In this paper, we introduce a generalization of frames called partial resolution squares. We are interested in constructing sets of complementary partial resolution squares for Steiner triple systems. Our main result is the existence of six complementary partial resolution squares for Steiner triple systems of order v which can be superimposed in a v × v array so that the resulting array is als...

Mutually unbiased bases encapsulate the concept of complementarity—the impossibility of simultaneous knowledge of certain observables—in the formalism of quantum theory. Although this concept is at the heart of quantum mechanics, the number of these bases is unknown except for systems of dimension being a power of a prime. We develop the relation between this physical problem and the mathematic...

An SOLS (self orthogonal Latin square) of order v with ni missing subSOLS (holes) of order hi (1 ≤ i ≤ k), which are disjoint and spanning (i.e. ∑k i=1 nihi = v), is called a frame SOLS and denoted by FSOLS(h n1 1 h n2 2 · · ·hk k ). In this paper, we prove that: if h ≥ 7, m ≥ 4, n ≥ 4 and n = 6, then there exists an FSOLS(h1); an FSOLS(h1) exists if h ≥ 7, h = 8, 12, 28 and m ≥ 17.

Suppose that n is even and a set of n 2 − 1 mutually orthogonal Latin squares of order n exists. Then we can construct a strongly regular graph with parameters (n, n 2 (n−1), n 2 ( 2 −1), n 2 ( 2 −1)), which is called a Latin square graph. In this paper, we give a sufficient condition of the Latin square graph for the existence of a projective plane of order n. For the existence of a Latin squa...

We give a construction for a new family of Group Divisible Designs (6s+ 2, 3, 4; 1, 2) using Mutually Orthogonal Latin Squares for all positive integers s. Consequently, we have proved that the necessary conditions are sufficient for the existence of GDD’s of block size four with three groups, λ1=2 and λ2=1.

We report on the complete computer search for a strongly regular graph with parameters (36,15,6,6) and chromatic number six. The result is that no such graph exists.

Two Latin squares of order n are r-orthogonal if their superposition produces exactly r distinct ordered pairs. If the second square is the transpose of the 3rst one, we say that the 3rst square is r-self-orthogonal, denoted by r-SOLS(n). It has been proved that the necessary condition for the existence of an r-SOLS(n) is n6 r6 n and r ∈ {n + 1; n − 1}. Zhu and Zhang conjectured that there is a...