Abstract
In the previous work, we introduce a notion of pre-difference sets in a finite group G defined by weaker conditions than the difference sets. In this paper we gave a construction of a pre-difference set in \(G=NA\) with A an abelian subgroup and N a subgroup satisfying \(N\cap A=\{e\}\), from a difference set in \(N\times A\). This gives a (16, 6, 2) pre-difference set in \(D_{16}\) and a (27, 13, 6) pre-difference set in UT(3, 3), where no non-trivial difference sets exist. We also give a product construction of pre-difference sets similar to Kesava Menon construction, which provides infinite series of pre-difference sets that are not difference sets. We show some necessary conditions for the existence of a pre-difference set in a group with index 2 subgroup. For the proofs, we use a rather simple framework “relation partitions,” which is obtained by dropping an axiom from association schemes. Most results are proved in that frame work.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bannai, E., Ito, T.: Algebraic Combinatorics I: Association Schemes. Benjamin/Cummings, Calfornia (1984)
Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10, i–vi, 1–97 (1973)
Deng, Y.: A note on difference sets in dihedral groups. Arch. Math. (Basel) 82, 4–7 (2004)
Fan, C.T., Siu, M.K., Ma, S.L.: Difference sets in dihedral groups and interlocking difference sets. Ars Combin. 20.A, 99–107 (1985)
GAP: NTL: GAP—groups, algorithms, programming—a system for computational discrete algebra. https://www.gap-system.org/. Accessed 20 July 2020
Jungnickel, D., Schmidt, B.: Difference Sets: an Update. London Mathematical Society Lecture Note Series, pp. 89–112. Cambridge University Press, Cambridge (1997). https://doi.org/10.1017/CBO9780511526114.010
Kajiura, H., Matsumoto, M., Okuda, T.: Approximation of integration over finite groups, difference sets and association schemes. arXiv:1903.00697
Kibler, R.: A summary of noncyclic difference sets, $k< 20$. J. Combin. Theory Ser. A 25, 62–67 (1978)
Menon, P.K.: On difference sets whose parameters satisfy a certain relation. Proc. AMS 13, 739–745 (1962)
Shiu, W.: Difference sets in groups containing subgroups of index 2. Ars Combin. 42, 199–205 (1996)
Turyn, R.J.: Character sums and difference sets. Pac. J. Math. 15, 319–346 (1965)
van Lint, J.H., Wilson, R.M.: A Course in Combinatorics, 2nd edn. Cambridge University Press, Cambridge (2001)
Zieschang, P.H.: An Algebraic Approach to Association Schemes. Lecture Notes in Mathematics, vol. 1628. Springer, Berlin (1996)
Acknowledgements
Genealogy: The supervisors of Hiroki Kajiura are Makoto Matsumoto and Takayuki Okuda. Enomoto is a supervisor of Makoto Matsumoto, and Bannai is a supervisor of Takayuki Okuda. The authors sincerely congratulate Bannai and Enomoto in their 75th birth-year.
Funding
Hiroki Kajiura is partially supported by JSPS Grant-in-Aid for JSPS Research Fellows Grant number JP19J21207, and Makoto Matsumoto by JSPS Grants-in-Aid for Scientific Research JP26310211 and JP18K03213, and Takayuki Okuda by JP16K17594, JP16K05132, JP16K13749, JP20K14310, and JP26287012.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
Not applicable. Any other lists are not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kajiura, H., Matsumoto, M. & Okuda, T. Non-existence and Construction of Pre-difference Sets, and Equi-Distributed Subsets in Association Schemes. Graphs and Combinatorics 37, 1531–1544 (2021). https://doi.org/10.1007/s00373-021-02279-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-021-02279-9