A lower bound on HMOLS with equal sized holes (Q2041120)

A holey Latin square of type \(h^n\) is an incomplete Latin square with an equipartition into \(n\) holes of a fixed size \(h\) of the set of symbols under consideration. Then, a set of HMOLS is a set of mutually orthogonal holey Latin squares of the same type. Let \(N(h^n)\) denote the maximum number of HMOLS of type \(h^n\). In this paper, for any \(\varepsilon>0\), the existence of a set of \(k>k_0(h,\varepsilon)\) HMOLS of type \(h^n\) is proved, whenever \(n\geq k^{(3+\varepsilon)\omega(h)k^2}\). Here, \(\omega(h)\) denotes the number of distinct prime factors of \(h\). Even if \(k_0(h,\epsilon)\) relies on some analytic number theory estimates, the proposed construction methods work for any value of \(k\). In this regard, the authors propose a direct construction of \(k\) HMOLS of type \(h^q\), for large prime powers \(q\). It is based on cyclotomic numbers in suitably large finite fields and on a natural extension of transversal designs to holey transversal designs. Then, this construction is recursively generalized for HMOLS with any number of holes. The authors also establish a lower bound on \(N(h^n)\) in terms of \(n\). More specifically, for any integer \(h\geq 2\) and any real number \(\delta>2\), they prove that \(N(h^n)\geq (\log n)^{1/\delta}\), for all \(n>n_0(h,\delta)\).