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: Ku, Cheng Yeaw, Wong, Kok Bin
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2021
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Online Access:https://hdl.handle.net/10356/150794
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Institution: Nanyang Technological University
Language: English
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spelling sg-ntu-dr.10356-1507942021-06-08T08:57:06Z On cross Parsons numbers Ku, Cheng Yeaw Wong, Kok Bin School of Physical and Mathematical Sciences Science::Mathematics Parsons Numbers Parsons Graphs 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). 2021-06-08T08:57:06Z 2021-06-08T08:57:06Z 2019 Journal Article Ku, C. Y. & Wong, K. B. (2019). On cross Parsons numbers. Graphs and Combinatorics, 35(1), 287-301. https://dx.doi.org/10.1007/s00373-018-1993-6 0911-0119 https://hdl.handle.net/10356/150794 10.1007/s00373-018-1993-6 2-s2.0-85057850676 1 35 287 301 en Graphs and Combinatorics © 2018 Springer Japan KK, part of Springer Nature. All rights reserved.
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic Science::Mathematics
Parsons Numbers
Parsons Graphs
spellingShingle Science::Mathematics
Parsons Numbers
Parsons Graphs
Ku, Cheng Yeaw
Wong, Kok Bin
On cross Parsons numbers
description 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).
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Ku, Cheng Yeaw
Wong, Kok Bin
format Article
author Ku, Cheng Yeaw
Wong, Kok Bin
author_sort Ku, Cheng Yeaw
title On cross Parsons numbers
title_short On cross Parsons numbers
title_full On cross Parsons numbers
title_fullStr On cross Parsons numbers
title_full_unstemmed On cross Parsons numbers
title_sort on cross parsons numbers
publishDate 2021
url https://hdl.handle.net/10356/150794
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