On cross Parsons numbers
Let Fq be the field of size q and SL(n, q) be the special linear group of order n over the field Fq. Assume that n is an even integer. Let Ai⊆SL(n,q) for i=1,2,…,k and |A1|=|A2|=⋯=|Ak|=l. The set {A1,A2,…,Ak} is called a k-cross (n, q)-Parsons set of size l, if for any pair of (i, j) with i≠j, A−B∈S...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/150794 |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | Let Fq be the field of size q and SL(n, q) be the special linear group of order n over the field Fq. Assume that n is an even integer. Let Ai⊆SL(n,q) for i=1,2,…,k and |A1|=|A2|=⋯=|Ak|=l. The set {A1,A2,…,Ak} is called a k-cross (n, q)-Parsons set of size l, if for any pair of (i, j) with i≠j, A−B∈SL(n,q) for all A∈Ai and B∈Aj. 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)≤q. Furthermore, m(3,2,q)=q if and only if q=4r for some positive integer r. We will also show that if n is a multiple of q−1, then m(q−1,n,q)≥q12n(n−1). |
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