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

Abstract

The concept of \(t\)-(v, \(\lambda\)) trades of block designs has been studied in detail. See, for example, A.~S. Hedayat (1990) and Billington (2003). Latin trades have also been extensively studied 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 \(t\)-(\(v, k\)) Latin trades.In this paper, 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 \(s_i = 2^{i+1}-2^{(t+1)-i} \), \(0 \leq i \leq t+1\), as critical points. 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 precisely determine \(S(3, 4)\).