Non-existence and Construction of Pre-difference Sets, and Equi-Distributed Subsets in Association Schemes

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.

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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.

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Correspondence to Makoto Matsumoto.

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Kajiura, H., Matsumoto, M. & Okuda, T. Non-existence and Construction of Pre-difference Sets, and Equi-Distributed Subsets in Association Schemes. Graphs and Combinatorics (2021). https://doi.org/10.1007/s00373-021-02279-9

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Keywords

  • Difference set
  • Group
  • Pre-difference set
  • Association Scheme
  • Relation-partition
  • Equi-distributed subset

Mathematics Subject Classification

  • 05B10 Difference sets
  • 05E30 Association schemes
  • Strongly regular graphs
  • 20D60 Arithmetic and combinatorial problems