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Frame difference families and resolvable balanced incomplete block designs

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Abstract

Frame difference families, which can be obtained via a careful use of cyclotomic conditions attached to strong difference families, play an important role in direct constructions for resolvable balanced incomplete block designs. We establish asymptotic existences for several classes of frame difference families. As corollaries new infinite families of 1-rotational \((pq+1,p+1,1)\)-RBIBDs over \({\mathbb {F}}_{p}^+ \times {\mathbb {F}}_{q}^+\) are derived, and the existence of \((125q+1,6,1)\)-RBIBDs is discussed. We construct (v, 8, 1)-RBIBDs for \(v\in \{624,\) \(1576,2976,5720,5776,10200,14176,24480\}\), whose existence were previously in doubt. As applications, we establish asymptotic existences for an infinite family of optimal constant composition codes and an infinite family of strictly optimal frequency hopping sequences.

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References

  1. Abel R.J.R., Buratti M.: Some progress on \((v,4,1)\) difference families and optical orthogonal codes. J. Comb. Theory Ser. A 106, 59–75 (2004).

    Article  MathSciNet  Google Scholar 

  2. Abel R.J.R., Buratti M.: Difference families. In: Colbourn C.J., Dinitz J.H. (eds.) CRC Handbook of Combinatorial Designs, pp. 392–410. CRC Press, Boca Raton (2007).

    Google Scholar 

  3. Abel R.J.R., Cavenagh N.: Concerning eight mutually orthogonal Latin squares. J. Comb. Des. 15, 255–261 (2007).

    Article  MathSciNet  Google Scholar 

  4. Abel R.J.R., Ge G., Yin J.: Resolvable and near-resolvable designs. In: Colbourn C.J., Dinitz J.H. (eds.) CRC Handbook of Combinatorial Designs, pp. 124–132. CRC Press, Boca Raton (2007).

    Google Scholar 

  5. Arasu K.T., Bhandari A.K., Ma S.L., Sehgal S.: Regular difference covers. Kyungpook Math. J. 45, 137–152 (2005).

    MathSciNet  MATH  Google Scholar 

  6. Bao J., Ji L.: Frequency hopping sequences with optimal partial hamming correlation. IEEE Trans. Inf. Theory 62, 3768–3783 (2015).

    Article  MathSciNet  Google Scholar 

  7. Bose R.C.: On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements. Cal. Math. Soc. Golden Jubilee Commemoration Volume, Part II 341–354 (1958–1959).

  8. Buratti M.: Hadamard partitioned difference families and their descendants. arXiv:1705.04716v2.

  9. Buratti M.: On resolvable difference families. Des. Codes Cryptogr. 11, 11–23 (1997).

    Article  MathSciNet  Google Scholar 

  10. Buratti M.: Old and new designs via difference multisets and strong difference families. J. Comb. Des. 7, 406–425 (1999).

    Article  MathSciNet  Google Scholar 

  11. Buratti M.: Cyclic designs with block size 4 and related optimal optical orthogonal codes. Des. Codes Cryptogr. 26, 111–125 (2002).

    Article  MathSciNet  Google Scholar 

  12. Buratti M., Finizio N.: Existence results for 1-rotational resolvable Steiner 2-designs with block size 6 or 8. Bull. Inst. Comb. 50, 29–44 (2007).

    MathSciNet  MATH  Google Scholar 

  13. Buratti M., Gionfriddo L.: Strong difference families over arbitrary graphs. J. Comb. Des. 16, 443–461 (2008).

    Article  MathSciNet  Google Scholar 

  14. Buratti M., Pasotti A.: Combinatorial designs and the theorem of Weil on multiplicative character sums. Finite Fields Appl. 15, 332–344 (2009).

    Article  MathSciNet  Google Scholar 

  15. Buratti M., Zuanni F.: \(G\)-invariantly resolvable Steiner 2-designs arising from 1-rotational difference families. Bull. Belg. Math. Soc. 5, 221–235 (1998).

    MATH  Google Scholar 

  16. Buratti M., Yan J., Wang C.: From a 1-rotational RBIBD to a partitioned difference family. Electron. J. Comb. 17, R139 (2010).

    MathSciNet  MATH  Google Scholar 

  17. Buratti M., Costa S., Wang X.: New \(i\)-perfect cycle decompositions via vertex colorings of graphs. J. Comb. Des. 24, 495–513 (2016).

    Article  MathSciNet  Google Scholar 

  18. Cai H., Zhou Z., Yang Y., Tang X.: A new construction of frequency hopping sequences with optimal partial Hamming correlation. IEEE Trans. Inf. Theory 60, 5782–5790 (2014).

    Article  MathSciNet  Google Scholar 

  19. Chang Y., Ji L.: Optimal (4up,5,1) optical orthogonal codes. J. Comb. Des. 12, 346–361 (2004).

    Article  MathSciNet  Google Scholar 

  20. Chen K., Wei R., Zhu L.: Existence of (q,7,1) difference families with \(q\) a prime power. J. Comb. Des. 10, 126–138 (2002).

    Article  MathSciNet  Google Scholar 

  21. Costa S., Feng T., Wang X.: New 2-designs from strong difference families. Finite Fields Appl. 50, 391–405 (2018).

    Article  MathSciNet  Google Scholar 

  22. Ding C., Yin J.: Combinatorial constructions of optimal constant-composition codes. IEEE Trans. Inf. Theory 51, 3671–3674 (2005).

    Article  MathSciNet  Google Scholar 

  23. Drake D.A.: Partial \(\lambda \)-geometries and generalized Hadamard matrices over groups. Can. J. Math. 31, 617–627 (1979).

    Article  MathSciNet  Google Scholar 

  24. Fan P., Darnell M.: Sequence Design for Communications Applications. Wiley, London (1996).

    Google Scholar 

  25. Ge G., Miao Y.: PBDs, frames, and resolvability. In: Colbourn C.J., Dinitz J.H. (eds.) CRC Handbook of Combinatorial Designs, pp. 261–270. CRC Press, Boca Raton (2007).

    Google Scholar 

  26. Greig M., Abel R.J.R.: Resolvable balance incomplete block designs with block size 8. Des. Codes Cryptogr. 11, 123–140 (1997).

    Article  MathSciNet  Google Scholar 

  27. Lempel A., Greenberger H.: Families of sequences with optimal Hamming-correlation properties. IEEE Trans. Inf. Theory 20, 90–94 (1974).

    Article  MathSciNet  Google Scholar 

  28. Li S., Wei H., Ge G.: Generic constructions for partitioned difference families with applications: a unified combinatorial approach. Des. Codes Cryptogr. 82, 583–599 (2017).

    Article  MathSciNet  Google Scholar 

  29. Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1997).

    MATH  Google Scholar 

  30. Lu X.: Improving the existence bounds for grid-block difference families. Graphs Comb. 33, 549–559 (2017).

    Article  MathSciNet  Google Scholar 

  31. Luo Y., Fu F., Vinck A.J.H., Chen W.: On constant-composition codes over \(Z_q\). IEEE Trans. Inf. Theory 49, 3010–3016 (2003).

    Article  Google Scholar 

  32. Momihara K.: Strong difference families, difference covers, and their applications for relative difference families. Des. Codes Cryptogr. 51, 253–273 (2009).

    Article  MathSciNet  Google Scholar 

  33. Pavlidou N., Vinck A.J.H., Yazdani J., Honary B.: Power line communications: state of the art and future trends. IEEE Commun. Mag. 41, 34–40 (2003).

    Article  Google Scholar 

  34. Stinson D.R.: Combinatorial Designs: Constructions and Analysis. Springer, New York (2004).

    MATH  Google Scholar 

  35. Yang L., Giannakis G.B.: Ultra-wideband communications: an idea whose time has come. IEEE Signal Process. Mag. 21, 26–54 (2004).

    Article  Google Scholar 

  36. Zhou Z., Tang X., Wu D., Yang Y.: Some new classes of zero-difference balanced functions. IEEE Trans. Inf. Theory 58, 139–145 (2012).

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank Professor Marco Buratti of Università di Perugia for his many valuable comments.

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Authors and Affiliations

  1. Dipartimento DICATAM, Università degli Studi di Brescia, Via Valotti 9, 25123, Brescia, Italy

    Simone Costa

  2. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Tao Feng

  3. Department of Mathematics, Ningbo University, Ningbo, 315211, People’s Republic of China

    Xiaomiao Wang

Authors
  1. Simone Costa
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  2. Tao Feng
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  3. Xiaomiao Wang
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Corresponding author

Correspondence to Xiaomiao Wang.

Additional information

Communicated by M. Buratti.

Supported by NSFC under Grant 11471032, and Fundamental Research Funds for the Central Universities under Grant 2016JBM071, 2016JBZ012 (T. Feng), NSFC under Grant 11771227, and Zhejiang Provincial Natural Science Foundation of China under Grant LY17A010008 (X. Wang)

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Costa, S., Feng, T. & Wang, X. Frame difference families and resolvable balanced incomplete block designs. Des. Codes Cryptogr. 86, 2725–2745 (2018). https://doi.org/10.1007/s10623-018-0472-7

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  • DOI: https://doi.org/10.1007/s10623-018-0472-7

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