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November  2021, 15(4): 633-646. doi: 10.3934/amc.2020086

Quasi-symmetric designs on $ 56 $ points

Vedran Krčadinac , and  Renata Vlahović Kruc

Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, HR-10000 Zagreb, Croatia

* Corresponding author: V. Krčadinac

Received  November 2019 Revised  February 2020 Published  November 2021 Early access  June 2020

Fund Project: This work has been fully supported by the Croatian Science Foundation under the project 6732

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.

Keywords: Quasi-symmetric design, symmetric design, automorphism group, construction method, self-orthogonal code.
Mathematics Subject Classification: 05B05, 94B25, 05C69.
Citation: Vedran Krčadinac, Renata Vlahović Kruc. Quasi-symmetric designs on $ 56 $ points. Advances in Mathematics of Communications, 2021, 15 (4) : 633-646. doi: 10.3934/amc.2020086
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.

[10]

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.10, 2018., http://www.gap-system.org

[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.

[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.

[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.

[10]

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.10, 2018., http://www.gap-system.org

[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.

[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.

[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.

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Table 1.  The known QSDs with $ v = 56 $
No. $ v $ $ k $ $ \lambda $ $ r $ $ b $ $ x $ $ y $ NQSD Ref.
47 56 16 18 66 231 4 8 $ \ge 3 $ [13]
48 56 15 42 165 616 3 6 0 [5]
49 56 12 9 45 210 0 3 ?
50 56 21 24 66 176 6 9 0 [5]
51 56 20 19 55 154 5 8 ?
52 56 16 6 22 77 4 6 $ \ge 2 $ [24,16]
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No. $ v $ $ k $ $ \lambda $ $ r $ $ b $ $ x $ $ y $ NQSD Ref.
47 56 16 18 66 231 4 8 $ \ge 3 $ [13]
48 56 15 42 165 616 3 6 0 [5]
49 56 12 9 45 210 0 3 ?
50 56 21 24 66 176 6 9 0 [5]
51 56 20 19 55 154 5 8 ?
52 56 16 6 22 77 4 6 $ \ge 2 $ [24,16]
Table 3.  Dimensions and weight distributions of self-orthogonal binary codes spanned by $ (56,16,6) $ QSDs
$ \dim $ $ a_0 $ $ a_8 $ $ a_{12} $ $ a_{16} $ $ a_{20} $ $ a_{24} $ $ a_{28} $
$ C_1 $ 26 1 91 2 016 152 425 2 939 776 16 194 619 28 531 008
$ C_2 $ 26 1 7 $ 2\,016 $ $ 155\,365 $ $ 2\,926\,336 $ $ 16\,224\,019 $ $ 28\,493\,376 $
$ C_3 $ 24 1 75 0 40 089 730 368 4 055 835 7 124 480
$ C_{4\hbox{-}6,9,10} $ 22 1 15 0 9 933 183 168 1 012 515 1 783 040
$ C_{7,11\hbox{-}13} $ 25 1 75 672 77 721 1 465 984 8 103 963 14 257 600
$ C_8 $ 25 1 75 960 75 417 1 474 048 8 087 835 14 277 760
$ C_{14} $ 22 1 15 0 10 701 178 560 1 024 035 1 767 680
$ C_{15} $ 23 1 15 288 19 917 361 216 2 040 867 3 544 000
$ C_{16} $ 23 1 15 96 19 917 365 056 2 028 579 3 561 280
$ C_{17} $ 24 1 75 160 39 833 728 704 4 062 235 7 115 200
$ C_{18} $ 22 1 15 64 9 677 183 424 1 012 771 1 782 400
$ C_{19,21,24} $ 22 1 15 16 10 061 182 080 1 015 459 1 779 040
$ C_{20,22} $ 22 1 15 64 10 445 178 816 1 024 291 1 767 040
$ C_{23} $ 25 1 75 1 280 74 905 1 470 720 8 100 635 14 259 200
$ C_{25} $ 25 1 75 992 77 209 1 462 656 8 116 763 14 239 040
$ C_{26} $ 27 1 139 4 992 307 161 5 848 832 32 477 083 56 941 312
$ C_{27} $ 27 1 99 4 304 305 873 5 872 320 32 406 731 57 039 072
$ C_{28,29} $ 27 1 99 4 112 307 409 5 866 944 32 417 483 57 025 632
$ C_{30} $ 26 1 147 1 008 158 529 2 920 512 16 231 467 28 485 536
$ C_{31,32,34,35} $ 27 1 147 3 696 309 057 5 862 976 32 423 979 57 018 016
$ C_{33,39} $ 27 1 147 4 976 307 009 5 849 664 32 475 179 56 943 776
$ C_{36} $ 26 1 75 2 240 153 241 2 931 200 16 218 395 28 498 560
$ C_{37,38} $ 27 1 75 4 416 305 817 5 871 616 32 408 859 57 036 160
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$ \dim $ $ a_0 $ $ a_8 $ $ a_{12} $ $ a_{16} $ $ a_{20} $ $ a_{24} $ $ a_{28} $
$ C_1 $ 26 1 91 2 016 152 425 2 939 776 16 194 619 28 531 008
$ C_2 $ 26 1 7 $ 2\,016 $ $ 155\,365 $ $ 2\,926\,336 $ $ 16\,224\,019 $ $ 28\,493\,376 $
$ C_3 $ 24 1 75 0 40 089 730 368 4 055 835 7 124 480
$ C_{4\hbox{-}6,9,10} $ 22 1 15 0 9 933 183 168 1 012 515 1 783 040
$ C_{7,11\hbox{-}13} $ 25 1 75 672 77 721 1 465 984 8 103 963 14 257 600
$ C_8 $ 25 1 75 960 75 417 1 474 048 8 087 835 14 277 760
$ C_{14} $ 22 1 15 0 10 701 178 560 1 024 035 1 767 680
$ C_{15} $ 23 1 15 288 19 917 361 216 2 040 867 3 544 000
$ C_{16} $ 23 1 15 96 19 917 365 056 2 028 579 3 561 280
$ C_{17} $ 24 1 75 160 39 833 728 704 4 062 235 7 115 200
$ C_{18} $ 22 1 15 64 9 677 183 424 1 012 771 1 782 400
$ C_{19,21,24} $ 22 1 15 16 10 061 182 080 1 015 459 1 779 040
$ C_{20,22} $ 22 1 15 64 10 445 178 816 1 024 291 1 767 040
$ C_{23} $ 25 1 75 1 280 74 905 1 470 720 8 100 635 14 259 200
$ C_{25} $ 25 1 75 992 77 209 1 462 656 8 116 763 14 239 040
$ C_{26} $ 27 1 139 4 992 307 161 5 848 832 32 477 083 56 941 312
$ C_{27} $ 27 1 99 4 304 305 873 5 872 320 32 406 731 57 039 072
$ C_{28,29} $ 27 1 99 4 112 307 409 5 866 944 32 417 483 57 025 632
$ C_{30} $ 26 1 147 1 008 158 529 2 920 512 16 231 467 28 485 536
$ C_{31,32,34,35} $ 27 1 147 3 696 309 057 5 862 976 32 423 979 57 018 016
$ C_{33,39} $ 27 1 147 4 976 307 009 5 849 664 32 475 179 56 943 776
$ C_{36} $ 26 1 75 2 240 153 241 2 931 200 16 218 395 28 498 560
$ C_{37,38} $ 27 1 75 4 416 305 817 5 871 616 32 408 859 57 036 160
Table 2.  Dimensions and weight distributions of self-orthogonal binary codes spanned by $ (56,16,18) $ QSDs
$ \dim $ $ a_0 $ $ a_8 $ $ a_{12} $ $ a_{16} $ $ a_{20} $ $ a_{24} $ $ a_{28} $
$ C_{1,2} $ 23 1 75 0 21 657 353 536 2 059 035 3 520 000
$ C_3 $ 19 1 0 0 1 722 19 936 134 085 212 800
$ C_4 $ 23 1 15 216 20 493 359 200 2 044 899 3 538 960
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$ \dim $ $ a_0 $ $ a_8 $ $ a_{12} $ $ a_{16} $ $ a_{20} $ $ a_{24} $ $ a_{28} $
$ C_{1,2} $ 23 1 75 0 21 657 353 536 2 059 035 3 520 000
$ C_3 $ 19 1 0 0 1 722 19 936 134 085 212 800
$ C_4 $ 23 1 15 216 20 493 359 200 2 044 899 3 538 960
Table 4.  Distribution of the known $ (56,16,6) $ QSDs and symmetric $ (78,22,6) $ designs by full automorphism group order
$ | \mathrm{Aut}| $ #$ (56,16,6) $ #$ (78,22,6) $
$ 168 $ $ 1 $ $ 2 $
$ 78 $ $ 0 $ $ 1 $
$ 48 $ $ 876 $ $ 1664 $
$ 24 $ $ 1 $ $ 378 $
$ 21 $ $ 1 $ $ 2 $
$ 16 $ $ 228 $ $ 456 $
$ 12 $ $ 303 $ $ 606 $
$ 6 $ $ 0 $ $ 32 $
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$ | \mathrm{Aut}| $ #$ (56,16,6) $ #$ (78,22,6) $
$ 168 $ $ 1 $ $ 2 $
$ 78 $ $ 0 $ $ 1 $
$ 48 $ $ 876 $ $ 1664 $
$ 24 $ $ 1 $ $ 378 $
$ 21 $ $ 1 $ $ 2 $
$ 16 $ $ 228 $ $ 456 $
$ 12 $ $ 303 $ $ 606 $
$ 6 $ $ 0 $ $ 32 $
Table 5.  An updated table of QSDs with $ v = 56 $
No. $ v $ $ k $ $ \lambda $ $ r $ $ b $ $ x $ $ y $ NQSD
47 56 16 18 66 231 4 8 $ \ge 4 $
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 $ \ge 1410 $
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No. $ v $ $ k $ $ \lambda $ $ r $ $ b $ $ x $ $ y $ NQSD
47 56 16 18 66 231 4 8 $ \ge 4 $
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 $ \ge 1410 $
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