Latin squares with bounded size of row prefix intersections

A latin square is a matrix of size n×nn×n with entries from the set {1,…,n}{1,…,n}, such that each row and each column is a permutation on {1,…,n}{1,…,n}. We show how to construct a latin square such that for any two distinct rows, the prefixes of length hh of the two rows share at most about h2/nh2/n elements. This upper bound is close to optimal when contrasted with a lower bound derived from the Second Johnson bound [6].