Embedding partial Latin squares in Latin squares with many mutually orthogonal mates

Abstract

We show that any partial Latin square of order n can be embedded in a Latin square of order at most 16n 2 which has at least 2n mutually orthogonal mates. We also show that for any t 2, a pair of orthogonal partial Latin squares of order n can be embedded into a set of t mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to n. Furthermore, the constructions that we provide show that MOLS(n 2 ) MOLS(n)+2, consequently we give a set of 9 MOLS(576). The maximum known size of a set of MOLS(576) was previously given as 8 in the literature.

Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results

“…Furthermore if P is idempotent, then B can be constructed to be idempotent. In [16] Donovan, Grannell and Yazıcı compared these results to the result given by Barber et.al. in [4] further interpreting Theorem 4.2 which states that, for any s ∈ N, there exists k 0 ∈ N such that for any n ∈ N, any set of s-MOPLS(n) can be embedded in a set of s-MOLS(m), for every m k 0 n. That there is such a k 0 is an important existence result because it gives a linear order embedding.…”

“…For s 2, the proof given in [4] requires that k 0 > 10 7 (s + 2) 3 /9 and, being an existence result, there is little information about the structure of the resulting set of MOLS. For s = 2 and small n, certainly n 113 and possibly much larger, [16] gives a tighter embedding than that of [4], and it more closely specifies the structure of the resulting pair of MOLS.…”

“…Theorem 4.10. [15,16] Let P and Q be a pair of MOPLS(n). Then P and Q can be embedded in a pair of MOLS(k 4 ) and any order greater than or equal to 3k 4 where 2 a = k 2n > 2 a−1 for some integer a.…”

“…More recently, Donovan, Grannell and Yazıcı [16] have capitalized on these techniques to develop a construction for embedding a PLS(n) in an Latin square which has many orthogonal mates, as well as embedding a pair of MOPLS(n) in a set of many MOLS. While we state the results here, we will leave a fuller description of the methods to Section 5 where we give new generalizations of these constructions.…”

“…Theorem 4.11. [16] Let P be a PLS(t), t 3. Then P can be embedded in B, an LS(n) with n 16t 2 , which belongs to a set of at least 2t MOLS(n 2 ).…”