## 1. INTRODUCTION

In this article, we display five mutually orthogonal Latin squares (MOLS) of orders 18 and 60. For v=60, these come from a (60, 6, 1) difference matrix, while for v=18, they will come from an (18, 7, 2) difference matrix with a few extra properties that enable us to convert it to a structure with index 1.

An *orthogonal array*, OAλ(k,v) is a k×λv2 array A=(ai,j) with entries from a set *V* of size *v* such that for any two rows i1,i2 of *A* and any x,y∈V there are exactly λ columns, *j* such that ai1,j=x and ai2,j=y. The parameter λ can be omitted if it equals 1.

Existence of an OA(k,v) is equivalent to existence of k−2 mutually Latin squares (MOLS) of order *v*, denoted as k−2 MOLS(*v*). If A1,A2,...,Ak−2 are k−2 MOLS(*v*) with entries from the set Iv={1,2,...,v}, then the *v*^{2} columns (i,j,A1(i,j),A2(i,j), ...,Ak−2(i,j))T (i,j=1,2,...,v) form an OA(k,v); this process is also reversible.

If *G* is an Abelian group of order *g*, then a (g,k,λ) *difference matrix* over *G* is a k×λg array *D* with entries from *G*, such that for any distinct i1,i2∈{1,2,...,k}, the multiset {di1,j−di2,j:j=1,2,...,λg} contains each element of *G* exactly λ times.

Several known MOLS of small orders have been obtained from difference matrices. A (g,k,1) difference matrix has *g* columns. If *D* is a (g,k,1) difference matrix over *G*, then the *g*^{2} columns a+C (a∈G, *C* a column of *D*) give an OA(k,g). This array can also be extended to an OA(k+1,g): if Ci is the *ith* column of *D*, then adding an extra row, containing gi in all columns of the form a+Ci (a∈G) gives an OA(k+1,g), and hence also k−1 MOLS(*g*).

## 2. TWO NEW DIRECT CONSTRUCTIONS

An example of a new difference matrix, which gives five new MOLS of order 60, is a (60, 6, 1) difference matrix given below. This matrix is over the group Z2×Z30. Ten initial columns of this matrix are given in the array *B* below. Sixty columns are obtained by replacing each column (a,b,c,d,e,f)T in this array by the six columns in the array *C* below. These 60 columns form the required (60, 6, 1) difference matrix.

B=(0,0)(0,0)(0,0)(0,0)(0,0)(0,0)(0,0)(0,0)(0,0)(0,0)(1,10)(1,6)(0,17)(0,7)(1,5)(0,9)(0,3)(1,13)(1,17)(0,13)(1,22)(1,1)(1,8)(0,9)(1,21)(1,29)(1,0)(0,2)(0,12)(1,15)(1,24)(1,1)(0,14)(0,0)(0,16)(0,18)(0,8)(0,28)(0,17)(0,7)(0,17)(0,7)(0,20)(0,1)(1,4)(0,26)(0,19)(0,28)(1,21)(0,6)(1,14)(1,9)(0,10)(0,27)(1,20)(0,11)(0,13)(1,12)(0,28)(1,18),C=acb−d−e−fbac−e−f−dcba−f−d−edef−a+(1,0)−c+(1,0)−b+(1,0)efd−b+(1,0)−a+(1,0)−c+(1,0)fde−c+(1,0)−b+(1,0)−a+(1,0). Note that the differences between pairs of points in the column (a,b,c,d,e,f)T are of four types, which we call types 1, 2, 3, and 4. Differences of type 1 are those between rows 1 and 2 in the difference matrix, while differences of types 2, 3, and 4 are differences between rows 1 and 4, 1 and 5, and 1 and 6, respectively. The differences of each of these types are as follows:

- Type 1: b−a, c−b, a−c, d−e, e−f, f−d.
- Type 2: d−a, f−b, e−c.
- Type 3: e−a, d−b, f−c.
- Type 4: f−a, e−b, d−c.

In addition if *x* is a difference of type 2, 3, or 4, then x+(1,0) is also, due to the structure of the matrix *C*. Thus adding (1, 0) to any point in any of the ten initial columns changes only the type 1 differences. Our approach to find this (60, 6, 1) difference matrix was to first find by computer the (30, 6, 2) difference matrix over *Z*_{30} (obtained by ignoring the first (*Z*_{2}) coordinate of each point). Suitable *Z*_{2} coordinates could then be found by hand calculation (this was not difficult, since as already mentioned, changing the *Z*_{2} coordinate of one or more points in any of the ten initial columns changes only the type 1 differences).

We now turn our attention to finding five MOLS(18). In [6], a new method was used to obtain four MOLS(14), or an OA(6,14) from a difference matrix of index larger than 1. Suppose G1,G2 are groups of orders g1,g2, G=G1×G2, and *D* is a (g1g2,k,g1) difference matrix over *G*. Suppose also, *D* possesses the extra property that for any two rows i1,i2 of *D*, any z∈G1 and any (x,y) in G=G1×G2, there is exactly one column *j* of *D* such that both D(i1,j)−D(i2,j)=(x,y) and D(i1,j) has first coordinate *z* (from *G*_{1}). When this occurs, developing *D* over *G*_{2} (but not over *G*_{1}) gives an OA(k,g1g2).

In [6], by taking g1=2, g2=7, Todorov obtained a (14, 6, 2) difference matrix with this property (and hence also an OA(6, 14) or four MOLS(14)). Here, Todorov used an exhaustive search, although he was able to discard matrices that gave orthogonal arrays “equivalent” to those already considered. Here instead, by taking g1=2, g2=9 we start with a (2, 7, 6) difference matrix *D*_{1} possessing the extra property in the previous paragraph, that is, for any x1,x2∈Z2, and any two rows i1,i2, there are three columns *j* in *D*_{1} such that D1(i1,j)=x1 and D1(i2,j)=x2. Let these three columns be c1,x1,x2,i1,i2, c2,x1.x2,i1,i2, and c3,x1,x2,i1,i2.

We next use a method similar to that used to obtain eight MOLS(36) in [4]. Here, we choose a vector *T* with seven entries, each from Z3×Z3; and finally, by computer, we have to try to find a 7 × 12 array *D*_{2} with entries from Z3×Z3 such that for any two rows i1,i2, and any x1,x2∈Z2, adding 0, T(i1)−T(i2) and 2(T(i1)−T(i2)) to the three values in the set {D2(i1,j)−D2(i2,j):j∈{c1,x1,x2,i1,i2,c2,x1.x2,i1,i2,c3,x1,x2,i1,i2} produces a nine-element set containing each element of Z3×Z3 once.

Finding a (2, 7, 6) difference matrix with the required property is no problem; see the first symbols in the 7 × 12 array *A* below. In fact, a (2, 11, 6) difference matrix with this property can be obtained by adding a column of zeros to the incidence matrix of a symmetric (11, 6, 3) BIBD or balanced incomplete block design. (More generally, adding a column of zeros to the incidence matrix of a symmetric (4n−1,2n,n) BIBD will give a (2,4n−1,2n) difference matrix with a similar property.) For our (2, 7, 6) difference matrix, we used the first seven rows of a cyclic (11, 6, 3) BIBD with initial block {0,1,3,4,5,9}.

We tried 45 choices for the vector *T*, but most of them together with the chosen (2, 7, 6) difference matrix *D*_{1} gave no solution for the matrix *D*_{2} over Z3×Z3. Only two gave a solution; one of these is given below. When *D*_{1} and *T* are fixed, a few restrictions on the entries enable us to carry out an exhaustive search for *D*_{2}. These are as follows:

- All entries in the first row and first column of
*D*_{2} are (0, 0). - The sum of all entries in each row of
*D*_{2} is (0, 0). - For each row
*i* of *D*_{2}, if ji is the first column *j* such that D1(i,j)=1, then D2(i,ji)=(0,0). This last restriction is possible because if we have a suitable matrix *D*_{2}, then for any *i*, adding a constant to all values D2(i,j) for which D1(i,j)=1 also produces a suitable matrix *D*_{2}. - In the second row of
*D*_{2}, all entries have second coordinate zero. This restriction is possible since any column *c* of *D*_{2} can be replaced by c+T, c+2T, and (since the second entry in column *T* was chosen as (0, 1)), the second entry in one of *c*, c+T, c+2T will have second coordinate zero.

We now display a suitable array *A* with entries (D1(i,j),D2(i,j)). For convenience, these entries are written as xyz where x∈Z2 and y,z∈Z3. Add the following three vectors to each of the 12 columns of this array *A*: ((0,0,0), (0, 0, *y*), (0, *y*, 0), (0,2y,y), (0,2y,2y), (0,y,2y), (0,0,2y))T for 0≤y≤2. That is, we have taken the vector *T* to be ((0, 0), (0, 1), (1, 0), (2, 1), (2, 2), (1, 2), (0, 2))^{T}. The resulting 7 × 36 array obtained is the required (18, 7, 2) difference matrix.

A=000100100000100100100000000000100000000020100100000120110110010020010120000100022102112001101120121001020002000002100002102122010111110121021001000021000100020112100021112102102012000000011010100010110122011121120111000100002022011121020122100010112112. It is natural to ask whether for k=8 or 9, an (18, *k*, 2) difference matrix with similar properties might be constructible, giving six or seven MOLS of order 18. However, only a small proportion of the choices tried for *D*_{1} and *T* (2 in 45) led to even a suitable (18, 7, 2) difference matrix. It would thus appear this task, even if possible, is likely to be quite difficult.

## 3. FOUR AND FIVE IDEMPOTENT MOLS

A transvsersal in a Latin square of order *v* is a set of *v* cells, one from each row and column, containing each symbol in the Latin square exactly once. k−2 MOLS(*v*) are said to be idempotent if they possesss a common transversal. This is equivalent to saying that in the corresponding OA(k,v), there is there is a set of *v* columns containing *v* different elements in each row.

The author would like to thank the referees for pointing out that the five MOLS(18) obtained in the previous section are not idempotent. More generally, the second referee showed there are at most 15 columns in the associated OA(7, 18) whose elements in each row are all distinct.

It also appears unlikely that the five MOLS(60) obtained can be made idempotent. The first referee found 43 columns in the associated OA(7, 60) with no repeated element in any row, but did not find more, although this search was incomplete due to computation time. In fact, there are just five values of *v* (v=58,62,66,68,74) for which five idempotent MOLS(*v*) are known, but six MOLS(*v*) are not. Five MOLS(*v*), but not five idempotent MOLS(*v*) are known for v=12,18,21,28,33,35,39,42,44,51,52,54,60, and all but three of these (v=18,42,54) come from (*v*, 6, 1) difference matrices.

We also point out that any set k−1 MOLS(*v*) obtained from a (v,k,1) difference matrix cannot be made idempotent unless the (v,k,1) difference matrix is extendable to a (v,k+1,1) difference matrix. The columns of the OA(k+1,v) obtained from a (v,k,1) difference matrix over an abelian group *G* are obtained by adding each element of *G* to each of the *v* columns of difference matrix, and then adding a (k+1)th row, in which two columns contain the same element if and only if they are generated by the same column of the difference matrix. Thus, if this OA contains *v* columns with *v* distinct entries in each row, then these *v* columns must all be generated by different columns in the different matrix. If this occurs, delete the (k+1)th row from these *v* columns and add an extra row, containing the identity element of *G* in each column. The result is a (v,k+1,1) difference matrix over *G*.

On the other hand, there is no *v* (except v=5) for which four MOLS(*v*) are known and four idempotent MOLS(*v*) are not. There are two known sets of four idempotent MOLS not mentioned in Table 3.88 of [3], namely for v=14,26. Those for v=14 were given in [6] while those for v=26 come from a quasi-difference matrix over *Z*_{25}, displayed in [2] (for more information on quasi-difference matrices, including a definition, see [5]). Any column of this quasi-difference matrix with no blank entry will give 25 columns in the associated OA(6, 26) with 25 distinct non-infinite elements in each row; there is also a 26th column containing an infinite element in each row. We also point out that a construction for four idempotent MOLS(30), which was not given directly in [3], can be found in [1].

For t∈{4,5}, the following theorem summarizes the known results for existence of *t* MOLS(*v*) and *t* idempotent MOLS(*v*), updating the results given in [3].

Theorem 3.1. - There exist four MOLS(
*v*) for all integers v>4, except for v=6 and possibly for v∈{10, 22}. - There exist four idempotent MOLS(
*v*) for all integers v>6, except possibly for v∈{10, 22}. - There exist five MOLS(
*v*) for all integers v>6, except possibly for v∈{10, 14, 15, 20, 22, 26, 30, 34, 38, 46}. - There exist five idempotent MOLS(
*v*) for all integers v>6, except possibly for v∈{10, 12, 14, 15, 18, 20, 21, 22, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 46, 51, 52, 54, 60}.