Journal of Statistical Planning andInference 98 (2001) 287–292

www.elsevier.com/locate/jspi

A new construction for e$cient semi-Latin squares

David Bedforda ;∗, Roger M. Whitakerb

aDepartment of Mathematics, Keele University, Keele, Sta�ordshire, ST5 5BG, UKbDepartment of Computer Science, Cardi� University, PO Box 916, Cardi�, Wales, CF24 3XF, UK

Received 10 September 2000; received in revised form 1 October 2000; accepted 14 November 2000

Abstract

An (n×n)=k semi-Latin square has n2 cells (called blocks) in a square array. Each block hask plots, and there are nk treatments which are allocated to plots such that each treatment occursonce in each row and once in each column. For n¿ 4, k �= 4, optimal examples of (n×n)=ksemi-Latin squares with k¿ n have been identi:ed only in the case where k is a multiple ofn−1 such that there exists a set of k mutually orthogonal Latin squares of order n. We give a newconstruction for (n×n)=k semi-Latin squares where n is a prime power, k is any value in the rangen6 k6 (�+ 1)(n− 1) provided there exist � integers ri such that 1¡r1¡r2¡ · · ·¡r�¡n− 1where ri and n − 1 are relatively prime. The semi-Latin squares produced, have the propertythat any pair of treatments occurs together in at most r� blocks. Our construction generalisesthat for Trojan squares in that we use Latin squares obtained from more than one complete setof mutually orthogonal Latin squares. For n= 4 with k = 5, n= 5 with k ∈{5; 6; 7}, n= 7 withk ∈{7; : : : ; 11}, n= 8 with k ∈{8; : : : ; 13} we :nd that the semi-Latin squares produced haveA, D and E values which meet those which are best known, but our designs have a smallermaximum concurrence. In particular, when k = 2(n−1), n∈{4; 5; 7; 8} and n= k = 4 our designsare A-, D- and E- optimal. We also :nd examples of our designs which outperform the bestknown A, D and E valued semi-Latin squares in the E′ criteria. c© 2001 Elsevier Science B.V.All rights reserved.

MSC: 62K05

Keywords: Concurrence; E$ciency factor; Latin square; Mutually orthogonal Latin squares;Optimal design; Semi-Latin square; Trojan square

1. Introduction

An (n×n)=k semi-Latin square has n2 cells (called blocks) arranged in a squarearray. Each block has k plots, and there are nk treatments which are allocated to plotssuch that each treatment occurs once in each row and once in each column. In Fig. 1,we give an example with n= 4 and k = 6.

∗ Corresponding author.E-mail address: d.bedford@maths.keele.ac.uk (D. Bedford).

0378-3758/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reserved.PII: S0378 -3758(00)00303 -7

288 D. Bedford, R.M. Whitaker / J. Statistical Planning and Inference 98 (2001) 287–292

Fig. 1. A (4×4)=6 semi-Latin square.

Traditionally, these designs have been used to compare the eFectiveness of treatmentsin agricultural :eld trials, and their use is well documented. The reader is referred tothe comprehensive papers (Bailey, 1988,1992) for full details of their application andinterpretation and to Raghavarao (1971) for terms used but not de:ned in this paper.

There are many ways of constructing semi-Latin squares (see Bailey, 1988). In thispaper, we concern ourselves with just two of these: superimposition and in?ation.Given a semi-Latin square, the concurrence of a pair of distinct treatments i and j(denoted �ij) is the number of times treatments i and j occur in the same block. Toconstruct a square by inHation, take any (n×n)=k0 semi-Latin square, where k0 is apositive divisor of k and replace each of the treatments by k=k0 new treatments. Theresultant structure is an (n×n)=k semi-Latin square where the values of the concurrencesare n and those of the original design. We may also form a semi-Latin square fromthe superimposition of Latin squares (or semi-Latin squares). Suppose we are given aset S of k Latin squares of order n, such that when any two are superimposed as toform an array of ordered pairs, no ordered pair occurs more than r times. Then we canwrite the k Latin squares on mutually disjoint alphabets and, upon superimposition,we obtain an (n×n)=k semi-Latin square with the property that �ij6 r, ∀i; j. If S is amutually orthogonal set of Latin squares, then the resultant semi-Latin square is calleda Trojan square by Darby and Gilbert (1958) and has the property that �ij6 1 forall i; j.

2. Optimality

To compare semi-Latin squares, we shall apply the A, D, E and E′ measures ofe$ciency to the block design obtained on ignoring rows and columns. A semi-Latinsquare which maximises the A measure of e$ciency among semi-Latin squares of thatsize is said to be A-optimal. D-, E- and E′-optimality are de:ned similarly. Cheng andBailey (1991) showed that every Trojan square is A-, D- and E-optimal. Trojan squaresdo not exist for k¿ n. In this case, a semi-Latin square which is A-, D- and E-optimalis known only where k is a multiple of n − 1 and n is a prime power. Here, Bailey(1992) showed that an A-, D- and E-optimal example can be constructed by inHatingan (n×n)=(n− 1) Trojan square. The problem of constructing e$cient examples withk¿ n where k is not a multiple of n− 1 remains open. However, Bailey (1992) madethe following conjecture.

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Conjecture 1 (Bailey, 1992). If {L1; L2; : : : ; Ln−1} is a set of mutually orthogonalLatin squares of order n and k = a(n − 1) + b with a¿ 1 and 16 b¡n − 1, and �is the superimposition of the (a + 1)-fold in?ations of L1; L2; : : : ; Lb with the a-foldin?ations of Lb+1; : : : ; Ln−1 then � is an A,D and E optimal semi-Latin square.

When k¡n it is widely conjectured that optimal examples lie among those with�ij6 1, but no concensus has emerged on the analogous result when k¿ n. Bailey(1992) showed that any semi-Latin square with k¿ n has �ij¿1 for some i; j. Pat-terson and Williams (1976) suggest the use of designs with �ij6 2. Notice that theconstruction given in Conjecture 1 implies that �ij = n for some i; j. Paterson (1983)suggests the application of designs which minimize the number of small cycles in thetreatment–concurrence graph. The only case which has been fully enumerated is thatfor n= k = 4 (see Bailey and Chigbu, 1997). For these parameters Chigbu (1995) hasestablished that the optimal semi-Latin squares have either �ij ∈{0; 1; 2}, �ij ∈{0; 1; 4},or �ij ∈{0; 1; 2; 4}.

In this paper, we are concerned with constructing optimal (n×n)=k semi-Latin squareswith k¿ n and �ij¡n. By generalising the concept of a Trojan square and superim-posing sets of MOLS written on mutually disjoint alphabets, we obtain a constructionwhich gives e$cient examples for small n. Our construction requires that n is a primepower and that there exist � integers ri such that 1¡r1¡r2¡ · · ·¡r�¡n−1, where riand n−1 are relatively prime. The semi-Latin squares produced have k = (�+1)(n−1)and maximum concurrence of r�.

3. The construction

Central to our method is the construction, due to Bose (1938), that for a primepower n, a set M0 = {L0;1; : : : ; L0; n−1} of MOLS of order n may be constructed fromthe :nite :eld of order n (denoted GF[n]) on setting the (x; y)th entry of L0; j as c ifand only if jy + x − c= 0.

Suppose we have all integers ri such that 1 = r0¡r1¡r2¡ · · ·¡r�¡n−1 with ri andn−1 relatively prime. De:ne sets of Latin squares Mi = {Li;1; : : : ; Li;n−1}, i= 1; : : : ; �;such that the (x; y)th entry of Li; j is c if and only if jyri +x−c= 0 in GF[n]. We claimthat each Mi is a set of MOLS. To see this, note that in order to locate the ordered pair(a; b) in the superimposition of Li; j1 and Li; j2 , j1 �= j2, we need to :nd x and y suchthat j1yri + x= a and j2yri + x= b. On eliminating x we obtain yri = (b− a)=(j2 − j1)which has a unique solution in GF[n] since ri and n− 1 are relatively prime.

Now consider the superimposition, �, of Li;s ∈Mi and Lj; t ∈Mj, where i¡j. Anordered pair (a; b) occurs in the (x; y)th cell of � if and only if (x; y)th cells ofLi;s; Lj; t , respectively, contain a; b, but

syri + x − a= tyrj + x − b⇔ tyrj − syri − (b− a) = 0:

This gives at most rj solutions for y (since a polynomial of degree m over any :eldhas at most m roots) implying that the pair (a; b) can occur in at most rj cells of �.

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Fig. 2. Six Latin squares of order 4 obtained from our construction.

Thus, on superimposing any pair of Latin squares from M0 ∪M1 ∪ · · · ∪M� no orderedpair of elements occurs more than r� times.

Writing the Latin squares from M0 ∪M1 ∪ · · · ∪M� on mutually disjoint alphabets,we note that the resultant Latin squares will superimpose to form a semi-Latin squarewith k = (�+ 1)(n− 1) having the property that �ij6 r�, ∀i; j.

In Fig. 2, we give an example of six Latin squares which are obtained using ourconstruction. These Latin squares then superimpose to form the semi-Latin square inFig. 1.

4. Examples for small n

As far as semi-Latin squares with k¿ n are concerned, examples with n small aremost likely to be of practical use to statisticians. For such parameters, we constructsemi-Latin squares by our method and assess them in terms of e$ciency with respectto the A;D and E criteria. We let a parameter triple (n; k; r�) denote the existence of an(n×n)=k semi-Latin square constructed by our method on extending an (n×n)=(n− 1)Trojan square.

For the parameter triples (4; 5; 2), (5; k; 3) with k ∈{5; 6; 7}, (7; k; 5) with k ∈{7; : : : ;11} and (8; k; 2) with k ∈{8; : : : ; 13}, the associated semi-Latin squares are very e$-cient. We tabulate our results in Fig. 3. Furthermore, we note that the canonical e$-ciency factors for these designs, to four decimal places, match those of the semi-Latinsquares constructed in Conjecture 1. We also note that the semi-Latin squares con-structed in Conjecture 1 are not the same as our examples. This is easily seen onobserving that we have �ij¡n for all i; j whilst those in Conjecture 1 have �ij = n forsome i; j, since they are formed from inHation.

For the parameter triples (4; 4; 2), (n; 2(n − 1); n − 2) n= 4; 5; 7, and (8; 14; 2), thesemi-Latin squares we construct are numerically A-, D- and E-optimal. The examplewith n= 4 and k = 6 is shown in Fig. 1 and the example with n= k = 4 is obtainedfrom this by deleting all entries from {w; x; y; z;W; X; Y; Z}. The example with n= k = 4was :rst given in Bailey (1992), then subsequently identi:ed as optimal (see Chigbu,1995). The remaining examples have e$ciency factors which match, to four decimalplaces, those of the (�+ 1)-fold inHated (n×n)=(n− 1) Trojan square identi:ed as A-,D- and E-optimal by Bailey (1992).

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Fig. 3. A comparison of A; D; E and E′ values.

When k is not a multiple of n − 1, the “inHated Trojan square” in Fig. 3 refers tothe semi-Latin square as constructed in Conjecture 1. Otherwise, when k is a multipleof n− 1, this refers to the two-fold inHated (n×n)=(n− 1) Trojan square identi:ed asA-, D- and E-optimal by Bailey (1992). As far as the E′ values are concerned, weobserve that for the following parameter values, our construction performs better thanthe corresponding inHated Trojan squares: (5; k; 3) for k = 7; 8, (7; k; 5) for k = 8; : : : ; 12,and (8; k; 2) for k = 8; : : : ; 14.

These results show that our construction provides new examples of A-, D- andE-optimal semi-Latin squares with k¿ n, which, in many cases, out perform inHatedTrojan squares in the E′-criteria. Although we have not proved this observation ingeneral, we have established it for parameters which are of practical use. Given these:ndings, we conjecture that the semi-Latin squares produced by our construction areA-, D- and E-optimal.

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Acknowledgements

The authors would like to thank Rosemary Bailey for providing helpful comments.

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