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August  2017, 11(3): 635-645. doi: 10.3934/amc.2017047

Self-dual codes with an automorphism of order 13

Nikolay Yankov 1, , Damyan Anev 1, and  Müberra Gürel 2,

1. 

Faculty of Mathematics and Informatics, Shumen University, 9700 Shumen, Bulgaria
2. 

Istanbul Aydin University, Istanbul, Turkey

Received  February 2016 Revised  January 2017 Published  August 2017

Fund Project: This paper was supported by Shumen University under Grant RD-08-107/06.02.2017

Using a method for constructing binary self-dual codes having an automorphism of odd prime order $p$ we classify, up to equivalence, all singly-even self-dual $[78,39,14]$, $[80,40,14]$, $[82,41,14],$ and $[84,42,14]$ codes as well as all doubly-even $[80,40,16]$ codes for $p=13$. The results show that there are exactly 1592 inequivalent binary self-dual $[78,39,14]$ codes with an automorphism of type $13-(6,0)$ and we found 6 new values of the parameter in the weight function thus increasing more than twice the number of known values. As for binary $[80,40]$ self-dual codes with an automorphism of type $13-(6,2)$ there are 162696 singly-even self-dual codes with minimum distance 14 and 195 doubly-even such codes with minimum distance 16. We also construct many new codes of lengths 82 and 84 with minimum distance 14. Most of the constructed codes for all lengths have weight enumerators for which the existence was not known before.

Keywords: Automorphism, code, self-dual, classification.
Mathematics Subject Classification: Primary: 11T71; Secondary: 94B05.
Citation: Nikolay Yankov, Damyan Anev, Müberra Gürel. Self-dual codes with an automorphism of order 13. Advances in Mathematics of Communications, 2017, 11 (3) : 635-645. doi: 10.3934/amc.2017047
References:
[1]

A. Baartmans and V. Yorgov, Some new extremal codes of lengths 76 and 78, IEEE Trans. Inf. Theory, 49 (2003), 1353-1354. doi: 10.1109/tit.2003.810653.

[2]

I. Bouyukliev, About the code equivalence, in Advances in Coding Theory and Cryptography, World Scient. Publ. Comp. , 2007,126–151. doi: 10.1142/9789812772022_0009.

[3]

R. Dontcheva and M. Harada, Extremal doubly-even [80, 40, 16] codes with an automorphism of order 19, Finite Fields Appl., 9 (2003), 157-167. doi: 10.1016/s1071-5797(02)00018-7.

[4]

G. Dorfer and H. Maharaj, Generalized AG codes and generalized duality, Finite Fields Appl., 9 (2003), 194-210. doi: 10.1016/s1071-5797(02)00027-8.

[5]

S. T. Dougherty, T. A. Gulliver and M. Harada, Extremal binary self-dual codes, IEEE Trans. Inf. Theory, 43 (1997), 2036-2047. doi: 10.1109/18.641574.

[6]

P. Gaborit and A. Otmani, Experimental constructions of self-dual codes, Finite Fields Appl., 9 (2003), 372-394. doi: 10.1016/s1071-5797(03)00011-x.

[7]

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4. 8. 6,2016, http://www.gap-system.org
[8]

T. A. Gulliver and M. Harada, Classification of extremal double circulant self-dual codes of lengths 74–88, Discr. Math., 306 (2006), 2064-2072. doi: 10.1016/j.disc.2006.05.004.

[9]

T. A. Gulliver, M. Harada and J.-L. Kim, Construction of new extremal self-dual codes, Discr. Math., 263 (2003), 81-91. doi: 10.1016/s0012-365x(02)00570-8.

[10]

M. Harada and A. Munemasa, On $s$-extremal singly even self-dual $[24k+8, 12k+4, 4k+2]$ codes preprint, arXiv: 1511.02972
[11]

M. Harada and A. Munemasa, Some restrictions on weight enumerators of singly even self-dual codes, IEEE Trans. Inf. Theory, 52 (2006), 1266-1269. doi: 10.1109/tit.2005.864416.

[12]

W. C. Huffman, Automorphisms of codes with applications to extremal doubly even codes of length 48, IEEE Trans. Inf. Theory, 28 (1982), 511-521. doi: 10.1109/tit.1982.1056499.

[13]

W. C. Huffman and V. S. Pless, Fundamentals of Error-Correcting Codes Cambridge Univ. Press, 2003. doi: 10.1017/CBO9780511807077.

[14]

S. Kapralov, R. Russeva and V. Radeva, New extremal doubly even [80, 40, 16] codes with an automorphism of order 13, in Proc. 8th Int. Workshop Algebr. Combin. Coding Theory, 2002,139–142.
[15]

E. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inf. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.

[16]

N. Yankov, Self-dual [62, 31, 12] and [64, 32, 12] codes with an automorphism of order 7, Adv. Math. Commun., 8 (2014), 73-81. doi: 10.3934/amc.2014.8.73.

[17]

N. Yankov, D. Anev and M. Gürel, Constructing the self-dual codes with an automorphism of order 13 with 6 cycles available at http://shu.bg/tadmin/upload/storage/2497.pdf
[18]

N. Yankov and M. H. Lee, Classification of self-dual codes of length 50 with an automorphism of odd prime order, Des. Codes Crypt., 74 (2015), 571-579. doi: 10.1007/s10623-013-9874-8.

[19]

N. Yankov and R. Russeva, Binary self-dual codes of lengths 52 to 60 with an automorphism of order 7 or 13, IEEE Trans. Inf. Theory, 57 (2011), 7498-7506. doi: 10.1109/TIT.2011.2155619.

[20]

V. Yorgov, Binary self-dual codes with automorphisms of odd order, Probl. Inf. Transm., 19 (1983), 260-270.

[21]

V. Yorgov, A method for constructing inequivalent self-dual codes with applications to length 56, IEEE Trans. Inf. Theory, 33 (1987), 77-82. doi: 10.1109/TIT.1987.1057273.

[22]

T. Zhang, J. Michel, T. Feng and G. Ge, On the existence of certain optimal self-dual codes with lengths between 74 and 116 Electr. J. Combin. 22 (2015), P4. 33.

show all references

References:
[1]

A. Baartmans and V. Yorgov, Some new extremal codes of lengths 76 and 78, IEEE Trans. Inf. Theory, 49 (2003), 1353-1354. doi: 10.1109/tit.2003.810653.

[2]

I. Bouyukliev, About the code equivalence, in Advances in Coding Theory and Cryptography, World Scient. Publ. Comp. , 2007,126–151. doi: 10.1142/9789812772022_0009.

[3]

R. Dontcheva and M. Harada, Extremal doubly-even [80, 40, 16] codes with an automorphism of order 19, Finite Fields Appl., 9 (2003), 157-167. doi: 10.1016/s1071-5797(02)00018-7.

[4]

G. Dorfer and H. Maharaj, Generalized AG codes and generalized duality, Finite Fields Appl., 9 (2003), 194-210. doi: 10.1016/s1071-5797(02)00027-8.

[5]

S. T. Dougherty, T. A. Gulliver and M. Harada, Extremal binary self-dual codes, IEEE Trans. Inf. Theory, 43 (1997), 2036-2047. doi: 10.1109/18.641574.

[6]

P. Gaborit and A. Otmani, Experimental constructions of self-dual codes, Finite Fields Appl., 9 (2003), 372-394. doi: 10.1016/s1071-5797(03)00011-x.

[7]

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4. 8. 6,2016, http://www.gap-system.org
[8]

T. A. Gulliver and M. Harada, Classification of extremal double circulant self-dual codes of lengths 74–88, Discr. Math., 306 (2006), 2064-2072. doi: 10.1016/j.disc.2006.05.004.

[9]

T. A. Gulliver, M. Harada and J.-L. Kim, Construction of new extremal self-dual codes, Discr. Math., 263 (2003), 81-91. doi: 10.1016/s0012-365x(02)00570-8.

[10]

M. Harada and A. Munemasa, On $s$-extremal singly even self-dual $[24k+8, 12k+4, 4k+2]$ codes preprint, arXiv: 1511.02972
[11]

M. Harada and A. Munemasa, Some restrictions on weight enumerators of singly even self-dual codes, IEEE Trans. Inf. Theory, 52 (2006), 1266-1269. doi: 10.1109/tit.2005.864416.

[12]

W. C. Huffman, Automorphisms of codes with applications to extremal doubly even codes of length 48, IEEE Trans. Inf. Theory, 28 (1982), 511-521. doi: 10.1109/tit.1982.1056499.

[13]

W. C. Huffman and V. S. Pless, Fundamentals of Error-Correcting Codes Cambridge Univ. Press, 2003. doi: 10.1017/CBO9780511807077.

[14]

S. Kapralov, R. Russeva and V. Radeva, New extremal doubly even [80, 40, 16] codes with an automorphism of order 13, in Proc. 8th Int. Workshop Algebr. Combin. Coding Theory, 2002,139–142.
[15]

E. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inf. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.

[16]

N. Yankov, Self-dual [62, 31, 12] and [64, 32, 12] codes with an automorphism of order 7, Adv. Math. Commun., 8 (2014), 73-81. doi: 10.3934/amc.2014.8.73.

[17]

N. Yankov, D. Anev and M. Gürel, Constructing the self-dual codes with an automorphism of order 13 with 6 cycles available at http://shu.bg/tadmin/upload/storage/2497.pdf
[18]

N. Yankov and M. H. Lee, Classification of self-dual codes of length 50 with an automorphism of odd prime order, Des. Codes Crypt., 74 (2015), 571-579. doi: 10.1007/s10623-013-9874-8.

[19]

N. Yankov and R. Russeva, Binary self-dual codes of lengths 52 to 60 with an automorphism of order 7 or 13, IEEE Trans. Inf. Theory, 57 (2011), 7498-7506. doi: 10.1109/TIT.2011.2155619.

[20]

V. Yorgov, Binary self-dual codes with automorphisms of odd order, Probl. Inf. Transm., 19 (1983), 260-270.

[21]

V. Yorgov, A method for constructing inequivalent self-dual codes with applications to length 56, IEEE Trans. Inf. Theory, 33 (1987), 77-82. doi: 10.1109/TIT.1987.1057273.

[22]

T. Zhang, J. Michel, T. Feng and G. Ge, On the existence of certain optimal self-dual codes with lengths between 74 and 116 Electr. J. Combin. 22 (2015), P4. 33.

Table 1.  The number of codes with different $A_{16}$
$A_{16}$ # $A_{16}$ # $A_{16}$ # $A_{16}$ # $A_{16}$ # $A_{16}$ # $A_{16}$ # $A_{16}$ #
14586 1 15002 359 15379 4280 15756 4993 16133 1458 16510 189 16887 16 17290 3
14612 1 15015 417 15392 4282 15769 4803 16146 1381 16523 144 16900 11 17303 4
14625 2 15028 528 15405 4380 15782 4902 16159 1198 16536 140 16913 11 17316 2
14651 1 15041 539 15418 4551 15795 4669 16172 1089 16549 144 16926 14 17342 1
14677 4 15054 608 15431 4766 15808 4414 16185 1111 16562 136 16939 11 17368 2
14690 1 15067 707 15444 4805 15821 4425 16198 1077 16575 113 16952 6 17394 2
14703 2 15080 735 15457 4842 15834 4215 16211 1007 16588 101 16965 12 17407 1
14716 3 15093 870 15470 5185 15847 4008 16224 856 16601 90 16978 7 17420 1
14729 9 15106 957 15483 5164 15860 3988 16237 876 16614 88 16991 7 17433 1
14742 4 15119 1031 15496 5366 15873 3865 16250 767 16627 70 17004 4 17446 3
14755 6 15132 1176 15509 5393 15886 3854 16263 762 16640 83 17017 3 17472 2
14768 8 15145 1277 15522 5480 15899 3611 16276 706 16653 59 17030 7 17485 2
14781 11 15158 1328 15535 5401 15912 3428 16289 662 16666 54 17043 8 17498 1
14794 18 15171 1557 15548 5568 15925 3369 16302 579 16679 59 17056 1 17511 1
14807 20 15184 1724 15561 5671 15938 3235 16315 541 16692 73 17069 4 17524 1
14820 37 15197 1809 15574 5617 15951 3027 16328 572 16705 50 17082 10 17537 1
14833 42 15210 1934 15587 5752 15964 2914 16341 508 16718 49 17095 3 17550 1
14846 30 15223 2092 15600 5577 15977 2751 16354 394 16731 41 17108 6 17563 2
14859 50 15236 2299 15613 5632 15990 2533 16367 438 16744 41 17121 4 17654 1
14872 53 15249 2439 15626 5669 16003 2531 16380 384 16757 38 17134 5 17693 2
14885 86 15262 2597 15639 5764 16016 2362 16393 312 16770 35 17147 2 17823 1
14898 99 15275 2848 15652 5552 16029 2328 16406 334 16783 33 17173 5 17888 1
14911 141 15288 2972 15665 5459 16042 2136 16419 279 16796 24 17186 3 18070 1
14924 152 15301 3066 15678 5353 16055 2075 16432 285 16809 19 17199 2 18083 1
14937 182 15314 3219 15691 5379 16068 1959 16445 243 16822 17 17212 2 18473 1
14950 199 15327 3487 15704 5482 16081 1812 16458 257 16835 23 17225 2 18655 1
14963 250 15340 3688 15717 5213 16094 1711 16471 241 16848 16 17251 3 18967 1
14976 271 15353 3875 15730 5359 16107 1597 16484 203 16861 13 17264 1 19071 1
14989 304 15366 4068 15743 5048 16120 1505 16497 208 16874 15 17277 3
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$A_{16}$ # $A_{16}$ # $A_{16}$ # $A_{16}$ # $A_{16}$ # $A_{16}$ # $A_{16}$ # $A_{16}$ #
14586 1 15002 359 15379 4280 15756 4993 16133 1458 16510 189 16887 16 17290 3
14612 1 15015 417 15392 4282 15769 4803 16146 1381 16523 144 16900 11 17303 4
14625 2 15028 528 15405 4380 15782 4902 16159 1198 16536 140 16913 11 17316 2
14651 1 15041 539 15418 4551 15795 4669 16172 1089 16549 144 16926 14 17342 1
14677 4 15054 608 15431 4766 15808 4414 16185 1111 16562 136 16939 11 17368 2
14690 1 15067 707 15444 4805 15821 4425 16198 1077 16575 113 16952 6 17394 2
14703 2 15080 735 15457 4842 15834 4215 16211 1007 16588 101 16965 12 17407 1
14716 3 15093 870 15470 5185 15847 4008 16224 856 16601 90 16978 7 17420 1
14729 9 15106 957 15483 5164 15860 3988 16237 876 16614 88 16991 7 17433 1
14742 4 15119 1031 15496 5366 15873 3865 16250 767 16627 70 17004 4 17446 3
14755 6 15132 1176 15509 5393 15886 3854 16263 762 16640 83 17017 3 17472 2
14768 8 15145 1277 15522 5480 15899 3611 16276 706 16653 59 17030 7 17485 2
14781 11 15158 1328 15535 5401 15912 3428 16289 662 16666 54 17043 8 17498 1
14794 18 15171 1557 15548 5568 15925 3369 16302 579 16679 59 17056 1 17511 1
14807 20 15184 1724 15561 5671 15938 3235 16315 541 16692 73 17069 4 17524 1
14820 37 15197 1809 15574 5617 15951 3027 16328 572 16705 50 17082 10 17537 1
14833 42 15210 1934 15587 5752 15964 2914 16341 508 16718 49 17095 3 17550 1
14846 30 15223 2092 15600 5577 15977 2751 16354 394 16731 41 17108 6 17563 2
14859 50 15236 2299 15613 5632 15990 2533 16367 438 16744 41 17121 4 17654 1
14872 53 15249 2439 15626 5669 16003 2531 16380 384 16757 38 17134 5 17693 2
14885 86 15262 2597 15639 5764 16016 2362 16393 312 16770 35 17147 2 17823 1
14898 99 15275 2848 15652 5552 16029 2328 16406 334 16783 33 17173 5 17888 1
14911 141 15288 2972 15665 5459 16042 2136 16419 279 16796 24 17186 3 18070 1
14924 152 15301 3066 15678 5353 16055 2075 16432 285 16809 19 17199 2 18083 1
14937 182 15314 3219 15691 5379 16068 1959 16445 243 16822 17 17212 2 18473 1
14950 199 15327 3487 15704 5482 16081 1812 16458 257 16835 23 17225 2 18655 1
14963 250 15340 3688 15717 5213 16094 1711 16471 241 16848 16 17251 3 18967 1
14976 271 15353 3875 15730 5359 16107 1597 16484 203 16861 13 17264 1 19071 1
14989 304 15366 4068 15743 5048 16120 1505 16497 208 16874 15 17277 3
Table 2.  The order of the automorphism groups of all $E_\sigma(C)^\ast$
$|\text{Aut}(C)|$ 13 26 39 52 78 156 234 468
# 317529 4314 42 167 41 8 1 1
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$|\text{Aut}(C)|$ 13 26 39 52 78 156 234 468
# 317529 4314 42 167 41 8 1 1
Table 3.  $\beta$ in $W_{78, 1}$ and $|\text{Aut}(C)|$ for $[78, 34, 14]$ self-dual codes with an automorphism of type $13-(6, 0)$
$|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\beta$ 13 26 39 78 $\beta$ 13 26 39 78
$\boldsymbol{-117}$ $\textbf{1}$ $\boldsymbol{-39}$ $\textbf{302}$
$\boldsymbol{-104}$ $\textbf{1}$ $-26$ 437 30
$-78$ 5 7 1 3 $\boldsymbol{-13}$ $\textbf{421}$
$\boldsymbol{-65}$ $\textbf{37}$ 0 171 18 5
$\boldsymbol{-52}$ $\textbf{137}$ $\textbf{14}$ $\textbf{2}$
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$|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\beta$ 13 26 39 78 $\beta$ 13 26 39 78
$\boldsymbol{-117}$ $\textbf{1}$ $\boldsymbol{-39}$ $\textbf{302}$
$\boldsymbol{-104}$ $\textbf{1}$ $-26$ 437 30
$-78$ 5 7 1 3 $\boldsymbol{-13}$ $\textbf{421}$
$\boldsymbol{-65}$ $\textbf{37}$ 0 171 18 5
$\boldsymbol{-52}$ $\textbf{137}$ $\textbf{14}$ $\textbf{2}$
Table 4.  The cardinality of the automorphism group for the doubly-even $[80, 40, 16]$ codes
$|\text{Aut}(C)|$ 13 26 78 246480
# 172 18 4 1
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$|\text{Aut}(C)|$ 13 26 78 246480
# 172 18 4 1
Table 5.  Number of codes with pairs $(\alpha, |\text{Aut}(C)|)$ for the singly-even $[80, 40, 14]$ self-dual codes
$|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\alpha$ 13 26 39 78 $\alpha$ 13 26 39 $\alpha$ 13 26
$-351$ 1 1 $-221$ 6192 26 $-104$ 7078 24
$-325$ 1 $-208$ 9815 45 $-91$ 3632 3
$-312$ 5 2 2 $-195$ 14379 32 1 $-78$ 1637 14
$-299$ 32 1 $-182$ 18952 44 $-65$ 604 1
$-286$ 121 4 2 $-169$ 22002 31 1 $-52$ 168 4
$-273$ 294 7 1 $-156$ 22611 42 $-39$ 50
$-260$ 702 11 3 $-143$ 20758 13 $-26$ 4
$-247$ 1653 12 $-130$ 16648 38
$-234$ 3326 21 2 $-117$ 11635 8
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$|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\alpha$ 13 26 39 78 $\alpha$ 13 26 39 $\alpha$ 13 26
$-351$ 1 1 $-221$ 6192 26 $-104$ 7078 24
$-325$ 1 $-208$ 9815 45 $-91$ 3632 3
$-312$ 5 2 2 $-195$ 14379 32 1 $-78$ 1637 14
$-299$ 32 1 $-182$ 18952 44 $-65$ 604 1
$-286$ 121 4 2 $-169$ 22002 31 1 $-52$ 168 4
$-273$ 294 7 1 $-156$ 22611 42 $-39$ 50
$-260$ 702 11 3 $-143$ 20758 13 $-26$ 4
$-247$ 1653 12 $-130$ 16648 38
$-234$ 3326 21 2 $-117$ 11635 8
Table 6.  Number of codes with pairs $(\alpha, |\text{Aut}(C)|)$, $\beta=0$ in $W_{82, 2}$ for the $[82, 41, 14]$ self-dual codes with an automorphism of type $13-(6, 4)$, $C_\pi=G_4$
$|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\alpha$ 13 26 $\alpha$ 13 26 39 $\alpha$ 13 26 39 $\alpha$ 13 26 39
$-169$ 1 $-260$ 3199 79 $-351$ 88751 312 1 $-442$ 3002 28
$-182$ 2 $-273$ 7856 137 $-364$ 82010 297 5 $-455$ 1064 15
$-195$ 4 2 $-286$ 16765 168 1 $-377$ 67138 223 1 $-468$ 304 2
$-208$ 17 8 $-299$ 30265 230 $-390$ 46912 174 2 $-481$ 63 5 1
$-221$ 88 13 $-312$ 48818 302 $-403$ 28921 143 2 $-494$ 20
$-234$ 411 19 $-325$ 68292 330 2 $-416$ 15253 88 1 $-507$ 6 2
$-247$ 1227 50 $-338$ 83418 318 3 $-429$ 7240 51 1 $-520$ 1
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$|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\alpha$ 13 26 $\alpha$ 13 26 39 $\alpha$ 13 26 39 $\alpha$ 13 26 39
$-169$ 1 $-260$ 3199 79 $-351$ 88751 312 1 $-442$ 3002 28
$-182$ 2 $-273$ 7856 137 $-364$ 82010 297 5 $-455$ 1064 15
$-195$ 4 2 $-286$ 16765 168 1 $-377$ 67138 223 1 $-468$ 304 2
$-208$ 17 8 $-299$ 30265 230 $-390$ 46912 174 2 $-481$ 63 5 1
$-221$ 88 13 $-312$ 48818 302 $-403$ 28921 143 2 $-494$ 20
$-234$ 411 19 $-325$ 68292 330 2 $-416$ 15253 88 1 $-507$ 6 2
$-247$ 1227 50 $-338$ 83418 318 3 $-429$ 7240 51 1 $-520$ 1
Table 7.  Number of codes with pairs $(\alpha, |\text{Aut}(C)|)$, $\beta=13$ in $W_{82, 2}$ for the $[82, 41, 14]$ self-dual codes with an automorphism of type $13-(6, 4)$, $C_\pi=G_4$
$|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\alpha$ 13 $\alpha$ 13 39 $\alpha$ 13 39 $\alpha$ 13 39
-325 1 -377 44 -429 148 -481 38
-338 3 -390 71 -442 133 -494 21
-351 4 -403 123 1 -455 100 -507 4 1
-364 26 -416 146 -468 59 1 -520 3
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$|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\alpha$ 13 $\alpha$ 13 39 $\alpha$ 13 39 $\alpha$ 13 39
-325 1 -377 44 -429 148 -481 38
-338 3 -390 71 -442 133 -494 21
-351 4 -403 123 1 -455 100 -507 4 1
-364 26 -416 146 -468 59 1 -520 3
Table 8.  Number of codes with pairs $(\alpha, |\text{Aut}(C)|)$, $\beta=0$ in $W_{82, 2}$ for the $[82, 41, 14]$ self-dual codes with an automorphism of type $13-(6, 4)$, $C_\pi=G_5$
$|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\alpha$ 13 26 39 $\alpha$ 13 26 39 $\alpha$ 13 26 39 78 $\alpha$ 13 26 39
$-209$ 4 $-287$ 3291 41 $-365$ 23481 92 $-443$ 26 1
$-222$ 14 $-300$ 6480 67 $-378$ 20367 89 $-456$ 447 14 1
$-235$ 47 3 $-313$ 11032 65 2 $-391$ 15274 76 $-469$ 156 3
$-248$ 196 13 $-326$ 16352 87 $-404$ 9827 52 $-482$ 36
$-261$ 557 15 $-339$ 20791 93 $-417$ 5746 50 1 $-495$ 9 1
$-274$ 1446 34 1 $-352$ 23588 117 2 $-430$ 2837 19 2 $-508$ 1
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$|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\alpha$ 13 26 39 $\alpha$ 13 26 39 $\alpha$ 13 26 39 78 $\alpha$ 13 26 39
$-209$ 4 $-287$ 3291 41 $-365$ 23481 92 $-443$ 26 1
$-222$ 14 $-300$ 6480 67 $-378$ 20367 89 $-456$ 447 14 1
$-235$ 47 3 $-313$ 11032 65 2 $-391$ 15274 76 $-469$ 156 3
$-248$ 196 13 $-326$ 16352 87 $-404$ 9827 52 $-482$ 36
$-261$ 557 15 $-339$ 20791 93 $-417$ 5746 50 1 $-495$ 9 1
$-274$ 1446 34 1 $-352$ 23588 117 2 $-430$ 2837 19 2 $-508$ 1
Table 9.  Number of codes with pairs $(\alpha, |\text{Aut}(C)|)$, $\beta=13$ in $W_{82, 2}$ for the $[82, 41, 14]$ self-dual codes with an automorphism of type $13-(6, 4)$, $C_\pi=G_5$
$|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\alpha$ 13 39 $\alpha$ 13 26 $\alpha$ 13 26 39
$-339$ 1 $-417$ 27 $-482$ 12 5
$-365$ 5 1 $-430$ 20 $-495$ 4
$-378$ 6 $-443$ 1218 23 $-508$ 1
$-391$ 7 $-456$ 20 $-521$ 2 1
$-404$ 20 2 $-469$ 17
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$|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\alpha$ 13 39 $\alpha$ 13 26 $\alpha$ 13 26 39
$-339$ 1 $-417$ 27 $-482$ 12 5
$-365$ 5 1 $-430$ 20 $-495$ 4
$-378$ 6 $-443$ 1218 23 $-508$ 1
$-391$ 7 $-456$ 20 $-521$ 2 1
$-404$ 20 2 $-469$ 17
Table 10.  Number of codes with pairs $(\alpha, |\text{Aut}(C)|)$, $W_{84, 2}$ for the $[84, 42, 14]$ self-dual codes with an automorphism of type $13-(6, 6)$, $C_\pi=G_7$
$|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\alpha$ 13 26 39 78 $\alpha$ 13 26 39 78 $\alpha$ 13 26 39
3038 2 3272 36692 171 3506 12054 126
3064 3 3298 56508 214 4 3532 5477 84 5
3090 32 1 3324 75114 280 3558 2154 51
3116 126 8 3350 87114 288 3584 742 46
3142 568 16 2 3376 87901 334 6 3610 202 12 1
3168 1602 31 1 3402 77847 314 3636 44 6
3194 4452 48 3428 59855 288 3662 13 4
3220 10558 70 3 3454 40586 214 5 3688 1 1
3246 21074 145 1 3480 24064 207 1
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$|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\alpha$ 13 26 39 78 $\alpha$ 13 26 39 78 $\alpha$ 13 26 39
3038 2 3272 36692 171 3506 12054 126
3064 3 3298 56508 214 4 3532 5477 84 5
3090 32 1 3324 75114 280 3558 2154 51
3116 126 8 3350 87114 288 3584 742 46
3142 568 16 2 3376 87901 334 6 3610 202 12 1
3168 1602 31 1 3402 77847 314 3636 44 6
3194 4452 48 3428 59855 288 3662 13 4
3220 10558 70 3 3454 40586 214 5 3688 1 1
3246 21074 145 1 3480 24064 207 1
Table 11.  Number of codes with pairs $(\alpha, |\text{Aut}(C)|)$, $W_{84, 2}$ for the $[84, 42, 14]$ self-dual codes with an automorphism of type $13-(6, 6)$, $C_\pi=G_9$
$|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\alpha$ 13 26 39 $\alpha$ 13 26 39 $\alpha$ 13 26 39 $\alpha$ 13 26
3040 1 3222 7240 51 1 3404 83418 318 3 3586 1227 50
3066 6 2 3248 15253 88 1 3430 68292 330 2 3612 411 19
3092 20 3274 28921 143 2 3456 48818 302 3638 88 13
3118 63 5 1 3300 46912 174 2 3482 30265 230 3664 17 8
3144 304 2 3326 67138 223 1 3508 16765 168 1 3690 4 2
3170 1064 15 3352 82010 297 5 3534 7856 137 3716 2
3196 3002 28 3378 88751 312 1 3560 3199 79 3742 1
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$|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$ $|\text{Aut}(C)|$
$\alpha$ 13 26 39 $\alpha$ 13 26 39 $\alpha$ 13 26 39 $\alpha$ 13 26
3040 1 3222 7240 51 1 3404 83418 318 3 3586 1227 50
3066 6 2 3248 15253 88 1 3430 68292 330 2 3612 411 19
3092 20 3274 28921 143 2 3456 48818 302 3638 88 13
3118 63 5 1 3300 46912 174 2 3482 30265 230 3664 17 8
3144 304 2 3326 67138 223 1 3508 16765 168 1 3690 4 2
3170 1064 15 3352 82010 297 5 3534 7856 137 3716 2
3196 3002 28 3378 88751 312 1 3560 3199 79 3742 1
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