Implementing the MOLS Table for n Up to 500

Abstract

Latin squares are an essential tool in the construction of combinatorial designs. Optimal solutions for problems such as scheduling problems and permutation arrays for powerline communication rely on the ability to construct sets of mutually orthogonal Latin squares (MOLS) that are as large as possible. Although constructions of suitable sets are known, they are scattered among a wide variety of sources, and can be both difficult to understand and contain errors. We describe our experience implementing the largest known sets of MOLS of order n, for n up to 500. We give a source for each construction, provide additional hints for the difficult cases, and correct some errors along the way. We also give constructions for new sets of MOLS of order n, where n is 486, 567, 622, 635, 754, 756, 764, 766, 774, 778, 802, 810, 822, 826, 894, 906, 916, 920 or 936.

1. Introduction

A combinatorial design is an arrangement of a set of elements (or points) into defined substructures (blocks) in such a way that desired combinatorial properties are satisfied. Designs have many applications in, for example, communications, cryptography and networking [1], optical orthogonal codes [2], treatment trial design [3] and genetic screening algorithms [4].

Latin squares [5] are a key element in many aspects of combinatorial designs [6]. For example, they are used to construct designs such as block designs [7,8,9,10,11,12], group divisible designs [13,14,15], Latin square designs [16] and Room squares [17,18]. They have many other applications, for example, for the creation of tournaments [19], channel access scheduling in wireless communication [20] and in the design of cryptographic schemes [21].

Sets of mutually orthogonal Latin squares (MOLS) are used to find optimal solutions to combinatorial problems, e.g., solutions to variations of the Schoolgirl problem of Kirkman [22]:

Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast.

The solution to this problem is to use a Kirkman Triple System, which is a type of Resolvable Balanced Incomplete Block Design (RBIBD). The existence of Kirkman Triple Systems for all suitable sizes was completely solved by Ray-Chaudhuri and Wilson in 1971 [23].

The Social Golfer problem [24,25,26,27] is a generalisation of Kirkman’s Schoolgirl problem. It originated from the following question posed in 1998 to sci.op-research:

32 golfers play golf once a week, and always in groups of 4. For how many weeks can they play such that no two players play together more than once in the same group?

In fact, the solution to this problem is ten weeks. It can be obtained using a combinatorial structure known as a Resolvable Group Divisible Design ($RGDD$). The design for this case was originally constructed by Shen in 1996 [28] and independently solved by Colbourn in 1999 [29]. In 2004, Aguado recognised that the design would solve the problem and published a solution [30].

The problem can be generalised [31] to that of scheduling $n=g\times s$ players into g groups of s players for w weeks so that any two players are assigned to the same group at most once in w weeks.

Both RBIBDs and RGDDs rely, for their construction, on sets of MOLS of sufficient size. See [32] for an excellent source of information on construction methods for this type of design. In recent work [27] we have introduced a variation of the Social Golfer problem for breakout room allocations for online meetings, and balanced organisation for group-work in classrooms. Many optimal solutions to this problem can be found using RBIBDs and RGDDs.

A set of r MOLS of order n is equivalent to a particular type of r-separable permutation array [33]. Permutation arrays [34] have applications in block ciphers [35] and for the correction of interference errors during data transmission over power lines [36,37]. The most recent advances in the generation of sets of MOLS have come from permutation arrays [38,39].

In order to find optimal solutions to the type of combinatorial problem described above, we often want to construct a set of MOLS for a given order n, where the set is as large as possible. We call a set of t MOLS of order n an optimal set if no set of $t+1$ MOLS of order n is known.

An original comprehensive list of the then current greatest lower bounds on the number of MOLS of side n, where $n\le 10,000$, was given in 1979 in [40]. The table was updated in 1996 in [41], and again in 2007 in [6] (specifically in the chapter on MOLS [42]). However, it is not always apparent which construction to use for a given n. The constructions appear in dozens of papers, some from as far back as the 1950s, and it can be time-consuming to identify the correct source.

In this paper, we provide insights from our experience implementing optimal sets of MOLS for $n\le 500$. We chose 500 as a suitable value to extend our investigation to, allowing us to study a wide range of constructions spread across multiple sources. In addition, we believe most practical applications are unlikely to require squares bigger than this.

We assume that the reader is familiar with terms from combinatorial design theory, referring the reader to [6] in the first instance for additional background (although other texts such as [32,43,44] would also be useful).

Combinatorial Design Theory, and in particular the construction of sets of MOLS, is inherently related to symmetry. For example, constructions often involve finite (Galois) groups to create transversal designs from difference arrays. The nature of finite groups prevents the repetition of pairs in different blocks/groups by avoiding undesired periodicity due to symmetry. The constructions using permutation arrays use a subgroup of the isometry group of the symmetric group $S_n$: i.e., a group of symmetries acting on the permutations of a set of size n.

2. Motivation and Contribution

Our motivation for this paper arose from the following observations made when implementing MOLS constructions for $n\le 500$:

• The source for some of the constructions can be hard to identify;
• Some of the constructions require additional explanation;
• Some of the constructions contain errors;
• The latest version of the MOLS table [42] is out of date.

We identified a need for a companion paper, i.e., a helpful reference for others implementing MOLS constructions, rather than to give a complete survey of the constructions themselves.

Our first contribution is to provide such a companion paper and address the issues listed above. We include references for each case and additional notes on constructions for which we had to seek information from further sources. We also include some fixes to a small number of errors and update the table with results published since the appearance of [42].

Our second contribution is to provide constructions for some new optimal sets of MOLS. We give a construction for a new set of eight MOLS of order 486 in Section 5.2 (Theorem 3), and in Section 7, we give some constructions for new optimal sets of MOLS for some values greater than 500. Specifically, we give the following:

• New sets of 8 MOLS for $n=567$ and $n=936$;
• New sets of 10 MOLS for
  $n\in \{622,635,754,756,764,766,774,778,802,810,822,826,894,906,916,920\}$.

3. Preliminaries

In this section, we give some basic required definitions. Further related background material can be found in [6,32,43,44].

Definition 1.

A Latin square of side n (or order n) is an $n\times n$ array in which each cell $L(a,b)$ contains a single symbol from an n-set S, such that each symbol occurs exactly once in each row and each column.

Definition 2.

Two Latin squares L and $L^{\prime }$ of the same order are orthogonal if $L(a,b)=L(c,d)$ and $L^{\prime }(a,b)=L^{\prime }(c,d)$ implies $a=c$ and $b=d$. A set of Latin squares $L_1,L_2,\dots ,L_m$ is mutually orthogonal (or are a set of MOLS), if for every $1\le i<j\le m$, $L_i$ and $L_j$ are orthogonal.

In this paper, we are concerned with the number of mutually orthogonal Latin squares (MOLS) known to exist for a given n. It is common to use the notation $N\left(n\right)$ to denote the size of the largest set of MOLS of order n. A set of t MOLS of order n is said to be maximal if it cannot be extended to a set of $t+1$ MOLS. We refer instead to a set of t MOLS of order n as an optimal set if no set of $t+1$ MOLS of order n is known. Note that in some cases, an optimal set of MOLS is maximal—when the set has size $n-1$, say. Some maximal sets of t MOLS are also known for small values of t. For instance on p. 189 of [6], a set of t-maximal MOLS of order n is given for each $(t,n)\in \left\{\right(1,5),(1,7),(2,7),(2,8),(3,8\left)\right\}$.

We include the following definition of a transversal design as constructions are often described in terms of them, rather than $N\left(n\right)$.

Definition 3.

A transversal design of order n and block size k, denoted $TD(k,n)$, is a triple $(V,G,B)$, where V is a set of $kn$ elements (or points); G is a partition of V into k classes, called groups, each of size n; B is a collection of k-subsets of V, called blocks; and every unordered pair of elements from V is contained in one group, or one block, but not both.

Note 1.

The existence of a $TD(k,n)$ is equivalent to the existence of $k-2$ MOLS of order n.

We now provide definitions of some other designs that are used in MOLS constructions. We use the terms points and blocks in the same way as they are used in Definition 3. Our definitions are adaptations of more general definitions given in [6].

Definition 4.

Let K be a set of integers. A pairwise balanced design ($PBD$) of order v with block sizes from K is a pair $(V,B)$, where V is a finite set of v points and B is a family of blocks from V that satisfy (1) if $b\in B$ then $\left|b\right|\in K$ and (2) every pair of distinct points occurs in exactly one block.

Definition 5.

A truncated transversal design is a transversal design from which some points have been removed from some of the groups (thereby truncating those groups). The points are also removed from any block they appear in.

Definition 6.

An incomplete transversal design ($ITD$) of order (or group size) n, block size k, and hole sizes $b_1,\dots ,b_s$, denoted as $ITD(k,n;b_1,\dots ,b_s)$, is a quadruple $(V,G,H,B)$, where:

• V is a set of $k\times n$ points;
• G is a partition of V into k groups of size n;
• H is a set of disjoint subsets $H_1,\dots ,H_s$ of V, called holes, with the property that for each $1\le i\le s$, the size of the intersection of $H_i$ with each group is $b_i$;
• B is a collection of blocks from V of size k;
• Every unordered pair of elements from V is either contained in a block and not contained in either a group or a hole; contained in a group (and possibly a hole); or contained only in a hole.

In

Table 1. MOLS table from [42], with recent improvements underlined. Value in entry with row label i and column label j represents the current lower bound for $N\left(n\right)$, where $n=i+j$.

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
0	∞	∞	1	2	3	4	1	6	7	8	2	10	5	12	4	4	15	16	5	18
20	4	5	3	22	7	24	4	26	5	28	4	30	31	5	4	6	8	36	4	5
40	7	40	5	42	5	6	4	46	10	48	6	5	5	52	8	6	7	7	5	58
60	5	60	5	8	63	7	5	66	5	6	6	70	7	72	5	7	6	6	6	78
80	9	80	8	82	6	6	6	6	7	88	6	7	6	6	6	6	10	96	6	8
100	8	100	6	102	7	7	6	106	9	108	6	6	13	112	6	7	6	8	6	6
120	7	120	6	6	6	124	6	126	127	7	6	130	6	7	6	7	7	136	6	138
140	6	7	6	10	10	7	6	7	6	148	6	150	7	8	8	7	6	156	7	6
160	9	7	6	162	6	7	6	166	7	168	6	8	6	172	6	6	14	9	6	178
180	6	180	6	6	7	9	6	10	6	8	6	190	7	192	6	7	6	196	6	198
200	7	8	6	7	6	8	6	8	14	11	10	210	6	7	6	7	7	8	6	10
220	6	12	6	222	13	8	6	226	6	228	6	7	7	232	6	7	6	7	6	238
240	7	240	6	242	6	7	6	12	7	7	6	250	6	12	9	7	255	256	6	12
260	6	8	8	262	7	8	7	10	7	268	7	270	15	16	6	13	10	276	6	9
280	7	280	6	282	6	12	6	7	15	288	6	6	6	292	6	6	7	10	10	12
300	7	7	7	7	15	15	6	306	7	7	7	310	7	312	7	10	7	316	7	10
320	15	15	6	16	8	12	6	7	7	9	6	330	7	8	7	6	8	336	6	7
340	6	10	10	342	7	7	6	346	6	348	8	12	18	352	6	9	7	9	6	358
360	8	360	6	7	7	10	6	366	15	15	7	15	7	372	7	15	7	13	7	378
380	7	12	7	382	15	15	7	15	7	388	7	16	7	8	7	7	8	396	7	7
400	15	400	7	15	11	8	7	15	8	408	7	13	8	12	10	9	18	15	7	418
420	7	420	7	15	7	16	6	7	7	10	6	430	15	432	6	15	6	18	7	438
440	7	15	7	442	7	13	7	11	15	448	7	15	7	7	7	15	7	456	7	16
460	7	460	7	462	15	15	7	466	8	8	7	15	7	15	10	18	7	15	6	478
480	15	15	6	15	8	7	8	486	7	15	6	490	6	16	6	7	15	15	6	498

, we give the current lower bound for $N\left(n\right)$ for $n<500$, from [6] (specifically from the chapter on MOLS [42]), updated to include results published since the publication of [6] (these new results are underlined). Note that we do not include the number of MOLS for $n=500$ (there are 7) for ease of presentation of the table.

4. Summary of Common Constructions

Although we do not attempt to reproduce most of the constructions referred to in this paper, we present some of the simplest constructions in this section.

Some useful general results, originally from [45], are given in Theorem 1. We state them as presented in [6].

Theorem 1 .

1. 

$N(n\times m)\ge min\left\{N\right(n),N(m\left)\right\}$

2. 

If $n=p^e$, for prime p and some $e>0$, then $N\left(n\right)=n-1$.

3. 

If $n=p_1^{e_1}{p}_2^{e_2}\dots p_k^{e_k}$, where each $p_i$ is a prime, then $N\left(n\right)\ge min\{p_i^{e_i}-1|i\in \{1,2,\dots ,k\}\}$.

Proofs (leading to constructions) of these results are given in [7] Lemmas 3.4 and 3.5 and Theorem 3.1, and in [43] Theorem 5 and Corollary 5.1.

Many constructions are based on PBDs [46,47]. These constructions can be strengthened when some blocks in the design form a clear set (i.e., they are disjoint) or form a separable design [47]. Further variations including when a subset of blocks is almost clear, and when different sets of blocks are clear or separable are given in [48], with a good explanation of how to use almost clear sets presented in [49].

Original recursive constructions due to Wilson [50] were generalised by Wojtas [51], Stinson [52] and Brouwer and van Rees [53] (Theorem 1.1).

The constructions we use in this paper are all variations of Wilson’s construction, which we include here as Theorem 2.

Theorem 2

([50] (Theorem 2.2)). Let $D=(V,G,B)$ be a $TD(k+l;q)$ which has been truncated so that the last l groups have sizes $h_1,\dots ,h_l$, respectively. If the following hold:

• For $1\le i\le l$ there exists a transversal design $TD(k,h_i)$;
• For any $x\in \{0,1,\dots ,l\}$ for which D has a block of size $k+x$ there exists an incomplete transversal design of order $m+x$, block size k and x holes of size 1,

then a $TD(k,mq+h_1+h_2+\dots +h_l)$ exists.

Proof. 

We will refer to the $TD(k,mq+h_1+h_2+\dots +h_l)$ as $T{D}^{\prime }=(V^{\prime },G^{\prime },B^{\prime })$. The points V of D consist of points in the non-truncated groups, $V_1$, and those in the truncated groups, $V_2$. The points $V^{\prime }$ of $T{D}^{\prime }$ consist of m copies of each point in $V_1$, and k copies of each point in $V_2$. For $1\le i\le k$, for (untruncated) group $G_i$ in D, the corresponding group of $T{D}^{\prime }$ contains the m copies of each point in $G\cup V_1$, and the i-th copy of every element in $V_2$. For every block b of D of size $k+x$, construct an ITD of order $m+x$, block size k and x holes of size k, whose points are the m copies of each element of $b\cup V_1$ and k copies of each element of $b\cup V_2$. Each hole of the ITD consists of the k copies of an element of $b\cup V_2$. Add the blocks of each such ITD to $B^{\prime }$. Finally, for each $1\le i\le l$, add the blocks of the $TD(k,h_i)$ constructed on the k copies of the $h_i$ points in the i-th truncated group. □

In the construction of Theorem 2, a supply of suitable ITDs is required so that the pairs between copies of the same point in a truncated group do not appear more than once (they are filled in once using the transversal designs $TD(k,h_i)$). Some simple variations of Theorem 2 avoid the need for $ITD$s other than those for which $x\le 3$, which are derived by deleting x parallel blocks from a transversal design. For $x\le 3$, a set of x parallel blocks in any $TD(k,n)$ with $k\le n$ and $n\ge 3$ always exists. These variations are used in many MOLS constructions and appear in slightly different forms in various papers. Our presentation is similar to that used for constructions 3.1–3.6 in [54] and we will refer to them in this paper as Constructions 1 to 6.

Constructions 1 and 2 are Theorem 2 with $l=1$ and $l=2$ (so $x\le 1$ and $x\le 2$), respectively. In Construction 3, r of the groups of the $TD(k+r,q)$ are truncated to size 1, and a single block intersects all of the groups. In this case, an ITD with $x=1$ is used for blocks that intersect one truncated group, and a $TD(k+r,q)$ for the block of size $q+r$. Construction 4 is similar, there are $r+1$ truncated groups, the first r of which have size 1 and the other has size s. A block intersects either all of the groups or all but the last group. Construction 5 requires a transversal design containing a particular type of thwart involving three of its groups. This is equivalent to a truncated transversal design where every block intersects at least one of three truncated groups. It follows that $x\in \{1,2,3\}$ in this case. Construction 6 uses a truncated $TD(k+r,q)$ with k groups of size q and r groups of size 1, where every block intersects at most 2 of the truncated groups (so $x\in \{0,1,2\}$). This is obtained using an oval in $PG(2,q)$, as suggested in [54]. We describe how this is achieved in Section 6.

Construction 1

([50], Theorem 2.3). Suppose $0\le r\le q$ and that there exist a $TD(k+1,q)$, a $TD(k,r)$ and a $TD(k,m+x)$ for $x\in \{0,1\}$. Then, there exists a $TD(k,v)$ for $v=mq+r$.

Construction 2

([50], Theorem 2.4). Suppose $0\le r,s\le q$ and there exist a $TD(k+2,q)$, a $TD(k,r)$, a $TD(k,s)$ and a $TD(k,m+x)$ for $x=0,1,2$. Then, there exists a $TD(k,v)$ for $v=mq+r+s$.

Construction 3

([51] Lemma 2.1). Suppose there exist a $TD(k+r,q)$ and a $TD(k,m+x)$ for $x=0,1,r$. Then, there exists a $TD(k,v)$ for $v=mq+r$.

Construction 4

([53] Theorem 1.1). Suppose $1\le s\le q-1$, and there exist a $TD(k+r+1,q)$, a $TD(k,s)$ plus a $TD(k,m+x)$ for $x=0,1,2,y$ where either $y=r$ or $y=r+1$. Then, there exists a $TD(k,v)$ for $v=mq+r+s$.

Construction 5

([55] Lemma 3.1). Suppose $0\le r,s,t<q$, $(q-r-1)(q-s)<t$, $r\le s$ and there exist a $TD(k+3,q)$, a $TD(k,w)$ for $w=r,s,t$ and a $TD(k,m+x)$ for $x=1,2,3$. Then, there exists a $TD(k,v)$ for $v=mq+r+s+t$.

Construction 6

([54] Construction 3.6). Suppose q is a prime power, $r\ge 0$ and there exist a $TD(k+r,q)$ and a $TD(k,m+x)$ for $x=0,1,2$. Then, there exists a $TD(k,v)$ for $v=mq+r$.

Another common construction technique is to use structures known as $V(m,t)$ vectors [18,53,56,57,58]. These are very useful, compact ways of creating quasi-difference matrices [6], from which sets of MOLS can often be constructed. The three smallest cases where the largest known number of MOLS comes from a $V(m,t)$ are $n=10$, $n=46$ and $n=82$.

5. Sources and Constructions

In this section, we present sources for the specific MOLS constructions that have been used to construct (optimal) sets of MOLS of size equal to the current lower bound for $N\left(n\right)$ shown in Table 1, for $n\le 500$. These are not necessarily the original sources (for example, some constructions from [48] originally appeared in [7] or [40]).

Since the MOLS table was last updated in [42], some improved values have been found, and we include those for $n<500$ in Table 1 (underlined). All other values, apart from $n=486$, are the same as they were in [42]. These improvements include $N\left(14\right)\ge 4$ [59]; $N\left(18\right)\ge 5$ and $N\left(60\right)\ge 5$[60]; $N\left(35\right)\ge 6$, $N\left(48\right)\ge 10$, $N\left(63\right)\ge 8$ and $N\left(96\right)\ge 8$ [38]; $N\left(54\right)\ge 8$; and (as a further improvement for $n=96$) $N\left(96\right)\ge 10$[39]. A difference matrix construction to show that $N\left(108\right)\ge 9$ is also given in [39]. We include the additional new result $N\left(486\right)\ge 8$ in our table (also underlined). It is a simple consequence of the result for $n=54$ (and we include it in our auxiliary theorem, Theorem 3).

A survey of techniques used for MOLS constructions for the 1996 edition of the handbook [41] was given in [61], which was later updated in [49]. That paper does not include a full description of how the individual entries in the MOLS table were found, although references are given in some cases. Locating the source of many constructions can be arduous.

We start by considering the easy cases—where either Theorem 1 still gives the best lower bound, or the bound has not improved since the the original MOLS table in [40]. Note that [40] includes suggestions for constructions, but not all parameter values.

5.1. Easy Cases

By convention, $N\left(n\right)$ for $n\le 1$ is said to be ∞, so these cases are omitted. In addition, in the light of Theorem 1, we do not consider cases where n is a prime power.

Cases where n is not a prime power but Theorem 1 still gives the best known value are shown in

Table 2. Current best lower bounds for $N\left(n\right)$ which can be obtained from Theorem 1 part 1. Each cell contains n (top) and $N\left(n\right)$ (bottom).

72	77	88	99	104	117	136	143	152	153	171	184	187	200	207	216	221	225	232
7	6	7	8	7	8	7	10	7	8	8	7	10	7	8	7	12	8	7
247	248	261	272	296	297	299	304	319	323	324	325	328	333	341	344	351	368	376
12	7	8	15	7	10	12	15	10	16	8	12	7	8	10	7	12	15	7
391	392	396	400	424	425	432	437	456	459	464	468	472	475	488	493	496
16	7	8	15	7	16	15	18	7	16	15	8	7	18	7	16	15

. In most of these cases, n is the product of two prime powers. However, in a few cases, one of the elements in the product is not a prime power (e.g., for $n=324$, $n=396$ and $n=468$, one of the elements is 36—for which there are 8 MOLS, which were not found until 2004 [62]).

Hanani [7] proved that $N\left(n\right)\ge 6$ for all $n>76$, $n\notin \{82,90\}$, mainly by using Constructions 1 and 2 with $m=7$. Lemma 1 includes all values of n in this range for which this is still the best lower bound (refer to the proof of Theorem 3.7 in [7] for the appropriate construction).

Lemma 1.

If $n\in C_1\cup C_2\cup \dots \cup C_7$, then the best current lower bound for $N\left(n\right)$ is 6 and a construction can be found in [7]:

• $C_1=\{85,86,87,92,93,94,95,98,102,106,110,111,114,116,118,119,122\}$,
• $C_2=\{123,124,126,130,132,134,138,140,142,146,148,150,156,159,162\}$,
• $C_3=\{164,166,170,172,174,175,178,180,182,183,186,188,190,194,196\}$,
• $C_4=\{198,202,204,206,212,214,218,220,222,226,228,230,234,236,238\}$,
• $C_5=\{242,244,246,250,252,258,260,274,278,282,284,286,290,291,292\}$,
• $C_6=\{294,295,306,322,326,330,335,338,340,346,348,354,358,362,366\}$,
• $C_7=\{426,430,434,436,478,482,490,492,494,498\}$.

Cases for which [40] still provides the best lower bound are shown in

Table 3. Current best lower bounds for $N\left(n\right)$, which are still as given (with appropriate construction) in [40]. Each cell contains n (top) and $N\left(n\right)$ (bottom).

91	105	115	129	133	141	145	147	155	161	165	168	177	192	195	203	209
7	7	7	7	7	7	7	7	7	7	7	7	9	7	7	7	11
213	215	217	219	231	235	237	240	245	249	255	259	275	279	287	301	305
7	7	8	10	7	7	7	7	7	7	7	12	13	9	7	7	15
309	312	315	320	321	327	339	345	350	363	369	371	375	377	381	384	385
7	7	10	15	15	7	7	7	8	7	15	15	15	13	12	15	15
387	395	399	403	404	407	411	412	413	414	415	417	423	427	435	440	441
15	7	7	15	11	15	13	8	12	10	9	15	15	7	15	7	15
445	447	448	451	453	455	465	471	473	477	480	481	483	485	489	495	497
13	11	15	15	7	15	15	15	15	15	15	15	15	7	15	7	15

. They were all constructed in [40] using one of three constructions due to Wilson [50] and Wojtas [63] (our Constructions 1 and 3) and Hall [64]. In fact, they can all be obtained using Constructions 1 and 3 using $m\in \{8,16,31\}$.

5.2. Other Cases

In Table 4, we provide a source for the most up to date largest lower bound for $N\left(n\right)$, for $n\le 500$ and where n was not already considered in Section 5.1.

Several of the constructions are from [6]. A single Latin square for $n=6$ is from Section III.1.3 (p. 137) (any of those listed in Table 1.18); two MOLS of side 10 are from Section VI.35 (p. 530). Other constructions for sets of MOLS are from Section III.3.4—i.e., [42], pages 163 to 171. The original sources for these constructions are cited in [42] and include [54,58,62,65,66,67,68,69,70,71,72,73,74,75,76,77].

We could not identify a source for seven MOLS of order $n=402$. This was known in 1996 (see [42]), so should not rely on the existence of seven MOLS of order 24, which were not found until 2004 [62]. There are several constructions for this case that do rely on seven MOLS of order 24, however (see an example in Theorem 3). We also include in Theorem 3 constructions for $n\in \{329,484\}$ for which we also could not identify the original source, and a construction for $n=486$ which is a new result using the updated optimal value for $n=54$ from [39].

Theorem 3.

The following results hold:

1. 

$N\left(329\right)\ge 9$;

2. 

$N\left(402\right)\ge 7$;

3. 

$N\left(484\right)\ge 8$;

4. 

$N\left(486\right)\ge 8$.

Proof .

(1) Use Greig’s $Q-X$ construction [14] with two points at infinity, explained in Section 4.4 of [61] with $q=27$ and $x=15$. (2) Use Construction 1 with $m=23$, $q=17$, $r=11$. (3) Use Construction 3 with $m=16$, $q=29$, $r=20$. (4) Use Theorem 1 with $m=9$ and $n=54$. □

Table 4. Sources for MOLS constructions where n is not considered in Section 5.1.

n	$\mathit{N}\left(\mathit{n}\right)$	Source
6, 10, 15, 20, 21, 22, 24, 26, 28, 30	1, 2, 4, 4, 5, 3, 7, 4, 5, 4	[42]
33, 34, 36, 38, 39, 40, 42, 44, 46	5, 4, 8, 4, 5, 7, 5, 5, 4
50, 51, 52, 55, 56, 62, 75, 80	6, 5, 5, 6, 7, 5, 7, 9
12	5	[78]
14	4	[59]
18, 60	5, 5	[60]
35, 48, 63	6, 10, 8	[38]
45, 54, 96, 108	6, 8, 10, 9	[39]
57, 69, 70, 74, 78, 84, 90, 135, 264	7, 6, 6, 5, 6, 6, 6, 7, 7	[48]
265, 267, 273, 280, 285, 288	8, 10, 16, 7, 12, 15
58, 66, 68	5, 5, 5	[7]
65	7	[47]
76	6	[79]
82, 100, 144, 154, 210, 262, 276, 298, 342, 474	8, 8, 10, 8, 10, 8, 8, 10, 10, 10	[57]
112, 160, 176, 208, 224, 352, 416	13, 9, 14, 14, 13, 18, 18	[68]
120	7	[46]
158, 185, 205, 254, 316, 355	7, 9, 8, 9, 7, 9	[14]
189, 253, 357, 469	8, 12, 9, 8	[80]
201, 336, 360, 365, 393, 429	8, 8, 8, 10, 8, 10	[81]
266, 268, 270, 300, 302, 303, 308, 310, 314,	all 7	[62]
318, 332, 334, 356, 364, 370, 372, 374, 378,
380, 382, 386, 388, 390, 394, 398, 406, 410,
418, 420, 422, 428, 438, 442, 444, 446, 450,
452, 454, 458, 460, 462, 466, 470, 476, 500
405	8	[62]
408	8	[55]
329, 402, 484, 486	9, 7, 8, 8	Theorem 3

Table 4. Sources for MOLS constructions where n is not considered in Section 5.1.

n	$\mathit{N}\left(\mathit{n}\right)$	Source
6, 10, 15, 20, 21, 22, 24, 26, 28, 30	1, 2, 4, 4, 5, 3, 7, 4, 5, 4	[42]
33, 34, 36, 38, 39, 40, 42, 44, 46	5, 4, 8, 4, 5, 7, 5, 5, 4
50, 51, 52, 55, 56, 62, 75, 80	6, 5, 5, 6, 7, 5, 7, 9
12	5	[78]
14	4	[59]
18, 60	5, 5	[60]
35, 48, 63	6, 10, 8	[38]
45, 54, 96, 108	6, 8, 10, 9	[39]
57, 69, 70, 74, 78, 84, 90, 135, 264	7, 6, 6, 5, 6, 6, 6, 7, 7	[48]
265, 267, 273, 280, 285, 288	8, 10, 16, 7, 12, 15
58, 66, 68	5, 5, 5	[7]
65	7	[47]
76	6	[79]
82, 100, 144, 154, 210, 262, 276, 298, 342, 474	8, 8, 10, 8, 10, 8, 8, 10, 10, 10	[57]
112, 160, 176, 208, 224, 352, 416	13, 9, 14, 14, 13, 18, 18	[68]
120	7	[46]
158, 185, 205, 254, 316, 355	7, 9, 8, 9, 7, 9	[14]
189, 253, 357, 469	8, 12, 9, 8	[80]
201, 336, 360, 365, 393, 429	8, 8, 8, 10, 8, 10	[81]
266, 268, 270, 300, 302, 303, 308, 310, 314,	all 7	[62]
318, 332, 334, 356, 364, 370, 372, 374, 378,
380, 382, 386, 388, 390, 394, 398, 406, 410,
418, 420, 422, 428, 438, 442, 444, 446, 450,
452, 454, 458, 460, 462, 466, 470, 476, 500
405	8	[62]
408	8	[55]
329, 402, 484, 486	9, 7, 8, 8	Theorem 3

6. Implementation of the MOLS Constructions

We have implemented constructions for optimal sets of MOLS of order n, for all $n\le 500$, and make observations and corrections in some specific cases below. In most cases, the implementations can be carried out on a standard laptop (we used Java on a MacBook Pro with a 1.4 GHz Quad-Core Intel Core i5 with 8 GB of memory)—albeit with some laborious typing in of base blocks/difference matrices, etc., in some cases. For the cases which involved permutation arrays (from [38,39]), we first used GAP [82] to generate the orbits, which were then saved and called from our Java class.

Brouwer’s paper [48] is very important and contains construction instructions for many cases. However the online journal version is missing its bibliography. A full version can be found at [83].

Case $n=45$ originally appeared in [42], but contained an error. The entry in the sixth row and fourth column should be 201, not 210. This has now been corrected in [39].

Case $n=69$ uses an almost clear set construction which is not fully described in [48]. Refer to [49] (explanation of) Theorem 4.1 for a good description of this method. The construction is similar to the latter half of the construction for $n=76$ given in [79].

For $n=90$, the construction is outlined in [48] (although the original source is a note by Wojtas that we could not obtain). Note that the outline in [48] incorrectly suggests to truncate a $TD(9,11)$, whereas one should in fact truncate a $TD(11,11)$. The blocks should all intersect at least one of the truncated groups. The method to construct such a truncated design (i.e., a transversal design with a suitable thwart) is from [55]. The truncated groups are the complements of those used in Construction 5. This method is also used in constructions for larger values of n, e.g., $n=266$.

The solutions for several values of n, where n is the product of a power of 2 and a prime power, are stated in [49] as being from [74]. However, [74] only gives the construction method and a few small examples. The full constructions actually appear in [68], but in some cases contain some errors. We have summarised the required changes in each case in

Table 5. Corrections to matrix $A^{*}$ in difference array constructions from [68]. Each change has the form ci, rj: $(a_1,b_1)\to (a_2,b_2)$ indicating that the entry in column i, row j should be changed from $(a_1,b_1)$ to $(a_2,b_2)$.

n	#MOLS	Page	Required Changes
112	13	107	c13, r11	:	$(3,3)\to (3,9)$
176	14	110	c9, r3	:	$(5,1)\to (5,0)$
			c9, r22	:	$(4,8)\to (2,Z)$
			c10, r22	:	$(7,12)\to (4,8)$
			c12, r20	:	$(2,Z)\to (2,0)$
208	>14	111	c3, r20	:	$(6,10)\to (6,9)$
			c11, r24	:	$(2,Z)\to (2,0)$
224	13	108	c4, r12	:	$(6,19)\to (6,12)$
352	18	113	c9, r7	:	$(4,25)\to (4,27)$
			c9, r18	:	$(3,0)\to (3,Z)$
			c15, r2	:	$(4,1)\to (4,0)$
			c19, r19	:	$(0,0)\to (0,Z)$
416	18	114	c3, r2	:	$(2,10)\to (2,18)$
			c4, r17	:	$(2,Z)\to (2,0)$
			c8, r20	:	$(1,20)\to (1,12)$
			c10, r25	:	$(1,0)\to (10,0)$
			c12, r2	:	$(1,5)\to (11,5)$
			c15, r19	:	$(6,21)\to (6,15)$
			c16, r19	:	$(5,14)\to (5,23)$
544	18	115	c4, r23	:	$(8,9)\to (8,21)$
			c5, r17	:	$(13,Z)\to (13,0)$
			c7, r7	:	$(2,Z)\to (2,0)$
			c18, r33	:	$(12,14)\to (12,15)$
			c19, r9	:	$(13,9)\to (15,25)$
640	9	106	c7, r2	:	$(0,25)\to (0,124)$
896	13	109	c5, r2	:	$(4,120)\to (4,49)$
			c14, r14	:	$(0,0)\to (0,Z)$

. In all cases we indicate changes to matrix $A^{*}$ and use the indexing used in [68] (i.e., rows and columns start from 1). Note that three of these values are not part of our current survey (i.e., they are greater than 500), but we include them for completeness.

The construction for $n=189$, $n=253$ and $n=357$ from [80] requires a partition of $PG(2,q^2)$ into disjoint Baer subplanes. A good description of this partitioning process can be found in [84], Section 2.5.

The thwart construction of cases 201, 336, 360, 365, 393 and 429 from [81], requires a particular type of blocking set in a projective plane, which is referred to as a Baer configuration. A description of how to construct such a set can be found in [85], Section 13.5.

For the case $n=205$, a construction for a $(205,\{9,13,17\left\}\right)$-$PBD$ using a Baer subplane is given in [14]. The eight MOLS come from the fact that the blocks of size 9 form a clear set.

For the case $n=254$ from [14], a $(254,\{11,13,16\left\}\right)$-$PBD$ is used. This $PBD$ is the complement of one constructed in [80] and uses a difference set. By constructing $PG(2,q)$ with points the set of coset representatives of $GF\left(q^3\right)$, a difference set is obtained as the set of points on any line (e.g., the line containing $(0,1)$ is a difference set in reduced form). This is described in [86]. The resulting $(q^2+q+1,q+1,1)$ block design is separable by construction which is useful in other MOLS constructions, e.g., $n=57$ and $n=273$.

Case $n=408$ uses a thwart construction from [55] that relies on the fact that, if q is an odd prime power which is either a power of 3 or congruent to 1 mod 3, then $PG(2,q)$ contains an affine subplane $AG(2,3)$. The construction of this plane can be found in [87].

Cases $n\in \{35,48,54,63,96\}$ are from [38,39] and require orbits to be generated from a finite set of permutations. This requires the use of a computer algebra system such as GAP [82] or MAGMA [88,89]. A GAP program is provided in the recent update [39] (which also includes case $n=54$ and an improvement for $n=96$). For each case, having read the provided GAP code into GAP, the relevant orbits should be extracted and saved. In some cases, the additional orbit created from the action of the cyclic group $E_n$ (as defined in [38]) on the identity permutation should also be generated and saved. The orbits can then be used to create the required separable permutation array and the MOLS. The permutation array-to-MOLS construction can be found in [33].

In [54], there are slight errors for $n=762$, $n=764$ and $n=766$, where m should be 16, not 17. In addition, the case $n=914$ should instead be $n=916$. These values do not, of course, appear in our Table 1, but we use them to generate similar, new results in Section 7. Their Construction 3.6 is our Construction 6 and uses an oval in $PG(2,q)$. To clarify, the oval is used to construct a truncated $TD(k+r,q)$ with k groups of size q and r groups of size 1, where every block intersects at most two of the truncated groups. If O is an oval in $PG(2,q)$ and $p\in O$, then the groups of the truncated design are formed from the $q+1$ lines $L_p$ containing p in the following way: delete the last $(q+1)-(k+r)$ lines in $L_p$; delete p from the first k lines in $L_p$; and delete p and all other points from the other lines in $L_p$ apart from those that are in $O\setminus \left\{p\right\}$ (removing all deleted points from all of the lines of $PG(2,q)$). The blocks are the (reduced) lines of $PG(2,q)$ that do not contain p.

Note that oval constructions for $PG(2,q)$ for q an odd/even prime power, can be found in [90] and [91] respectively.

7. Some Improved Results for 500 < n < 1000

From new results obtained since the updated MOLS table in [42] (i.e., those underlined in Table 1), combined with recursive methods, it is possible to update the table for values greater than 500. We include a few such improvements in Theorem 4.

Theorem 4 .

1. 

$N\left(567\right)\ge 8$,

2. 

$N\left(n\right)\ge 10$

for $n\in \{622,635,754,756,764,766,774,778,802,810,822,826,894,906,916,920\}$,

3. 

$N\left(936\right)\ge 8$.

Proof .

(1) Follows from the fact that $567=63\times 9$, then apply Theorem 1. (2) These all follow as a consequence of the fact that $N\left(48\right)\ge 10$, using the same constructions and parameters as given for the corresponding 8 MOLS cases in [54] (making the amendments suggested in Section 6 where required). (3) This follows from the fact that $936=8\times 9\times 13$ and from Theorem 2.2 in [92] which states that, for integers a, b, c with $a\le b\le c$, $N\left(abc\right)\ge min\left\{N\right(a)+1,N(b),N(c\left)\right\}$. □

8. Conclusions and Future Work

Constructions for optimal sets of MOLS have been developed over many years and appear in a plethora of publications. Some of the references we cite in this paper are from many years ago and required a concerted effort to trace. We have implemented constructions for optimal sets of MOLS for all orders n up to 500. In doing so, we have located sources for the constructions, identified parts which require additional explanation, and corrected a small number of errors. The resources we accessed were overwhelmingly accurate and informative—the very small number of errors we found is testament to the quality of the original research. We have deliberately included as few constructions and data in this paper as possible to avoid introducing our own errors (or replacing old errors with new ones). We have included some new sets of MOLS, using recursive methods combined with some improved lower bounds for $N\left(n\right)$ published since the last update to the MOLS table in [42].

Of course, the real credit is due to the originators of the research that led to the constructions that we were able to implement. We believe that our paper will serve as a useful reference and guide for future researchers who not only want to use the existing constructions but also to extend them so that the MOLS table can be updated. Our hope is that our observations and corrections will allow others to implement the constructions while avoiding some of the difficulties we encountered along the way.

There are several interesting cases where finding larger optimal sets seems possible. For example, 22 remains the only value of n (apart from $n=4$) for which three MOLS of order n are known but four are not. It is possible that four MOLS of order 22 could be obtained by the method in [59], using a special kind of $(22,6,2)$ difference matrix. In [93], three MOLS of order 22 were found using this method.

In future work, we intend to continue our implementation of MOLS constructions. This will include implementing existing constructions for $n>500$, and devising constructions for new optimal sets of MOLS. Due to the recursive nature of many MOLS constructions, finding new optimal sets for smaller values of n will have implications for larger values. Having a definitive guide to the MOLS table will allow us to apply the constructions in our related work on combinatorial design constructions, for example, for solutions to Social Golfer-related problems [27].