Constructions of Pairs of Orthogonal Latin Cubes

A pair of orthogonal latin cubes of order $q$ is equivalent to an MDS code with distance $3$ or to an ${\rm OA}_1(3,5,q)$ orthogonal array. We construct pairs of orthogonal latin cubes for a sequence of previously unknown orders $q_i=16(18i-1)+4$ and $q'_i=16(18i+5)+4$. The minimal new obtained parameters of orthogonal arrays are ${\rm OA}_1(3,5,84)$. Keywords: latin square, latin cube, MOLS, MDS code, block design, Steiner system, orthogonal arrayComment: New pairs of orthogonal latin 3-cubes are available on the website https://ieee-dataport.org/open-access/graeco-latin-cube

Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results

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 $n^t$

Some new conjugate orthogonal Latin squares

AbstractWe present some new conjugate orthogonal Latin squares which are obtained from a direct method of construction of the starter-adder type. Combining these new constructions with earlier results of K. T. Phelps and the first author, it is shown that a (3, 2, 1)- (or (1, 3, 2)-) conjugate orthogonal Latin square of order v exists for all positive integers v ≠ 2, 6. It is also shown that a (3, 2, 1)- (or (1, 3, 2)-) conjugate orthogonal idempotent Latin square of order v exists for all positive integers v ≠ 2, 3, 6 with one possible exception v = 12, and this result can be used to enlarge the spectrum of a certain class of Mendelsohn designs and provide better results for problems on embedding

Mutually orthogonal latin squares with large holes

Two latin squares are orthogonal if, when they are superimposed, every ordered pair of symbols appears exactly once. This definition extends naturally to `incomplete' latin squares each having a hole on the same rows, columns, and symbols. If an incomplete latin square of order $n$ has a hole of order $m$, then it is an easy observation that $n \ge 2m$. More generally, if a set of $t$ incomplete mutually orthogonal latin squares of order $n$ have a common hole of order $m$, then $n \ge (t+1)m$. In this article, we prove such sets of incomplete squares exist for all $n,m \gg 0$ satisfying $n \ge 8(t+1)^2 m$

Existence of frame SOLS of type anb1

AbstractAn SOLS (self-orthogonal latin square) of order v with ni missing sub-SOLS (holes) of order hi (1⩽i⩽k), which are disjoint and spanning (i.e. ∑i=1knihi=v), is called a frame SOLS and denoted by FSOLS(h1n1h2n2 ⋯hknk). It has been proved that for b⩾2 and n odd, an FSOLS(anb1) exists if and only if n⩾4 and n⩾1+2b/a. In this paper, we show the existence of FSOLS(anb1) for n even and FSOLS(an11) for n odd

Embedding partial Latin squares in Latin squares with many mutually orthogonal mates

In this paper it is shown that any partial Latin square of order $n$ can be embedded in a Latin square of order at most $16n^2$ which has at least $2n$ mutually orthogonal mates. Further, for any $t\geq 2$, it is shown that a pair of orthogonal partial Latin squares of order $n$ can be embedded in a set of $t$ mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to $n$. A consequence of the constructions is that, if $N(n)$ denotes the size of the largest set of MOLS of order $n$, then $N(n^2)\geq N(n)+2$. In particular, it follows that $N(576)\ge 9$, improving the previously known lower bound $N(576)\ge 8$

On the number of latin hypercubes, pairs of orthogonal latin squares and MDS codes

Abstract The logarithm of the number of latin d-cubes of order n is Θ(n d ln n). The logarithm of the number of pairs of orthogonal latin squares of order n is Θ(n 2 ln n). Similar estimations are obtained for systems of mutually strong orthogonal latin d-cubes

Group Divisible Codes and Their Application in the Construction of Optimal Constant-Composition Codes of Weight Three

The concept of group divisible codes, a generalization of group divisible designs with constant block size, is introduced in this paper. This new class of codes is shown to be useful in recursive constructions for constant-weight and constant-composition codes. Large classes of group divisible codes are constructed which enabled the determination of the sizes of optimal constant-composition codes of weight three (and specified distance), leaving only four cases undetermined. Previously, the sizes of constant-composition codes of weight three were known only for those of sufficiently large length.Comment: 13 pages, 1 figure, 4 table