Possible volumes of t-(v, t + 1) Latin trades

Details Possible volumes of t-(v, t + 1) Latin trades Possible volumes of t-(v, t + 1) Latin trades

2009-05-02

arxiv.org/abs/0905.0191v1

Mathematics

Combinatorics

12 pages

Scientific paper

The concept of $t$-$(v,k)$ trades of block designs previously has been studied in detail. See for example A. S. Hedayat (1990) and Billington (2003). Also Latin trades have been studied in detail under various names, see A. D. Keedwell (2004) for a survey. Recently Khanban, Mahdian and Mahmoodian have extended the concept of Latin trades and introduced $\Ts{t}{v}{k}$. Here we study the spectrum of possible volumes of these trades, $S(t,k)$. Firstly, similarly to trades of block designs we consider $(t+2)$ numbers $s_i=2^{t+1}-2^{(t+1)-i}$, $0\leq i\leq t+1$, as critical points and then we show that $s_i\in S(t,k)$, for any $0\leq i\leq t+1$, and if $s\in (s_i,s_{i+1}),0\leq i\leq t$, then $s\notin S(t,t+1)$. As an example, we determine S(3,4) precisely.

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