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.

In this paper, we propose a notion for high-dimensional generalizations of mutually orthogonal Latin squares (MOLS) and mutually orthogonal diagonal Latin squares (MODLS), called mutually dimensionally orthogonal d-cubes (MOC) and mutually dimensionally orthogonal diagonal d-cubes (MODC). Systematic constructions for MOC and MODC by using polynomials over finite fields are investigated. In particular, for 3-dimensional cubes, the results for the maximum possible number of MODC are improved by adopting the proposed construction.

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.

Orthogonal arrays have important applications in statistics and computer science, as well as in coding theory. In this letter, a new construction method of symmetric orthogonal arrays of strength t is proposed, which is a concatenation of two orthogonal partitions according to a latin square. As far as we know, this is a new construction of symmetric orthogonal arrays of strength t, where t is a given integer. Based on the different latin squares, we also study the enumeration problem of orthogonal partitions, and a lower bound on the count of orthogonal partitions is derived.

BLOCKSUM, also known as KEISANBLOCK in Japanese, is a Latin square filling type puzzle, such as Sudoku. In this paper, we prove that the decision problem whether a given instance of BLOCKSUM has a solution or not is NP-complete.

Combinatorial designs are normally used to construct visual cryptographic schemes. For such schemes two parameters are very important viz. pixel expansion and contrast. Optimizing both is a very hard problem. The schemes having optimal contrast tend to use a high pixel expansion. The focus of the paper is to construct schemes for which pixel expansion is modest and the contrast is close to optimality. Here the tool is latin squares that haven't been used earlier for this purpose.