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A recursive construction of doubly resolvable Steiner quadruple systems

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Abstract

Two resolutions of the same 3-design are said to be orthogonal when each parallel class of one resolution has at most one block in common with each parallel class of the other resolution. If a Steiner quadruple system has two mutually orthogonal resolutions, the design is called doubly resolvable and denoted by DRSQS. In this paper, we define almost doubly resolvable candelabra quadruple system and then to get a recursive construction of DRSQS, i.e., for \(n\ge 16\), if there is a DRSQS(n), then there exists a DRSQS\((14n-12)\) and a DRSQS\((16n-12)\).

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References

  1. Bose R.C., Shrikhande S.S., Parker E.T.: Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler’s conjecture. Can. J. Math. 12, 189–203 (1960).

    Article  MathSciNet  Google Scholar 

  2. Booth T.R.: A resolvable quadruple system of order 20. Ars Comb. 5, 121–125 (1978).

    MathSciNet  Google Scholar 

  3. de Vries H.L.: On orthogonal resolutions of the classical Steiner quadruple system SQS(16). Des. Codes Cryptogr. 48, 287–292 (2008).

    Article  MathSciNet  Google Scholar 

  4. Dinitz J.H., Stinson D.R.: Room squares and related designs. In: Dinitz J.H., Stinson D.R. (eds.) Contemporary Design Theory: A Collection of Surveys, pp. 593–631. Wiley, New York (1992).

    Google Scholar 

  5. Greenwell D.L., Lindner C.C.: Some remarks on resolvable quadruple systems. Ars Comb. 6, 215–221 (1978).

    MathSciNet  Google Scholar 

  6. Hartman A.: Resolvable Steiner quadruple systems. Ars Comb. 9, 263–273 (1980).

    MathSciNet  Google Scholar 

  7. Hartman A.: Tripling quadruple systems. Ars Comb. 10, 255–309 (1980).

    MathSciNet  Google Scholar 

  8. Hartman A.: Doubly and orthogonally resolvable quadruple systems, Ars Combin., Combinatorial Mathematics, VII, Proc. Seventh Australian Conf., Lecture Notes in Math., Univ. Newcastle, Newcastle,: vol. 829. Springer, Berlin, pp. 157–164 (1979).

  9. Hartman A.: The existence of resolvable Steiner quadruple systems. J. Comb. Theory A 44, 182–206 (1987).

    Article  MathSciNet  Google Scholar 

  10. Hartman A., Phelps K.T.: Steiner quadruple systems. In: Dinitz J.H., Stinson D.R. (eds.) Contemporary Design Theory, pp. 205–240. Weiley, New York (1992).

    Google Scholar 

  11. Hung S.H.Y., Mendelsohn N.S.: On howell designs. J. Comb. Theory A 16, 174–198 (1974).

    Article  MathSciNet  Google Scholar 

  12. Ji L.: An improvement on H design. J. Comb. Des. 17, 25–35 (2009).

    Article  MathSciNet  Google Scholar 

  13. Ji L.: A complete solution to existence of H designs. J. Comb. Des. 27, 75–81 (2019).

    Article  MathSciNet  Google Scholar 

  14. Ji L., Zhu L.: Resolvable Steiner quadruple systems for the last 23 orders. SIAM J. Discret. Math. 19, 420–432 (2005).

    Article  MathSciNet  Google Scholar 

  15. Lindner C.C., Rodger C.A.: Design Theory. CRC Press, Boca Raton (2009).

    Google Scholar 

  16. Meng Z.: Doubly resolvable H designs. Graphs Comb. 32, 2563–2574 (2016).

    Article  MathSciNet  Google Scholar 

  17. Meng Z.: Doubly resolvable Steiner quadruple systems and related designs. Des. Codes Cryptogr. 84, 325–343 (2017).

    Article  MathSciNet  Google Scholar 

  18. Meng Z., Du B.: Resolvable candelabra quadruple systems with three groups. J. Comb. Des. 19, 247–267 (2011).

    Article  MathSciNet  Google Scholar 

  19. Meng Z., Zhang B., Wu Z.: Constructions of doubly resolvable Steiner quadruple systems. Des. Codes Cryptogr. 89, 781–795 (2021).

    Article  MathSciNet  Google Scholar 

  20. Mills W.H.: On the existence of H designs. Congr. Numer. 79, 129–141 (1990).

    MathSciNet  Google Scholar 

  21. Stern G., Lenz H.: Steiner triple systems with given subspaces, another proof of the Doyen-Wilson Theorem. Bull. Un. Mal. Ital. A 17, 109–114 (1980).

    MathSciNet  Google Scholar 

  22. Zhang X., Ge G.: Existence of resolvable H-designs with group sizes 2, 3, 4 and 6. Des. Codes Cryptogr. 55, 81–101 (2010).

    Article  MathSciNet  Google Scholar 

  23. Zhang X., Ge G.: H-designs with the properties of resolvability or (1,2)-resolvability. Des. Codes Cryptogr. 57, 225–256 (2010).

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments and suggestions.

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Authors and Affiliations

  1. College of Information Engineering, Shandong Youth University of Political Science, Jinan, 250103, People’s Republic of China

    Zhaoping Meng

  2. Smart Healthcare Big Data Engineering and Ubiquitous Computing Characteristic Laboratory in Universities of Shandong, Jinan, 250103, People’s Republic of China

    Zhaoping Meng

  3. School of Mathematics, Qilu Normal University, Jinan, 250103, People’s Republic of China

    Qingling Gao

  4. School of Electronic and Information Engineering, Fujian Polytechnic Normal University, Fuzhou, 350300, People’s Republic of China

    Zhanggui Wu

Authors
  1. Zhaoping Meng
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  2. Qingling Gao
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  3. Zhanggui Wu
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Correspondence to Zhaoping Meng.

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Communicated by C. J. Colbourn.

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Zhaoping Meng was supported by Shandong Provincial Natural Science Foundation Grant No. ZR2021MA057. Zhanggui Wu was supported by Education and Scientific Research Project for Young and Middle-aged Teachers of Fujian Province Grant No. JAT170685.

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Meng, Z., Gao, Q. & Wu, Z. A recursive construction of doubly resolvable Steiner quadruple systems. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623-024-01356-3

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  • DOI: https://doi.org/10.1007/s10623-024-01356-3

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