Abstract
Let \(F_q\) be the field of size q and SL(n, q) be the special linear group of order n over the field \(F_q\). Assume that n is an even integer. Let \({\mathcal {A}}_i\subseteq SL(n,q)\) for \(i=1,2,\dots , k\) and \(\vert {\mathcal {A}}_1\vert =\vert \mathcal A_2\vert =\cdots =\vert {\mathcal {A}}_k\vert =l\). The set \(\{\mathcal A_1,{\mathcal {A}}_2,\dots ,{\mathcal {A}}_k\}\) is called a k-cross (n, q)-Parsons set of size l, if for any pair of (i, j) with \(i\ne j\), \(A-B\in SL(n,q)\) for all \(A\in {\mathcal {A}}_i\) and \(B\in {\mathcal {A}}_j\). Let m(k, n, q) be the largest integer l for which there is a k-cross (n, q)-Parsons set of size l. The integer m(k, n, q) will be called the k-cross (n, q)-Parsons numbers. In this paper, we will show that \(m(3,2,q)\le q\). Furthermore, \(m(3,2,q)= q\) if and only if \(q=4^r\) for some positive integer r. We will also show that if n is a multiple of \(q-1\), then \(m(q-1,n,q)\ge q^{\frac{1}{2}n(n-1)}\).
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Acknowledgements
We would like to thank the anonymous referees for the comments and suggestions that helped us make several improvements to this paper. This project is supported by University of Malaya Research Grant GPF025B-2018.
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Ku, C.Y., Wong, K.B. On Cross Parsons Numbers. Graphs and Combinatorics 35, 287–301 (2019). https://doi.org/10.1007/s00373-018-1993-6
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DOI: https://doi.org/10.1007/s00373-018-1993-6