Existence of Five MOLS of Orders 18 and 60

Authors


Abstract

In this article, we provide direct constructions for five mutually orthogonal Latin squares (MOLS) of orders math formula and 60. For math formula, these come from a new (60, 6, 1) difference matrix. For math formula, the required construction is obtained by combining two different methods that were used in the constructions of four MOLS(14) and eight MOLS(36).

1. INTRODUCTION

In this article, we display five mutually orthogonal Latin squares (MOLS) of orders 18 and 60. For math formula, these come from a (60, 6, 1) difference matrix, while for math formula, 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, math formula is a math formula array math formula with entries from a set V of size v such that for any two rows math formula of A and any math formula there are exactly λ columns, j such that math formula and math formula. The parameter λ can be omitted if it equals 1.

Existence of an OAmath formula is equivalent to existence of math formula mutually Latin squares (MOLS) of order v, denoted as math formula MOLS(v). If math formula are math formula MOLS(v) with entries from the set math formula, then the v2 columns math formula math formula math formula form an OAmath formula; this process is also reversible.

If G is an Abelian group of order g, then a math formula difference matrix over G is a math formula array D with entries from G, such that for any distinct math formula, the multiset math formula contains each element of G exactly λ times.

Several known MOLS of small orders have been obtained from difference matrices. A (math formula difference matrix has g columns. If D is a math formula difference matrix over G, then the g2 columns math formula (math formula, C a column of D) give an OAmath formula. This array can also be extended to an OAmath formula: if math formula is the ith column of D, then adding an extra row, containing math formula in all columns of the form math formula math formula gives an OAmath formula, and hence also math formula 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 math formula. Ten initial columns of this matrix are given in the array B below. Sixty columns are obtained by replacing each column math formula in this array by the six columns in the array C below. These 60 columns form the required (60, 6, 1) difference matrix.

display math

Note that the differences between pairs of points in the column math formula 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: math formula, math formula, math formula, math formula, math formula, math formula.
  •  Type 2: math formula, math formula, math formula.
  •  Type 3: math formula, math formula, math formula.
  •  Type 4: math formula, math formula, math formula.

In addition if x is a difference of type 2, 3, or 4, then math formula 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 Z30 (obtained by ignoring the first (Z2) coordinate of each point). Suitable Z2 coordinates could then be found by hand calculation (this was not difficult, since as already mentioned, changing the Z2 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 math formula from a difference matrix of index larger than 1. Suppose math formula are groups of orders math formula, math formula, and D is a math formula difference matrix over G. Suppose also, D possesses the extra property that for any two rows math formula of D, any math formula and any math formula in math formula, there is exactly one column j of D such that both math formula and math formula has first coordinate z (from G1). When this occurs, developing D over G2 (but not over G1) gives an math formula.

In [6], by taking math formula math formula, 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 math formula, math formula we start with a (2, 7, 6) difference matrix D1 possessing the extra property in the previous paragraph, that is, for any math formula, and any two rows math formula, there are three columns j in D1 such that math formula and math formula. Let these three columns be math formula, math formula and math formula.

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 math formula; and finally, by computer, we have to try to find a 7 × 12 array D2 with entries from math formula such that for any two rows math formula, and any math formula, adding 0, math formula and math formula to the three values in the set math formula produces a nine-element set containing each element of math formula 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 math formula BIBD will give a math formula 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 math formula.

We tried 45 choices for the vector T, but most of them together with the chosen (2, 7, 6) difference matrix D1 gave no solution for the matrix D2 over math formula. Only two gave a solution; one of these is given below. When D1 and T are fixed, a few restrictions on the entries enable us to carry out an exhaustive search for D2. These are as follows:

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

We now display a suitable array A with entries math formula. For convenience, these entries are written as math formula where math formula and math formula. Add the following three vectors to each of the 12 columns of this array A: ((0,0,0), (0, 0, y), (0, y, 0), math formula math formula math formula math formula for math formula. 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.

display math

It is natural to ask whether for math formula 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 D1 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. math formula MOLS(v) are said to be idempotent if they possesss a common transversal. This is equivalent to saying that in the corresponding OAmath formula, 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 (math formula) 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 math formula, and all but three of these math formula come from (v, 6, 1) difference matrices.

We also point out that any set math formula MOLS(v) obtained from a math formula difference matrix cannot be made idempotent unless the math formula difference matrix is extendable to a math formula difference matrix. The columns of the OAmath formula obtained from a math formula 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 math formulath 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 math formulath row from these v columns and add an extra row, containing the identity element of G in each column. The result is a math formula difference matrix over G.

On the other hand, there is no v (except math formula) 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 math formula. Those for math formula were given in [6] while those for math formula come from a quasi-difference matrix over Z25, 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 math formula, 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.
  1. There exist four MOLS(v) for all integers math formula, except for math formula and possibly for math formula, 22}.
  2. There exist four idempotent MOLS(v) for all integers math formula, except possibly for math formula, 22}.
  3. There exist five MOLS(v) for all integers math formula, except possibly for math formula, 14, 15, 20, 22, 26, 30, 34, 38, 46}.
  4. There exist five idempotent MOLS(v) for all integers math formula, except possibly for math formula, 12, 14, 15, 18, 20, 21, 22, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 46, 51, 52, 54, 60}.

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