Quasi-symmetric designs

A table of feasible parameters for exceptional quasi-symmetric 2-designs with 2k ≤ v ≤ 70 appears in the survey [15]. Here is an updated table with links to the actual designs. We only list parameters for which designs are known to exist, and add three more rows to the table from [15]. Where possible, the designs and the references are given in chronological order. For example, the list of (56,16,6) QSDs starts with the first known example from [16], the next design is from [14], and the remaining designs are from [10]. Other references can be found in [15].

vkλr bxyNQSD Reference
21 6 4 16 56 0 2 1[15]
21 7 12 40 120 1 3 1[15]
22 6 5 21 77 0 2 1[15]
22 7 16 56 176 1 3 1[15]
23 7 21 77 253 1 3 1[15]
28 12 11 27 63 4 6 ≥89559[11,5,9,17]
31 7 7 35 155 1 3 5[15]
36 16 12 28 63 6 8 ≥522079[11,5,9]
45 9 8 44 220 1 3 1[15]
49 9 6 36 196 1 3 ≥44 [15]
56 16 6 22 77 4 6 ≥1410[16,14,10]
56 16 18 66 231 4 8 ≥4[9,10]
63 15 35 155 651 3 7 ≥1[15]
64 24 46 126 336 8 12 ≥30264 [7,4]
66 30 29 65 143 12 15 ≥10000[2,13,9]
78 22 6 22 78 6 6 ≥3141[6,16,14,3,10]
78 36 30 66 143 15 18 ≥10000[2,13,9]
217 7 1 36 1116 0 1 ≥4[12,1,9]

In [8], quasi-symmetric (36,16,12) QSDs with an automorphism group isomorphic to the Frobenius group of order 21 were classified and tested for embeddability in symmetric (64,28,12) designs as residuals. The total number of QSDs is 921 and 116 of them are non-embeddable. A list of incidence matrices is available here, with the non-embeddable QSDs appearing first.

References

  1. S. Bagchi, B. Bagchi, Designs from pairs of finite fields. I. A cyclic unital U(6) and other regular Steiner 2-designs, J. Combin. Theory Ser. A 52 (1989), 51-61.
  2. C. Bracken, G. McGuire, H. Ward, New quasi-symmetric designs constructed using mutually orthogonal Latin squares and Hadamard matrices, Des. Codes Cryptogr. 41 (2006), no. 2, 195-198.
  3. D. Crnkovic, D. Dumicic Danilovic, S. Rukavina, On symmetric (78,22,6) designs and related self-orthogonal codes, Util. Math. 109 (2018), 227-253.
  4. D. Crnkovic, B.G. Rodrigues, S. Rukavina, V. D. Tonchev, Quasi-symmetric 2-(64,24,46) designs derived from AG(3,4), Discrete Math. 340 (2017), no. 10, 2472-2478.
  5. Y. Ding, S. Houghten, C. Lam, S. Smith, L. Thiel, V. D. Tonchev, Quasi-symmetric 2-(28,12,11) designs with an automorphism of order 7, J. Combin. Des. 6 (1998), no. 3, 213-223.
  6. Z. Janko, 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.
  7. D. Jungnickel, V. D. Tonchev, Maximal arcs and quasi-symmetric designs, Des. Codes Cryptogr. 77 (2015), no. 2-3, 365-374.
  8. V. Krcadinac, Non-embeddable quasi-residual quasi-symmetric designs, Appl. Algebra Engrg. Comm. Comput. 33 (2022), no. 4, 477-483.
  9. V. Krcadinac, R. Vlahovic, New quasi-symmetric designs by the Kramer-Mesner method, Discrete Math. 339 (2016), no. 12, 2884-2890.
  10. V. Krcadinac, R. Vlahovic Kruc, Quasi-symmetric designs on 56 points, Adv. Math. Commun. 15 (2021), no. 4, 633-646.
  11. C. Lam, L. Thiel, V. D. Tonchev, On quasi-symmetric 2-(28,12,11) and 2-(36,16,12) designs, Des. Codes Cryptogr. 5 (1995), no. 1, 43-55.
  12. R. Mathon, Constructions for cyclic Steiner 2-designs, Ann. Discrete Math. 34 (1987), 353-362.
  13. T. P. McDonough, V. C. Mavron, H. N. Ward, Amalgams of designs and nets, Bull. Lond. Math. Soc. 41 (2009), no. 5, 841-852.
  14. A. Munemasa, V. D. Tonchev, A new quasi-symmetric 2-(56,16,6) design obtained from codes, Discrete Math. 284 (2004), no. 1-3, 231-234.
  15. M. S. Shrikhande, Quasi-symmetric designs, in: The Handbook of Combinatorial Designs, Second Edition (eds. C.J. Colbourn and J.H. Dinitz), CRC Press, 2007, pp. 578-582.
  16. 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), no. 1, 49-75.
  17. R. Vlahovic Kruc, V. Krcadinac, Quasi-symmetric 2-(28,12,11) designs with an automorphism of order 5, Glas. Mat. Ser. III 58 (2023), No. 2, 159-166.

Vedran Krcadinac,