330, 1920]$ $1786$ $[8330, 6953]$ $[8330, 6952]$ $[8330, 6952]$ $826$ $[8330, 2023]$ $[8330, 2022]$ $[8330, 2022]$ $1345$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$ $2185$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$ $3745$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$ $3025$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$ $2065$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$ $841$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$ $1561$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$ $2401$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$ $721$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$ $105$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$ $2289$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$ $3009$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$ $3849$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$ $2169$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$ $945$ $[8330, 3978]$ $[8330, 3978]$ $[8330, 3977]$ $2505$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$ $1785$ $[8330, 2058]$ $[8330, 2058]$ $[8330, 2057]$ $825$ $[8330, 6410]$ $[8330, 6410]$ $[8330, 6409]$ $1344$ $[8330, 6272]$ $[8330, 6273]$ $[8330, 6272]$ $2184$ $[8330, 5083]$* $[8330, 5084]$* $[8330, 5083]$ $2904$ $[8330, 51]$* $[8330, 52]$* $[8330, 51]$ $3744$ $[8330, 2702]$ $[8330, 2703]$ $[8330, 2702]$ $3024$ $[8330, 6374]$ $[8330, 6375]$ $[8330, 6374]$ $2064$ $[8330, 2600]$ $[8330, 2601]$ $[8330, 2600]$ $840$ $[8330, 2651]$* $[8330, 2652]$* $[8330, 2651]$ $1560$ $[8330, 6323]$* $[8330, 6324]$* $[8330, 6323]$ $2400$ $[8330, 5134]$ $[8330, 5135]$ $[8330, 5134]$ $1680$ $[8330,102]$ $[8330,103]$ $[8330,102]$ $720$ $[8330, 5032]$ $[8330, 5033]$ $[8330, 5032]$

Table 9.  Non-trivial binary codes of the pairwise non-isomorphic $1$-designs on 2058 points and 8330 blocks

 $k$ $C_{k}''$ ${\rm Aut}(C_k'')$ $\bar{C}_{k}''$ ${\rm Aut}(\bar{C}_{k}'')$ $E_{k}''$ $840$ $[2058,731]$ ${\rm He}$ $[2058,732]$ ${\rm He}$ $[2058,731]$ $882$ $[2058, 52]$ ${\rm He}$ $[2058, 51]$ ${\rm He}$ $[2058, 51]$ $336$ $[2058,680]$ ${\rm He}\textrm{:}2$ $[2058,681]$ ${\rm He}\textrm{:}2$ $[2058,680]$ $378$ $[2058,103]$ ${\rm He}\textrm{:}2$ $[2058,102]$ ${\rm He}\textrm{:}2$ $[2058,102]$ $42$ $[2058,783]$ ${\rm He}\textrm{:}2$ $[2058,782]$ ${\rm He}:2$ $[2058,782]$

Table 10.  Non-trivial binary codes of the pairwise non-isomorphic $1$-designs on 8330 points and 2058 blocks

 $k$ ${C_k''}^t$ ${\rm Aut}({C_k''}^t)$ ${\bar{C}_k}^{''t}$ ${\rm Aut}({\bar{C}_k}^{''t})$ $840$ $[8330,731]$ ${\rm He}$ $[8330,732]$ ${\rm He}$ $882$ $[8330, 52]$ ${\rm He}$ $[8330, 51]$ ${\rm He}$ $336$ $[8330,680]$ ${\rm He}\textrm{:}2$ $[8330,681]$ ${\rm He}\textrm{:}2$ $378$ $[8330,103]$ ${\rm He}\textrm{:}2$ $[8330,102]$ ${\rm He}\textrm{:}2$ $42$ $[8330,783]$ ${\rm He}\textrm{:}2$ $[8330,782]$ ${\rm He}\textrm{:}2$

Table 11.  Non-trivial binary pairwise non-equivalent codes of the orbit matrices of the symmetric self-orthogonal $1$-designs on 2058 points

 $[294,111]$ $[294,110]$ $[686,261]$ $[686,260]$ $[294, 10,126]$ $[294, 6,120]$ $[686, 18,224]$ $[686, 17,224]$ $[294, 93]$ $[294, 98]$ $[686,227]$ $[686,226]$ $[294,101]$ $[294,105]$ $[686,243]$ $[686,244]$ $[294, 18, 96]$ $[294, 13,112]$ $[686, 34,192]$ $[686, 35,126]$

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

 $[294,104]$ $[294,105]$ $[686,243]$ $[686,244]$ $[294, 7,120]$ $[294, 6,120]$ $[686, 18,224]$ $[686, 17,224]$ $[294, 98]$ $[294, 99]$ $[686,226]$ $[686,227]$ $[294, 13,112]$ $[294, 12,112]$ $[686, 35,126]$ $[686, 34,192]$ $[294,111]$ $[294,110]$ $[686,261]$ $[686,260]$

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

 $[490, 43]$ $[490, 44]$ $[1190,101]$ $[1190,102]$ $[490, 4,210]$ $[490, 3,240]$ $[1190, 10,554]$ $[1190, 9,564]$ $[490, 40]$ $[490, 41]$ $[1190, 92]$ $[1190, 93]$ $[490, 7, 90]$ $[490, 6,224]$ $[1190, 19,360]$ $[1190, 18,360]$ $[490, 47]$ $[490, 46]$ $[1190,111]$ $[1190,110]$

Table 14.  Non-trivial binary pairwise non-equivalent codes of the orbit matrices of the self-orthogonal symmetric $1$-designs on 8330 points

 $[490, 47]$ $[490, 46]$ $[1190,111]$ $[1190,110]$ $[490, 41]$ $[490, 40]$ $[1190, 99]$ $[1190, 98]$ $[490, 44]$ $[490, 43]$ $[1190,105]$ $[1190,104]$ $[490, 3,240]$ $[490, 4,210]$ $[1190, 6,568]$ $[1190, 7,568]$ $[490, 6,224]$ $[490, 7, 90]$ $[1190, 12,512]$ $[1190, 13,512]$

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)

 $[154, 75, 8]$ $[154, 74, 8]$ $[42, 15, 6]$ $[42, 14, 8]$ $[154, 12, 42]$ $[154, 11, 48]$ $[42, 4, 18]$ $[42, 3, 24]$ $[154, 57, 12]$ $[154, 56, 12]$ $[42, 9, 12]$ $[42, 8, 12]$ $[154, 65, 8]$ $[154, 66, 8]$ $[42, 11, 8]$ $[42, 12, 8]$ $[154, 20, 32]$ $[154, 21, 32]$ $[42, 6, 16]$ $[42, 7, 16]$

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)

 $[952,352]$ $[1008,384]$ $[952, 20,368]$ $[1008, 24,336]$ $[952,312]$ $[1008,336]$ $[952,332]$ $[1008,360]$ $[952, 40,224]$ $[1008, 48]$

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)

 $[346, 75, 26]$ $[346, 74, 32]$ $[42, 15, 6]$ $[42, 14, 8]$ $[346, 57, 32]$ $[346, 56, 32]$ $[42, 9, 12]$ $[42, 8, 12]$ $[346, 66, 26]$ $[346, 65, 32]$ $[42, 12, 8]$ $[42, 11, 8]$ $[346, 11,152]$ $[346, 12,106]$ $[42, 3, 24]$ $[42, 4, 18]$ $[346, 20, 96]$ $[346, 21, 96]$ $[42, 6, 16]$ $[42, 7, 16]$

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)

 $[3992,352]$ $[4144,384]$ $[3992,312]$ $[4144,336]$ $[3992,332]$ $[4144,360]$ $[3992, 20, 1776]$ $[4144, 24, 1776]$ $[3992, 40]$ $[4144, 48]$

Table 19.  Strongly regular graphs constructed from the support designs of the codes of small dimension

 SRG Automorphism group Code $w$ $(21, 10, 3, 6)$ $S_7$ $[42, 11]$ $8$ $[42, 12]$ $8$ $[42, 12]$ $34$ $[1190, 12]$ $512$ $[1190, 13]$ $512$ $[1190, 13]$ $678$ $(28, 12, 6, 4)$ $S_8$ $[42, 8]$ $12$ $[42, 9]$ $12$ $[42, 9]$ $30$ $[1008, 24]$ $660$ $(49, 12, 5, 2)$ $(S_7\times S_7){:}2$ $[294, 10]$ $138$ $[294, 10]$ $156$ $[1190, 9]$ $636$ $[1190, 10]$ $554$ $[1190, 10]$ $636$ $[1190, 18]$ $360$ $[1190, 19]$ $360$ $[1190, 19]$ $830$ $(49, 18, 7, 6)$ $((7{:}3\times 7{:}3){:}2){:}2$ $[294, 10]$ $138$ $[294, 10]$ $156$ $[294, 18]$ $96$ $[1190, 9]$ $636$ $[1190, 10]$ $554$ $[1190, 10]$ $636$ $[1190, 18]$ $360$ $[1190, 19]$ $360$ $[1190, 19]$ $830$ $(56, 10, 0, 2)$ $(L_3(4){:}2){:}2$ $[154, 20]$ $32$ $[154, 21]$ $32$ $[154, 21]$ $122$ $(63, 30, 13, 15)$ $2^6{:}L_3(2)$ $[1008, 24]$ $336$ $[4144, 24]$ $2512$ $(105, 32, 4, 12)$ $((L_3(4){:}3){:}2){:}2$ $[346, 20]$ $96$ $[346, 21]$ $96$ $[346, 21]$ $250$ $(112, 30, 2, 10)$ $(U_4(3){:}4){:}2$ $[952, 20]$ $416$ $[3992, 20]$ $2080$ $(120, 42, 8, 18)$ $(L_3(4){:}2){:}3$ $[346, 20]$ $112$ $[346, 21]$ $112$ $[346, 21]$ $234$ $[952, 20]$ $588$ $[3992, 20]$ $2284$

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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