# Optimal semi-Latin squares

If the cells in a semi-Latin square have an effect over and above the effects of the rows and columns then not all semi-Latin squares are equally efficient. An inflated Latin square is worst possible, because the contrasts between the letters of the original Latin square are completely confounded with contrasts between cells.

Optimal semi-Latin squares are known in some cases.

• Cheng and Bailey, 1991 showed that Trojan squares are optimal.
• Bailey, 1988 showed that if k= r(n-1) then an r-fold inflation of a Trojan square is optimal.
• Bailey, 1992 showed that if n=3 and k is odd then an optimal square can be formed by superposing almost equal numbers of each of a pair of mutually orthogonal Latin squares.

For some values of n and k, optimal semi-Latin squares have been found by exhaustive search. For k < n th search has been restricted to simple semi-Latin squares, also called SOMAs. When k = n-1 then any SOMA must be Trojan, so there is no SOMA unless there is a projective plane of order n.

• Chigbu, 1995 found all optimal semi-Latin squares for n=k=4.
• Bailey and Royle, 1997 found the optimal SOMA for n=6 and k=2.
• Soicher, 1999 found the optimal SOMA for n=6 and k=3, and that there is no SOMA for n=6 and k=4,

Below are optimal semi-Latin squares for small numbers of letters (v = nk). They are colour-coded according to the result which shows optimality. Unless otherwise stated, they are optimal for all the following critieria:
 A maximizes the harmonic mean of the canonical efficiency factors; equivalently, minimizes the average variance of estimators of simple contrasts D maximizes the geometric mean of the canonical efficiency factors; equivalently, minimizes the volume of the ellipsoid of confidence around the estimates of treatment effects E maximizes the minimum of the canonical efficiency factors E' maximizes the minimum of the efficiency factors for simple contrasts; equivalently, minimizes the maximum variance of estimators of simple contrasts

 v=6, n=3, k=2
 A B C D E F C F E B A D E D A F C B

 v=8, n=4, k=2
 A B C D E F G H C H A F G D E B E D G B A H C F G F E H C B A D

 v=9, n=3, k=3
 A B C D E F G H I D E I G H C A B F G H F A B I D E C

 v=10, n=5, k=2
 A B C D E F G H I J I H A J C B E D G F G D I F A H C J E B E J G B I D A F C H C F E H G J I B A D

 v=12, n=3, k=4
 A B C D E F G H I J K L E F K L I J C D A B G H I J G H A B K L E F C D

 v=12, n=4, k=3
 A B C D E F G H I J K L D K I A H L J E C G B F G E L J B I A K F D H C J H F G K C D B L A E I

 v=12, n=6, k=2
 A B C D E F G H I J K L F G A I D K B C H L E J C H F L A G J K B E D I J L B K H I A E D G C F I K E H C J D L A F B G D E D J B L F I C K A H
is A-optimal and D-optimal
 A B C D E F G H I J K L C J F G A H I L D K B E F K A I C L D E B G H J G L H K B D A J C E F I E I B L G J C K F H A D D H E J I K B F A L C G
is E-optimal
 A L D J E K F G B H C I I E B G F D C L A K J H F J I K C H B E D L A G B K A H J G D I C F E L C G F L A I K H E J D B D H E C B L A J G I F K
is E'-optimal

 v=14, n=7, k=2
 A B C D E F G H I J K L M N M J A L C N E B G D I F K H K D M F A H C J E L G N I B I L K N M B A D C F E H G J G F I H K J M L A N C B E D E N G B I D K F M H A J C L C H E J G L I N K B M D A F

 v=15, n=3, k=5
 A B C D E F G H I J K L M N O F G H N O K L M D E A B C I J K L M I J A B C N O F G H D E

 v=15, n=5, k=3
 A B C D E F G H I J K L M N O M K I A N L D B O G E C J H F J E O M H C A K F D N I G B L G N F J B I M E L A H O D K C D H L G K O J N C M B F A E I

 v=16, n=4, k=4
 A B C D E F G H I J K L M N O P E F O L A B K P M N G D I J C H I J G P M N C L A B O H E F K D M N K H I J O D E F C P A B G L

 v=16, n=8, k=2
 A B C D E F G H I J K L M N O P C F A H G B E D K N I P O J M L E J G L A N C P M B O D I F K H G N E P C J A L O F M H K B I D I H K F M D O B A P C N E L G J K D I B O H M F C L A J G P E N M P O N I L K J E H G F A D C B O L M J K P I N G D E B C H A F

 v=18, n=3, k=6
 A B C D E F G H I J K L M N O P Q R G H I P Q R M N O D E F A B C J K L M N O J K L A B C P Q R G H I D E F

 v=18, n=6, k=3
 A L U F K V B G W C H Y D I Z E J X C I V B J U H L Z E F W G K X A D Y D J W A E Z C K U I L X B F Y G H V E K Y H I W A F X D G U J L V B C Z F G Z C D X I J Y A B V E H U K L W B H X G L Y D E V J K Z A C W F I U

 v=18, n=9, k=2
 A B C D E F G H I J K L M N O P Q R E D A F C B K J G L I H Q P M R O N C F E B A D I L K H G J O R Q N M R M H O I Q L A N C P E R G B I D K F Q I M L O H E P A R C N K D G F I B O L Q H M I C R E N A P I F K B G D G N I P K R M B O D Q F A H C I E L K P G R I N Q D M F O B E I A L C H I R K N G P O F Q B M D C L E H A I

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Modified 3/5/00