### A bound on correlation immunity.

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We consider two classes of latin squares that are prolongations of Cayley tables of finite abelian groups. We will show that all squares in the first of these classes are confirmed bachelor squares, squares that have no orthogonal mate and contain at least one cell though which no transversal passes, while none of the squares in the second class can be included in any set of three mutually orthogonal latin squares.

A latin trade is a subset of a latin square which may be replaced with a disjoint mate to obtain a new latin square. A $k$-homogeneous latin trade is one which intersects each row, each column and each entry of the latin square either $0$ or $k$ times. In this paper, we show that a construction given by Cavenagh, Donovan and Drápal for $3$-homogeneous latin trades in fact classifies every minimal $3$-homogeneous latin trade. We in turn classify all $3$-homogeneous latin trades. A corollary is that any $3$-homogeneous...

G. B. Belyavskaya and G. L. Mullen showed the existence of a complement for a $k$-tuple of orthogonal $n$-ary operations, where $k<n$, to an $n$-tuple of orthogonal $n$-ary operations. But they proposed no method for complementing. In this article, we give an algorithm for complementing a $k$-tuple of orthogonal $n$-ary operations to an $n$-tuple of orthogonal $n$-ary operations and an algorithm for complementing a $k$-tuple of orthogonal $k$-ary operations to an $n$-tuple of orthogonal $n$-ary operations. Also we find some...

Using results of Altshuler and Negami, we present a classification of biembeddings of symmetric configurations of triples in the torus or Klein bottle. We also give an alternative proof of the structure of 3-homogeneous Latin trades.

We describe, in a constructive way, a family of commutative loops of odd order, $n\ge 7$, which have no nontrivial subloops and whose multiplication group is isomorphic to the alternating group ${\mathcal{A}}_{n}$.

We review results for the embedding of orthogonal partial Latin squares in orthogonal Latin squares, comparing and contrasting these with results for embedding partial Latin squares in Latin squares. We also present a new construction that uses the existence of a set of $t$ mutually orthogonal Latin squares of order $n$ to construct a set of $2t$ mutually orthogonal Latin squares of order ${n}^{t}$.

Our purpose is to determine the complete set of mutually orthogonal squares of order d, which are not necessary Latin. In this article, we introduce the concept of supersquare of order d, which is defined with the help of its generating subgroup in $${\mathbb{F}}_{d}\times {\mathbb{F}}_{d}$$ . We present a method of construction of the mutually orthogonal supersquares. Further, we investigate the orthogonality of extraordinary supersquares, a special family of squares, whose generating subgroups are extraordinary. The extraordinary subgroups...