Abstract
Linked systems of symmetric group divisible designs of type II are introduced, and several examples are obtained from affine resolvable designs and a variant of mutually orthogonal Latin squares. Furthermore, an equivalence between such symmetric group divisible designs and some association schemes with 5-classes is provided.
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Notes
If \(k=\lambda _1\), then its incidence matrix A satisfies that \(A=\bar{A}\otimes J_n\) for some incidence matrix \(\bar{A}\) of a symmetric design.
In [11] the theorems are valid under the assumption \(k>\lambda _1\). If \(k=\lambda _1\), then linked systems of symmetric group divisible designs are \(A_{i,j}\otimes J_n\) where \(A_{i,j}\)’s are linked systems of symmetric designs.
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Acknowledgements
The authors would like to thank the referees for many useful comments, especially for suggesting a change that significantly shortened the proof of Proposition 3.2. Hadi Kharaghani is supported by an NSERC Discovery Grant. Sho Suda is supported by JSPS KAKENHI Grant Number 15K21075, 18K03395.
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Kharaghani, H., Suda, S. Linked systems of symmetric group divisible designs of type II. Des. Codes Cryptogr. 87, 2341–2360 (2019). https://doi.org/10.1007/s10623-019-00622-z
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DOI: https://doi.org/10.1007/s10623-019-00622-z
Keywords
- Association scheme
- Symmetric group divisible design
- Hadamard matrix
- Affine resolvable design
- Mutually orthogonal Latin square