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Constructing linked systems of relative difference sets via Schur rings

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Abstract

In the present paper, we study relative difference sets (RDSs) and linked systems of them. It is shown that a closed linked system of RDSs is always graded by a group. Based on this result, we also define a product of RDS linked systems sharing the same grading group. Further, we generalize the Davis-Polhill-Smith construction of a linked system of RDSs. Finally, we construct new linked system of RDSs in a Heisenberg group over a finite field and family of RDSs in an extraspecial p-group of exponent \(p^2\). All constructions of new RDSs and their linked systems make usage of cyclotomic Schur rings.

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Notes

  1. In what follows we prefer to use the word “linked” instead of “linking”.

  2. We use here notation proposed in [17].

  3. Although our definition of closedness is formally different form the one given in [12], it is equivalent to the original.

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  1. Ben Gurion University of the Negev, Beer Sheva, Israel

    Mikhail Muzychuk & Grigory Ryabov

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  1. Mikhail Muzychuk
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  2. Grigory Ryabov
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M.M. and G.R. wrote the main manuscript text.

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Correspondence to Grigory Ryabov.

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Communicated by M. Buratti.

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The second author is supported by The Israel Science Foundation (project No. 87792731).

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Muzychuk, M., Ryabov, G. Constructing linked systems of relative difference sets via Schur rings. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623-024-01406-w

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