The Mean King’s problem with mutually unbiased bases is reconsidered for arbitrary d-level systems. Hayashi, Horibe and Hashimoto [Phys. Rev. A 71, 052331 (2005)] related the problem to the existence of a maximal set of d − 1 mutually orthogonal Latin squares, in their restricted setting that allows only measurements of projection-valued measures. However, we then cannot find a solution to the ...

An antilattice is an algebraic structure based on the same set of axioms as a lattice except that two commutativity for ∧ and ∨ are replaced by anticommutative counterparts. In this paper, we study certain classes antilattices, including elementary (no nontrivial subantilattices), odd subantilattices order [Formula: see text]), simple congruences) irreducible (not expressible direct product). f...

Two Latin squares of order v are r-orthogonal if their superposition produces exactly r distinct ordered pairs. If the second square is the (3, 2, 1)-conjugate of the first one, we say that the first square is (3, 2, 1)conjugate r-orthogonal, denoted by (3, 2, 1)-r-COLS(v). The nonexistence of (3, 2, 1)-r-COLS(v) for r ∈ {v + 2, v + 3, v + 5} has been proved by Zhang and Xu [Int. J. Combin. Gra...

A latin square of order n is an n×n array of n symbols in which each symbol occurs exactly once in each row and column. A transversal of such a square is a set of n entries containing no pair of entries that share the same row, column or symbol. Transversals are closely related to the notions of complete mappings and orthomorphisms in (quasi)groups, and are fundamental to the concept of mutuall...

Using Hadamard matrices and mutually orthogonal Latin squares, we construct two new quasi-symmetric designs, with parameters 2 − (66, 30, 29) and 2− (78, 36, 30). These are the first examples of quasisymmetric designs with these parameters. The parameters belong to the families 2− (2u2−u, u2−u, u2−u−1) and 2− (2u2 +u, u2, u2−u) which are related to Hadamard parameters. The designs correspond to...

Wheel structures of the Orthogonal Latin Squares (OLS) polytope (PI) are presented in [2]. The current work focuses on the families of valid inequalities arising from wheels and proves that certain among them are facet-defining for PI . For two of these families we provide efficient separation procedures. We also present results regarding odd-hole inequalities, which essentially form a larger c...

Cycle switches are the simplest changes which can be used to alter latin squares, and as such have found many applications in the generation of latin squares. They also provide the simplest examples of latin interchanges or trades in latin square designs. In this paper we construct graphs in which the vertices are classes of latin squares. Edges arise from switching cycles to move from one clas...

Keywords: Self-orthogonal Latin square Strongly symmetric Magic square Yang Hui type a b s t r a c t In this paper, a strongly symmetric self-orthogonal diagonal Latin square of order n with a special property (* SSSODLS(n)) is introduced. It is proved that a * SSSODLS(n) exists if and only if n ≡ 0 (mod 4) and n ̸ = 4. As an application, it is shown that there exists a Yang Hui type magic squar...

We prove a conjecture by Garbe et al. [arXiv:2010.07854] showing that Latin square is quasirandom if and only the density of every pattern . This result best possible in sense cannot be replaced with or for any n.