On the structure of (v,k,t) trades
type: Item
instance of: scholarly article
title: On the structure of (v,k,t) trades
zbMATH Open document ID: 0593.62071
DOI: 10.1016/0378-3758(86)90131-X
author: H. L. Hwang
published in: Journal of Statistical Planning and Inference
publication date: 1986-01-01T00:00:00Z
review text: Let V be a set of v treatments. Then \(v\Sigma\) t is the collection of all t-subsets of V. Let k,t be positive integers such that \(t<k<v\). The elements of \(v\Sigma\) k are called blocks. A (v,k,t) trade of volume s consists of two disjoint collections \(T_ 1\) and \(T_ 2\) each consisting of s blocks such that for every element of \(v\Sigma\) t the number of blocks containing this element is the same in both \(T_ 1\) and \(T_ 2\). If s equals zero the trade is said to be void. The set of treatments covered by \(T_ 1\) is the same as that for \(T_ 2\) and this common set is called the foundation of the trade. The author motivates the study of the structure of (v,k,t) trades by observing that when a t-(v,k,\(\lambda)\) design D contains the collection \(T_ 1\) of a (v,k,t) trade, then replacing the blocks of \(T_ 1\) in D by the blocks in \(T_ 2\) of the trade yields another t-(v,k,\(\lambda)\) design. The following results are proved: (i) The minimum foundation size of a nonvoid (v,k,t) trade is \(k+t+1.\) (ii) For \(v\geq k+t+1\), the minimum volume of a nonvoid (v,k,t) trade is \(2^ t.\) (iii) The nonvoid (v,k,t) trades with the minimum foundation size \(k+t+1\) and minimum volume \(2^ t\) exist and have a unique structure. (iv) For any v,k and t, \(t\geq 2\), there is no (v,k,t) trade of volume \(2^ t+1\).
Mathematics Subject Classification ID: 62K10
Mathematics Subject Classification ID: 62K99
zbMATH DE Number: 3954091
zbMATH Keywords: t-designs
zbMATH Keywords: (v,k,t) trade
zbMATH Keywords: foundation of the trade
zbMATH Keywords: minimum volume
reviewed by: Hariharan K. Iyer
MaRDI profile type: Publication
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cites work: The family of t-designs. II
full work available at URL: 0378-3758(86)90131-x
OpenAlex ID: W2063211067
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description: scientific article; zbMATH DE number 3954091