On the existence of 3-way k-homogeneous Latin trades

A μ-way Latin trade of volume s is a collection of μ partial Latin squares T1,T2,…,Tμ, containing exactly the same s filled cells, such that, if cell (i,j) is filled, it contains a different entry in each of the μ partial Latin squares, and such that row i in each of the μ partial Latin squares contains, set-wise, the same symbols, and column j likewise. It is called a μ-wayk-homogeneous Latin trade if, in each row and each column, Tr, for 1≤r≤μ, contains exactly k elements, and each element appears in Tr exactly k times. It is also denoted as a (μ,k,m) Latin trade, where m is the size of the partial Latin squares.We introduce some general constructions for μ-way k-homogeneous Latin trades, and specifically show that, for all k≤m, 6≤k≤13, and k=15, and for all k≤m, k=4,5 (except for four specific values), a 3-way k-homogeneous Latin trade of volume km exists. We also show that there is no (3,4,6) Latin trade and there is no (3,4,7) Latin trade. Finally, we present general results on the existence of 3-way k-homogeneous Latin trades for some modulo classes of m.