In this paper, we improve the existence results for holey self-orthogonal Latin squares with orthogonal mates (HSOLSSOMs), especially for 6 il. We able to construct three new SOLSSOMs of orders 46, and existence of which is previously unknown.

An orthogonal latin square graph is a graph whose vertices are latin squares of the same order, adjacency being synonymous with orthogonality. We are interested in orthogonal latin square graphs in which each square is orthogonal to the Cayley table M of a group G and is obtained from M by permuting columns. These permutations, regarded as permutations of G, are orthomorphisms of G and the grap...

In this paper we construct a class of regular Low Density Parity Check (LDPC) codes derived from Latin squares. The parity check matrices of these codes are constructed by permuting shift-orthogonal Latin squares of order n in blockrows and block-columns. I show that the constructed LDPC codes are self-orthogonal and their minimum and stopping distances are bounded. This helps us to construct a...

In an earlier paper, [A.D. Keedwell, Australas J. Combin. 47 (2010), 227–238], we proved that complete sets of orthogonal diagonal Sudoku latin squares exist of all orders p, where p is a prime. We also showed that complete sets of orthogonal Sudoku latin squares which are left semidiagonal exist of all orders p, s > 1, and we conjectured that these may be right semi-diagonal also but we were n...

The enumeration of self-orthogonal Latin squares (SOLS) of a given order seems to be an open problem in the literature on combinatorial designs. The existence of at least one SOLS is guaranteed for any order except 2, 3 and 6, but it is not known how many of these squares of a given order exist. In this talk we present enumeration tables of unequal SOLS, idempotent SOLS, isomorphism classes of ...

Maximal sets of s mutually orthogonal Latin squares of order v are constructed for in nitely many new pairs (s; v). c © 1999 Published by Elsevier Science B.V. All rights reserved

This paper investigates the existence of holey self-orthogonal Latin squares with a symmetric orthogonal mate of type 4u (briefly HSOLSSOM(4u)). For u > 0, the necessary conditions for existence of such an HSOLSSOM are (1) u must be even, and (2) u ≤ (4n−4)/3, and either (n, u) = (4, 4) or n ≥ 5. We show that these conditions are sufficient except possibly (1) for 36 cases with n ≤ 37, (2) for ...

Using the tools of algebraic coding theory, we give a new proof of the nonexistence of two mutually orthogonal Latin squares of order 6.

A latin square is a bachelor square if it does not possess an orthogonal mate; equivalently, it does not have a decomposition into disjoint transversals. We define a latin square to be a confirmed bachelor square if it contains an entry through which there is no transversal. We prove the existence of confirmed bachelor squares for all orders greater than three. This resolves the existence quest...