Abstract
We give two new constructions of almost difference sets. The first is a generic construction of \((q^{2}(q+1),\,q(q^{2}-1),\,q(q^{2}-q-1),\,q^{2}-1)\) almost difference sets in certain groups of order \(q^{2}(q+1)\) (q is an odd prime power) having (\(\mathbb {F}_{q^{2}},\,+)\) as a subgroup. This construction yields several infinite families of almost difference sets, many of which occur in nonabelian groups. The second construction yields \((4p,\,2p+1,\,p,\,p-1)\) almost difference sets in dihedral groups of order 4p where \(p\equiv 3 \ (\mathrm{mod} \ 4)\) is a prime. Moreover, it turns out that some of the infinite families of almost difference sets obtained produce Cayley graphs which are Ramanujan graphs.
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Acknowledgements
The authors would like to thank Professor Qing Xiang for his suggestions which led to some of the work in this manuscript. We would also like to thank the anonymous referees for their valuable comments which helped to improve this paper.
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Communicated by K. T. Arasu.
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The authors were supported by the National Science Foundation of China under Grant Nos. 11601220 and 61672015.
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Michel, J., Wang, Q. Almost difference sets in nonabelian groups. Des. Codes Cryptogr. 87, 1243–1251 (2019). https://doi.org/10.1007/s10623-018-0519-9
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DOI: https://doi.org/10.1007/s10623-018-0519-9