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

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, Mandian and Mahmoodian have extended the concept of Latin trades and introduced t-(v, k) Latin trades. 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 <= i <= t + 1, as critical points and then we show that s(i) is an element of S(t, k), for any 0 <= i <= t + 1, and if s is an element of (s(i), s(i+1)), 0 <= i <= t, then s is not an element of S(t, t + 1). As an example, we determine S(3, 4) precisely.