Superimposed codes is a special combinatorial structure that has many applications in information theory, data communication and cryptography. On the other hand, mutually orthogonal latin squares is a beautiful combinatorial object that has deep connection with design theory. In this paper, we draw a connection between these two structures. We give explicit construction of mutually orthogonal l...

We consider a pair of MOLS (mutually orthogonal Latin squares) having holes, corresponding to missing sub-MOLS, which are disjoint and spanning. If the two squares are mutual transposes, we say that we have SOLS (self-orthogonal Latin squares) with holes. It is shown that a pair of SOLS with n holes of size h ≥ 2 exist if and only if n ≥ 4 and it is also shown that a pair of SOLS with n holes o...

To date very few results are known on the critical sets for a set of Mutually Orthogonal Latin Squares(MOLS). In this paper, we consider Orthogonal Array OA(n, k + 2, n, 2) constructed from k mutually orthogonal cyclic latin squares of order n and obtain bounds on the possible sizes of the minimal critical sets. In particular, for n = 7 we exhibit a critical set, thereby improving the bound rep...

We present the numbers of isotopy classes and main classes of Latin squares, and the numbers of isomorphism classes of quasigroups and loops, up to order 10. The best previous results were for Latin squares of order 8 (Kolesova, Lam and Thiel, 1990), quasigroups of order 6 (Bower, 2000) and loops of order 7 (Brant and Mullen, 1985). The loops of order 8 have been independently found by “QSCGZ” ...