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Constructions of linear codes with small hulls from association schemes
Quasi-symmetric designs on $ 56 $ points
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, HR-10000 Zagreb, Croatia |
Computational techniques for the construction of quasi-symmetric block designs are explored and applied to the case with $ 56 $ points. One new $ (56,16,18) $ and many new $ (56,16,6) $ designs are discovered, and non-existence of $ (56,12,9) $ and $ (56,20,19) $ designs with certain automorphism groups is proved. The number of known symmetric $ (78,22,6) $ designs is also significantly increased.
References:
[1] |
T. Beth, D. Jungnickel and H. Lenz, Hanfried Design Theory. Vol. II, 2nd edition, Cambridge University Press, Cambridge, 1999. |
[2] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system Ⅰ: The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[3] |
A. E. Brouwer,
The uniqueness of the strongly regular graph on 77 points, J. Graph Theory, 7 (1983), 455-461.
doi: 10.1002/jgt.3190070411. |
[4] |
A. E. Brouwer,
Uniqueness and nonexistence of some graphs related to $M_22$, Graphs Combin., 2 (1986), 21-29.
doi: 10.1007/BF01788073. |
[5] |
A. R. Calderbank,
Geometric invariants for quasisymmetric designs, J. Combin. Theory Ser. A, 47 (1988), 101-110.
doi: 10.1016/0097-3165(88)90044-1. |
[6] |
D. Crnković, D. Dumičić Danilović and S. Rukavina,
On symmetric (78, 22, 6) designs and related self-orthogonal codes, Util. Math., 109 (2018), 227-253.
|
[7] |
D. Crnković, B. G. Rodrigues, S. Rukavina and V. D. Tonchev,
Quasi-symmetric $2$-$(64, 24, 46)$ designs derived from $AG(3, 4)$, Discrete Math., 340 (2017), 2472-2478.
doi: 10.1016/j.disc.2017.06.008. |
[8] |
Y. Ding, S. Houghten, C. Lam, S. Smith, L. Thiel and V. D. Tonchev,
Quasi-symmetric $2$-$(28, 12, 11)$ designs with an automorphism of order $7$, J. Combin. Des., 6 (1998), 213-223.
|
[9] |
I. A. Faradžev, Constructive enumeration of combinatorial objects, in Problemes combinatoires et théorie des graphes, Colloq. Internat. CNRS, Paris, 1978,131–135. Google Scholar |
[10] |
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.10, 2018., http://www.gap-system.org Google Scholar |
[11] |
Z. Janko and T. Van Trung,
Construction of a new symmetric block design for $(78, 22, 6)$ with the help of tactical decompositions, J. Combin. Theory Ser. A, 40 (1985), 451-455.
doi: 10.1016/0097-3165(85)90107-4. |
[12] |
V. Krčadinac,
Steiner $2$-designs $S(2, 4, 28)$ with nontrivial automorphisms, Glas. Mat. Ser. III, 37(57) (2002), 259-268.
|
[13] |
V. Krčadinac and R. Vlahović,
New quasi-symmetric designs by the Kramer-Mesner method, Discrete Math., 339 (2016), 2884-2890.
doi: 10.1016/j.disc.2016.05.030. |
[14] |
B. D. McKay,
Isomorph-free exhaustive generation, J. Algorithms, 26 (1998), 306-324.
doi: 10.1006/jagm.1997.0898. |
[15] |
B. D. McKay and A. Piperno,
Practical graph isomorphism, Ⅱ, J. Symbolic Comput., 60 (2014), 94-112.
doi: 10.1016/j.jsc.2013.09.003. |
[16] |
A. Munemasa and V. D. Tonchev,
A new quasi-symmetric $2$-$(56, 16, 6)$ design obtained from codes, Discrete Math., 284 (2004), 231-234.
doi: 10.1016/j.disc.2003.11.036. |
[17] |
A. Neumaier, Regular sets and quasisymmetric 2-designs, in Combinatorial Theory (Schloss Rauischholzhausen, 1982), Lecture Notes in Math., Vol. 969, Springer, Berlin-New York, 1982,258–275. |
[18] |
S. Niskanen and P. R. J. Östergård, Cliquer User's Guide, Version 1.0, Communications Laboratory, Helsinki University of Technology, Espoo, Finland, Tech. Rep. T48, 2003. Google Scholar |
[19] |
P. R. J. Östergård,
A fast algorithm for the maximum clique problem, Discrete Appl. Math., 120 (2002), 197-207.
doi: 10.1016/S0166-218X(01)00290-6. |
[20] |
R. M. Pawale, M. S. Shrikhande and S. M. Nyayate, Conditions for the parameters of the block graph of quasi-symmetric designs, Electron. J. Combin., 22 (2015), Paper 1.36, 30 pp.
doi: 10.37236/3954. |
[21] |
R. C. Read, Every one a winner or how to avoid isomorphism search when cataloguing combinatorial configurations, in Ann. Discrete Math., 2, 1978,107–120.
doi: 10.1016/S0167-5060(08)70325-X. |
[22] |
M. S. Shrikhande, Quasi-symmetric designs, in The CNC Handbook of Combinatorial Designs, Second Edition, CRC Press, Boca Raton, FL, 2007,578–582. Google Scholar |
[23] |
M. S. Shrikhande and S. S. Sane, Quasi-symmetric Designs, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511665615.![]() ![]() |
[24] |
V. D. Tonchev,
Embedding of the Witt-Mathieu system $S(3, 6, 22)$ in a symmetric $2$-$(78, 22, 6)$ design, Geom. Dedicata, 22 (1987), 49-75.
doi: 10.1007/BF00183053. |
show all references
References:
[1] |
T. Beth, D. Jungnickel and H. Lenz, Hanfried Design Theory. Vol. II, 2nd edition, Cambridge University Press, Cambridge, 1999. |
[2] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system Ⅰ: The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[3] |
A. E. Brouwer,
The uniqueness of the strongly regular graph on 77 points, J. Graph Theory, 7 (1983), 455-461.
doi: 10.1002/jgt.3190070411. |
[4] |
A. E. Brouwer,
Uniqueness and nonexistence of some graphs related to $M_22$, Graphs Combin., 2 (1986), 21-29.
doi: 10.1007/BF01788073. |
[5] |
A. R. Calderbank,
Geometric invariants for quasisymmetric designs, J. Combin. Theory Ser. A, 47 (1988), 101-110.
doi: 10.1016/0097-3165(88)90044-1. |
[6] |
D. Crnković, D. Dumičić Danilović and S. Rukavina,
On symmetric (78, 22, 6) designs and related self-orthogonal codes, Util. Math., 109 (2018), 227-253.
|
[7] |
D. Crnković, B. G. Rodrigues, S. Rukavina and V. D. Tonchev,
Quasi-symmetric $2$-$(64, 24, 46)$ designs derived from $AG(3, 4)$, Discrete Math., 340 (2017), 2472-2478.
doi: 10.1016/j.disc.2017.06.008. |
[8] |
Y. Ding, S. Houghten, C. Lam, S. Smith, L. Thiel and V. D. Tonchev,
Quasi-symmetric $2$-$(28, 12, 11)$ designs with an automorphism of order $7$, J. Combin. Des., 6 (1998), 213-223.
|
[9] |
I. A. Faradžev, Constructive enumeration of combinatorial objects, in Problemes combinatoires et théorie des graphes, Colloq. Internat. CNRS, Paris, 1978,131–135. Google Scholar |
[10] |
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.10, 2018., http://www.gap-system.org Google Scholar |
[11] |
Z. Janko and T. Van Trung,
Construction of a new symmetric block design for $(78, 22, 6)$ with the help of tactical decompositions, J. Combin. Theory Ser. A, 40 (1985), 451-455.
doi: 10.1016/0097-3165(85)90107-4. |
[12] |
V. Krčadinac,
Steiner $2$-designs $S(2, 4, 28)$ with nontrivial automorphisms, Glas. Mat. Ser. III, 37(57) (2002), 259-268.
|
[13] |
V. Krčadinac and R. Vlahović,
New quasi-symmetric designs by the Kramer-Mesner method, Discrete Math., 339 (2016), 2884-2890.
doi: 10.1016/j.disc.2016.05.030. |
[14] |
B. D. McKay,
Isomorph-free exhaustive generation, J. Algorithms, 26 (1998), 306-324.
doi: 10.1006/jagm.1997.0898. |
[15] |
B. D. McKay and A. Piperno,
Practical graph isomorphism, Ⅱ, J. Symbolic Comput., 60 (2014), 94-112.
doi: 10.1016/j.jsc.2013.09.003. |
[16] |
A. Munemasa and V. D. Tonchev,
A new quasi-symmetric $2$-$(56, 16, 6)$ design obtained from codes, Discrete Math., 284 (2004), 231-234.
doi: 10.1016/j.disc.2003.11.036. |
[17] |
A. Neumaier, Regular sets and quasisymmetric 2-designs, in Combinatorial Theory (Schloss Rauischholzhausen, 1982), Lecture Notes in Math., Vol. 969, Springer, Berlin-New York, 1982,258–275. |
[18] |
S. Niskanen and P. R. J. Östergård, Cliquer User's Guide, Version 1.0, Communications Laboratory, Helsinki University of Technology, Espoo, Finland, Tech. Rep. T48, 2003. Google Scholar |
[19] |
P. R. J. Östergård,
A fast algorithm for the maximum clique problem, Discrete Appl. Math., 120 (2002), 197-207.
doi: 10.1016/S0166-218X(01)00290-6. |
[20] |
R. M. Pawale, M. S. Shrikhande and S. M. Nyayate, Conditions for the parameters of the block graph of quasi-symmetric designs, Electron. J. Combin., 22 (2015), Paper 1.36, 30 pp.
doi: 10.37236/3954. |
[21] |
R. C. Read, Every one a winner or how to avoid isomorphism search when cataloguing combinatorial configurations, in Ann. Discrete Math., 2, 1978,107–120.
doi: 10.1016/S0167-5060(08)70325-X. |
[22] |
M. S. Shrikhande, Quasi-symmetric designs, in The CNC Handbook of Combinatorial Designs, Second Edition, CRC Press, Boca Raton, FL, 2007,578–582. Google Scholar |
[23] |
M. S. Shrikhande and S. S. Sane, Quasi-symmetric Designs, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511665615.![]() ![]() |
[24] |
V. D. Tonchev,
Embedding of the Witt-Mathieu system $S(3, 6, 22)$ in a symmetric $2$-$(78, 22, 6)$ design, Geom. Dedicata, 22 (1987), 49-75.
doi: 10.1007/BF00183053. |
26 | 1 | 91 | 2 016 | 152 425 | 2 939 776 | 16 194 619 | 28 531 008 | |
26 | 1 | 7 | ||||||
24 | 1 | 75 | 0 | 40 089 | 730 368 | 4 055 835 | 7 124 480 | |
22 | 1 | 15 | 0 | 9 933 | 183 168 | 1 012 515 | 1 783 040 | |
25 | 1 | 75 | 672 | 77 721 | 1 465 984 | 8 103 963 | 14 257 600 | |
25 | 1 | 75 | 960 | 75 417 | 1 474 048 | 8 087 835 | 14 277 760 | |
22 | 1 | 15 | 0 | 10 701 | 178 560 | 1 024 035 | 1 767 680 | |
23 | 1 | 15 | 288 | 19 917 | 361 216 | 2 040 867 | 3 544 000 | |
23 | 1 | 15 | 96 | 19 917 | 365 056 | 2 028 579 | 3 561 280 | |
24 | 1 | 75 | 160 | 39 833 | 728 704 | 4 062 235 | 7 115 200 | |
22 | 1 | 15 | 64 | 9 677 | 183 424 | 1 012 771 | 1 782 400 | |
22 | 1 | 15 | 16 | 10 061 | 182 080 | 1 015 459 | 1 779 040 | |
22 | 1 | 15 | 64 | 10 445 | 178 816 | 1 024 291 | 1 767 040 | |
25 | 1 | 75 | 1 280 | 74 905 | 1 470 720 | 8 100 635 | 14 259 200 | |
25 | 1 | 75 | 992 | 77 209 | 1 462 656 | 8 116 763 | 14 239 040 | |
27 | 1 | 139 | 4 992 | 307 161 | 5 848 832 | 32 477 083 | 56 941 312 | |
27 | 1 | 99 | 4 304 | 305 873 | 5 872 320 | 32 406 731 | 57 039 072 | |
27 | 1 | 99 | 4 112 | 307 409 | 5 866 944 | 32 417 483 | 57 025 632 | |
26 | 1 | 147 | 1 008 | 158 529 | 2 920 512 | 16 231 467 | 28 485 536 | |
27 | 1 | 147 | 3 696 | 309 057 | 5 862 976 | 32 423 979 | 57 018 016 | |
27 | 1 | 147 | 4 976 | 307 009 | 5 849 664 | 32 475 179 | 56 943 776 | |
26 | 1 | 75 | 2 240 | 153 241 | 2 931 200 | 16 218 395 | 28 498 560 | |
27 | 1 | 75 | 4 416 | 305 817 | 5 871 616 | 32 408 859 | 57 036 160 |
26 | 1 | 91 | 2 016 | 152 425 | 2 939 776 | 16 194 619 | 28 531 008 | |
26 | 1 | 7 | ||||||
24 | 1 | 75 | 0 | 40 089 | 730 368 | 4 055 835 | 7 124 480 | |
22 | 1 | 15 | 0 | 9 933 | 183 168 | 1 012 515 | 1 783 040 | |
25 | 1 | 75 | 672 | 77 721 | 1 465 984 | 8 103 963 | 14 257 600 | |
25 | 1 | 75 | 960 | 75 417 | 1 474 048 | 8 087 835 | 14 277 760 | |
22 | 1 | 15 | 0 | 10 701 | 178 560 | 1 024 035 | 1 767 680 | |
23 | 1 | 15 | 288 | 19 917 | 361 216 | 2 040 867 | 3 544 000 | |
23 | 1 | 15 | 96 | 19 917 | 365 056 | 2 028 579 | 3 561 280 | |
24 | 1 | 75 | 160 | 39 833 | 728 704 | 4 062 235 | 7 115 200 | |
22 | 1 | 15 | 64 | 9 677 | 183 424 | 1 012 771 | 1 782 400 | |
22 | 1 | 15 | 16 | 10 061 | 182 080 | 1 015 459 | 1 779 040 | |
22 | 1 | 15 | 64 | 10 445 | 178 816 | 1 024 291 | 1 767 040 | |
25 | 1 | 75 | 1 280 | 74 905 | 1 470 720 | 8 100 635 | 14 259 200 | |
25 | 1 | 75 | 992 | 77 209 | 1 462 656 | 8 116 763 | 14 239 040 | |
27 | 1 | 139 | 4 992 | 307 161 | 5 848 832 | 32 477 083 | 56 941 312 | |
27 | 1 | 99 | 4 304 | 305 873 | 5 872 320 | 32 406 731 | 57 039 072 | |
27 | 1 | 99 | 4 112 | 307 409 | 5 866 944 | 32 417 483 | 57 025 632 | |
26 | 1 | 147 | 1 008 | 158 529 | 2 920 512 | 16 231 467 | 28 485 536 | |
27 | 1 | 147 | 3 696 | 309 057 | 5 862 976 | 32 423 979 | 57 018 016 | |
27 | 1 | 147 | 4 976 | 307 009 | 5 849 664 | 32 475 179 | 56 943 776 | |
26 | 1 | 75 | 2 240 | 153 241 | 2 931 200 | 16 218 395 | 28 498 560 | |
27 | 1 | 75 | 4 416 | 305 817 | 5 871 616 | 32 408 859 | 57 036 160 |
23 | 1 | 75 | 0 | 21 657 | 353 536 | 2 059 035 | 3 520 000 | |
19 | 1 | 0 | 0 | 1 722 | 19 936 | 134 085 | 212 800 | |
23 | 1 | 15 | 216 | 20 493 | 359 200 | 2 044 899 | 3 538 960 |
23 | 1 | 75 | 0 | 21 657 | 353 536 | 2 059 035 | 3 520 000 | |
19 | 1 | 0 | 0 | 1 722 | 19 936 | 134 085 | 212 800 | |
23 | 1 | 15 | 216 | 20 493 | 359 200 | 2 044 899 | 3 538 960 |
# |
# |
|
# |
# |
|
No. | NQSD | |||||||
47 | 56 | 16 | 18 | 66 | 231 | 4 | 8 | |
48 | 56 | 15 | 42 | 165 | 616 | 3 | 6 | 0 |
49 | 56 | 12 | 9 | 45 | 210 | 0 | 3 | ? |
50 | 56 | 21 | 24 | 66 | 176 | 6 | 9 | 0 |
51 | 56 | 20 | 19 | 55 | 154 | 5 | 8 | ? |
52 | 56 | 16 | 6 | 22 | 77 | 4 | 6 |
No. | NQSD | |||||||
47 | 56 | 16 | 18 | 66 | 231 | 4 | 8 | |
48 | 56 | 15 | 42 | 165 | 616 | 3 | 6 | 0 |
49 | 56 | 12 | 9 | 45 | 210 | 0 | 3 | ? |
50 | 56 | 21 | 24 | 66 | 176 | 6 | 9 | 0 |
51 | 56 | 20 | 19 | 55 | 154 | 5 | 8 | ? |
52 | 56 | 16 | 6 | 22 | 77 | 4 | 6 |
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