Latin squares are a classical and well-studied topic of discrete mathematics, and recently Takeuti and Adachi (IACR ePrint, 2023) proposed (2, n)-threshold secret sharing based on mutually orthogonal Latin squares (MOLS). Hence efficient constructions of as large sets of MOLS as possible are also important from practical viewpoints. In this letter, we determine the maximum number of MOLS among a known class of Latin squares defined by weighted sums. We also mention some known property of Latin squares interpreted via the relation to secret sharing and a connection of Takeuti-Adachi’s scheme to Shamir’s secret sharing scheme.

Orthogonal arrays and orthogonal partitions have great significance in communications and coding theory. In this letter, by using a generalized orthogonal partition, Latin squares and orthogonal Latin squares, we present an iterative construction method of orthogonal arrays of strength t and orthogonal partitions. As an application of the method, more orthogonal arrays of strength t and orthogonal partitions than the existing methods can be constructed.

The orthogonal array is an important object in combinatorial design theory, and it is applied to many fields, such as computer science, coding theory and cryptography etc. This paper mainly studies the existence of the mixed orthogonal arrays of strength two with seven factors and presents some new constructions. Consequently, a few new mixed orthogonal arrays are obtained.