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

TL;DR

This paper investigates the possible volumes of extended Latin trades, identifying critical points and establishing the spectrum of volumes for certain parameters, with a detailed example for specific cases.

Contribution

It introduces the spectrum of volumes for extended Latin trades, extending previous concepts and providing precise results for particular parameter sets.

Findings

Identifies critical volume points for Latin trades.

Shows certain intervals contain no valid trade volumes.

Precisely determines the spectrum for S(3,4).

Abstract

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.