Abstract
In this paper, by employing Weil’s Theorem on multiplicative character sums, an intermediate algebraic consequence on the existence bound of an element satisfying certain cyclotomic conditions in a finite field is proposed. In many cases, this approach improves the bound due to Buratti and Pasotti (Finite Fields Appl 15(3), 332–344, 2009), which can be widely used for showing the asymptotic existence of combinatorial designs with point-regular automorphisms. Moreover, this approach is applied to improving the existence bound for grid-block difference families, which can be regarded as generalizations of difference families.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abel, R.J.R., Buratti, M.: Difference families. In: Colbourn, C.J., Dinitz, J.H. (eds.) Handbook of Combinatorial Designs, vol. Chap. VI. 16, 2nd edn, pp. 392–410. CRC Press, Boca Raton (2006)
Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)
Buratti, M., Pasotti, A.: Combinatorial designs and the theorem of Weil on multiplicative character sums. Finite Fields Appl. 15(3), 332–344 (2009)
Chen, K., Wei, R., Zhu, L.: Existence of \((q, 7, 1)\) difference families with \(q\) a prime power. J. Comb. Des. 10(2), 126–138 (2002)
Chen, K., Zhu, L.: Existence of \((q, 6, 1)\) difference families with \(q\) a prime power. Des. Codes Cryptogr. 15(2), 167–173 (1998)
Chen, K., Zhu, L.: Improving Wilson’s bound on difference families. Util. Math. 55, 189–200 (1999)
Du, D.Z., Hwang, F.: Pooling Designs and Nonadaptive Group Testing: Important Tools for DNA Sequencing. World Scientific, Singapore (2006)
Fu, H.L., Hwang, F., Jimbo, M., Mutoh, Y., Shiue, C.: Decomposing complete graphs into \(K_r \times K_c\)’s. J. Stat. Plan. Inference 119(2), 225–236 (2004)
Lamken, E.R., Wilson, R.M.: Decompositions of edge-colored complete graphs. J. Comb. Theory Ser. A 89(2), 149–200 (2000)
Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (1997)
Lu, X.N., Satoh, J., Jimbo, M.: Grid-block difference families and related combinatorial structures. Manuscript submitted for publication (2016)
Wang, C., Colbourn, C.J.: The existence of \((K_2 \times K_6)\)-designs. Graphs Comb. 29(5), 1557–1567 (2013)
Wilson, R.M.: Cyclotomy and difference families in elementary abelian groups. J. Number Theory 4(1), 17–47 (1972)
Acknowledgements
This work was supported in part by JSPS under Grant-in-Aid for JSPS Fellows No. 26011700.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lu, XN. Improving the Existence Bounds for Grid-Block Difference Families. Graphs and Combinatorics 33, 549–559 (2017). https://doi.org/10.1007/s00373-017-1776-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-017-1776-5