Semi-Latin squares, permutation groups, and statistical optimality - - PowerPoint PPT Presentation

Semi-Latin squares, permutation groups, and statistical

• ptimality

Leonard Soicher

Queen Mary, University of London

Ottawa-Carleton Discrete Mathematics Days: 13-14 May 2011

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 1 / 28

Introduction

This talk is about an interaction between certain areas of combinatorial design theory, the statistical theory of optimal designs, computation, and permutation groups.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 2 / 28

Introduction

This talk is about an interaction between certain areas of combinatorial design theory, the statistical theory of optimal designs, computation, and permutation groups. In particular, we shall be interested in semi-Latin squares, which are a generalisation of Latin squares, and are used in the design of comparative experiments.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 2 / 28

Introduction

This talk is about an interaction between certain areas of combinatorial design theory, the statistical theory of optimal designs, computation, and permutation groups. In particular, we shall be interested in semi-Latin squares, which are a generalisation of Latin squares, and are used in the design of comparative experiments. Of particular interest will be “uniform” semi-Latin squares, which generalise complete sets of mutually orthogonal Latin squares.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 2 / 28

Block designs

Definition

A block design is an ordered pair (V , B), such that V is a finite non-empty set of points, and B is a (disjoint from V ) finite non-empty multiset (or collection) of non-empty subsets of V called blocks.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 3 / 28

Block designs

Definition

A block design is an ordered pair (V , B), such that V is a finite non-empty set of points, and B is a (disjoint from V ) finite non-empty multiset (or collection) of non-empty subsets of V called blocks.

Definition

Let t be a non-negative integer. A t-design, or more specifically a t-(v, k, λ) design, is a block design (V , B) such that v = |V |, each block has the same size k, and each t-subset of V is contained in the same positive number λ of blocks.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 3 / 28

Block designs

Definition

A block design is an ordered pair (V , B), such that V is a finite non-empty set of points, and B is a (disjoint from V ) finite non-empty multiset (or collection) of non-empty subsets of V called blocks.

Definition

Let t be a non-negative integer. A t-design, or more specifically a t-(v, k, λ) design, is a block design (V , B) such that v = |V |, each block has the same size k, and each t-subset of V is contained in the same positive number λ of blocks.

Example

The block design (V , B), with V = {1, . . . , 7} and B = [{1, 2, 4}, {2, 3, 5}, {3, 4, 6}, {4, 5, 7}, {1, 5, 6}, {2, 6, 7}, {1, 3, 7}] is a 2-(7, 3, 1) design (and also a 1-(7, 3, 3) design and a 0-(7, 3, 7) design).

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 3 / 28

Statistical design theory

Block designs are of interest to both pure mathematicians and statisticians.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 4 / 28

Statistical design theory

Block designs are of interest to both pure mathematicians and statisticians. They are used by statisticians in the design of comparative experiments: the points represent “treatments” to be compared, and blocks represent sets of homogenous “experimental units” in which treatments are placed.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 4 / 28

Statistical design theory

Block designs are of interest to both pure mathematicians and statisticians. They are used by statisticians in the design of comparative experiments: the points represent “treatments” to be compared, and blocks represent sets of homogenous “experimental units” in which treatments are placed. Statisticians sometimes allow points to be repeated within a block, which here we do not.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 4 / 28

Statistical design theory

Block designs are of interest to both pure mathematicians and statisticians. They are used by statisticians in the design of comparative experiments: the points represent “treatments” to be compared, and blocks represent sets of homogenous “experimental units” in which treatments are placed. Statisticians sometimes allow points to be repeated within a block, which here we do not. There are various measures of the “efficiency” of a block design for the design of an experiment, and we discuss these now in the context

• f 1-designs.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 4 / 28

The (scaled) information matrix

Let ∆ be a 1-(v, k, r) design, with v ≥ 2.

Definition

The concurrence matrix of ∆ is the v × v matrix whose rows and columns are indexed by the points, and whose (α, β)-entry is the number of blocks containing both α and β.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 5 / 28

The (scaled) information matrix

Let ∆ be a 1-(v, k, r) design, with v ≥ 2.

Definition

The concurrence matrix of ∆ is the v × v matrix whose rows and columns are indexed by the points, and whose (α, β)-entry is the number of blocks containing both α and β.

Definition

The scaled information matrix of ∆ is F(∆) := Iv − (rk)−1Λ, where Λ is the concurrence matrix of ∆.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 5 / 28

Canonical efficiency factors

The eigenvalues of F(∆) are all real and lie in [0, 1].

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 6 / 28

Canonical efficiency factors

The eigenvalues of F(∆) are all real and lie in [0, 1]. F(∆) has constant row-sum 0, so the all-1 vector is an eigenvector with corresponding eigenvalue 0.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 6 / 28

Canonical efficiency factors

The eigenvalues of F(∆) are all real and lie in [0, 1]. F(∆) has constant row-sum 0, so the all-1 vector is an eigenvector with corresponding eigenvalue 0. It can be shown that the remaining eigenvalues are all non-zero if and

• nly if ∆ is connected (i.e. its point-block incidence graph is

connected).

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 6 / 28

Canonical efficiency factors

The eigenvalues of F(∆) are all real and lie in [0, 1]. F(∆) has constant row-sum 0, so the all-1 vector is an eigenvector with corresponding eigenvalue 0. It can be shown that the remaining eigenvalues are all non-zero if and

• nly if ∆ is connected (i.e. its point-block incidence graph is

connected).

Definition

Omitting the zero eigenvalue corresponding to the all-1 vector, the eigenvalues δ1 ≤ δ2 ≤ · · · ≤ δv−1 of F(∆) are called the canonical efficiency factors of ∆.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 6 / 28

Efficiency measures for 1-designs

Let ∆ be a 1-(v, k, r) design with v ≥ 2 and canonical efficiency factors δ1 ≤ · · · ≤ δv−1.

Definition

If ∆ is not connected, then A∆ = D∆ = E∆ := 0. Otherwise, we define these efficiency measures by A∆ := (v − 1)/

v−1

• i=1

1/δi, D∆ := v−1

• i=1

δi 1/(v−1) , E∆ := δ1 = min{δ1, . . . , δv−1}.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 7 / 28

On efficiency measures

If k = v (so ∆ has complete blocks) then each of its canonical efficiency factors is equal to 1 and so are each of the efficiency measures above.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 8 / 28

On efficiency measures

If k = v (so ∆ has complete blocks) then each of its canonical efficiency factors is equal to 1 and so are each of the efficiency measures above. If k < v (so ∆ has incomplete blocks), we want to minimize the loss

• f “information” due to blocking, and want the above efficiency

measures to be as close to 1 as possible.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 8 / 28

On efficiency measures

If k = v (so ∆ has complete blocks) then each of its canonical efficiency factors is equal to 1 and so are each of the efficiency measures above. If k < v (so ∆ has incomplete blocks), we want to minimize the loss

• f “information” due to blocking, and want the above efficiency

measures to be as close to 1 as possible. To test conjectures and rank designs we have classified, we need to be able to compare efficiency measures exactly. I can do this using algebraic computation in GAP, and this functionality should be included in the next release of the DESIGN package for GAP.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 8 / 28

Statistical optimality

Let ∆ be a 1-(v, k, r) design with v ≥ 2 and canonical efficiency factors δ1 ≤ · · · ≤ δv−1.

Definition

∆ is A-optimal in a class C of 1-(v, k, r) designs containing ∆ if A∆ ≥ AΓ for each Γ ∈ C. D-optimal and E-optimal are defined similarly.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 9 / 28

Statistical optimality

Let ∆ be a 1-(v, k, r) design with v ≥ 2 and canonical efficiency factors δ1 ≤ · · · ≤ δv−1.

Definition

∆ is A-optimal in a class C of 1-(v, k, r) designs containing ∆ if A∆ ≥ AΓ for each Γ ∈ C. D-optimal and E-optimal are defined similarly.

Definition

∆ is Schur-optimal in a class C of 1-(v, k, r) designs containing ∆ if for each design Γ ∈ C, with canonical efficiency factors γ1 ≤ · · · ≤ γv−1, we have ℓ

i=1 δi ≥ ℓ i=1 γi, for ℓ = 1, . . . , v − 1.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 9 / 28

On Schur-optimality

A Schur-optimal design need not exist within a given class C of 1-(v, k, r) designs, but when it does, that design is optimal in C with respect to a very wide range of statistical optimality criteria, including being A- D- and E-optimal (Giovagnoli and Wynn (1981)).

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 10 / 28

On Schur-optimality

A Schur-optimal design need not exist within a given class C of 1-(v, k, r) designs, but when it does, that design is optimal in C with respect to a very wide range of statistical optimality criteria, including being A- D- and E-optimal (Giovagnoli and Wynn (1981)). If ∆ is a 2-(v, k, λ) design, then the canonical efficiency factors of ∆ are all equal (to v(k − 1)/((v − 1)k)), from which it follows that ∆ is Schur-optimal in the class of all 1-(v, k, λ(v − 1)/(k − 1)) designs. (However, a 2-design may well not exist with the properties we are interested in.)

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 10 / 28

Semi-Latin squares

Definition

An (n × n)/k semi-Latin square is an n × n array S, whose entries are k-subsets of an nk-set Ω, the set of symbols for S, such that each symbol is in exactly one entry in each row and exactly one entry in each column of

• S. The entry in row i and column j is called the (i, j)-entry of S and is

denoted by S(i, j). To avoid trivialities, we assume throughout that n > 1, k > 0.

Example

Here is a (5 × 5)/3 semi-Latin square: 1 2 11 3 4 12 5 6 13 7 8 14 9 10 15 3 5 15 1 7 11 2 9 12 4 10 13 6 8 14 4 6 14 2 8 15 1 10 11 3 9 12 5 7 13 7 9 13 5 10 14 3 8 15 2 6 11 1 4 12 8 10 12 6 9 13 4 7 14 1 5 15 2 3 11

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 11 / 28

On semi-Latin squares

Note that an (n × n)/1 semi-Latin square is (essentially) the same thing as a Latin square of order n.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 12 / 28

On semi-Latin squares

Note that an (n × n)/1 semi-Latin square is (essentially) the same thing as a Latin square of order n. Semi-Latin squares have many applications, including the design of agricultural experiments, consumer testing, and message authentication.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 12 / 28

On semi-Latin squares

Note that an (n × n)/1 semi-Latin square is (essentially) the same thing as a Latin square of order n. Semi-Latin squares have many applications, including the design of agricultural experiments, consumer testing, and message authentication. Two (n × n)/k semi-Latin squares are isomorphic if one can be

• btained from the other by a sequence of one or more of: a row

permutation, a column permutation, transposing, and renaming symbols.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 12 / 28

On semi-Latin squares

Note that an (n × n)/1 semi-Latin square is (essentially) the same thing as a Latin square of order n. Semi-Latin squares have many applications, including the design of agricultural experiments, consumer testing, and message authentication. Two (n × n)/k semi-Latin squares are isomorphic if one can be

• btained from the other by a sequence of one or more of: a row

permutation, a column permutation, transposing, and renaming symbols. Semi-Latin squares exist in profusion, and a good choice of semi-Latin square for a given application can be very important.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 12 / 28

Underlying block design

Let S be an (n × n)/k semi-Latin square. If we ignore the row and column structure of S, we obtain its underlying block design ∆(S), whose points are the symbols of S and whose blocks are the entries of S.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 13 / 28

Underlying block design

Let S be an (n × n)/k semi-Latin square. If we ignore the row and column structure of S, we obtain its underlying block design ∆(S), whose points are the symbols of S and whose blocks are the entries of S. Note that ∆(S) is a 1-(nk, k, n) design.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 13 / 28

Underlying block design

Let S be an (n × n)/k semi-Latin square. If we ignore the row and column structure of S, we obtain its underlying block design ∆(S), whose points are the symbols of S and whose blocks are the entries of S. Note that ∆(S) is a 1-(nk, k, n) design. Following the analysis of Bailey (1992), the A-, D-, and E-measures

• f S are those of ∆(S), and S is optimal with respect to a statistical
• ptimality criterion if ∆(S) is optimal with respect to that criterion in

the class of underlying block designs of (n × n)/k semi-Latin squares.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 13 / 28

Inflation

Definition

Let s be a positive integer. An s-fold inflation of an (n × n)/k semi-Latin square is obtained by replacing each symbol α in the semi-Latin square by s symbols σα,1, . . . , σα,s, such that σα,i = σβ,j if and only if α = β and i = j. The result is an (n × n)/(ks) semi-Latin square.

Example

Here is a (3 × 3)/2 semi-Latin square formed by a 2-fold inflation of a Latin square of order 3: 1 4 2 5 3 6 3 6 1 4 2 5 2 5 3 6 1 4

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 14 / 28

Superposition

Definition

The superposition of an (n × n)/k semi-Latin square with an (n × n)/ℓ semi-Latin square (with disjoint symbol sets) is performed by superimposing the first square upon the second, resulting in an (n × n)/(k + ℓ) semi-Latin square.

Example

Here is a (3 × 3)/2 semi-Latin square which is the superposition of two (mutually orthogonal) Latin squares of order 3: 1 4 2 5 3 6 3 5 1 6 2 4 2 6 3 4 1 5

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 15 / 28

Some optimality results

Cheng and Bailey (1991) proved that a superposition of k MOLS of

• rder n (with pairwise disjoint symbol sets) is A-, D-, and E-optimal.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 16 / 28

Some optimality results

Cheng and Bailey (1991) proved that a superposition of k MOLS of

• rder n (with pairwise disjoint symbol sets) is A-, D-, and E-optimal.

Bailey (1992) proved that for all s ≥ 1, an s-fold inflation of the superpostion of n − 1 MOLS of order n is A-, D-, and E-optimal.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 16 / 28

Some optimality results

Cheng and Bailey (1991) proved that a superposition of k MOLS of

• rder n (with pairwise disjoint symbol sets) is A-, D-, and E-optimal.

Bailey (1992) proved that for all s ≥ 1, an s-fold inflation of the superpostion of n − 1 MOLS of order n is A-, D-, and E-optimal. I have performed various searches, classifications and rankings of semi-Latin squares using GAP and its GRAPE and DESIGN packages. This includes the complete classification (up to isomorphism) of the (4 × 4)/k semi-Latin squares with k ≤ 10, and the ranking of these squares with respect to their A-, D-, and E-measures.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 16 / 28

SOMA(k, n)s

Definition

A SOMA(k, n) is an (n × n)/k semi-Latin square with the property that any two distinct symbols occur together in at most one entry. The superposition of k MOLS of order n (with pairwise disjoint symbol sets) is a SOMA(k, n).

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 17 / 28

SOMA(k, n)s

Definition

A SOMA(k, n) is an (n × n)/k semi-Latin square with the property that any two distinct symbols occur together in at most one entry. The superposition of k MOLS of order n (with pairwise disjoint symbol sets) is a SOMA(k, n). When a SOMA(k, n) exists, it is believed that for the A-, D-, and E-measures, optimal (n × n)/k semi-Latin squares can be found amongst the SOMA(k, n)s.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 17 / 28

SOMA(k, n)s

Definition

A SOMA(k, n) is an (n × n)/k semi-Latin square with the property that any two distinct symbols occur together in at most one entry. The superposition of k MOLS of order n (with pairwise disjoint symbol sets) is a SOMA(k, n). When a SOMA(k, n) exists, it is believed that for the A-, D-, and E-measures, optimal (n × n)/k semi-Latin squares can be found amongst the SOMA(k, n)s. The SOMA(k, 6)s have been classified and ranked for k = 2, 3 (Bailey and Royle (1997), Soicher). There is no SOMA(k, 6) when k > 3, but I am determining efficient (6 × 6)/k semi-Latin squares for k = 4, . . . , 10.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 17 / 28

Almost as good a three MOLS of order 10?

It is not known whether there are three MOLS of order 10, but (assuming certain groups of symmetries) I have discovered and ranked various SOMA(3, 10)s and SOMA(4, 10)s. (I don’t know whether a SOMA(k, 10) exists with 4 < k < 8.)

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 18 / 28

Almost as good a three MOLS of order 10?

It is not known whether there are three MOLS of order 10, but (assuming certain groups of symmetries) I have discovered and ranked various SOMA(3, 10)s and SOMA(4, 10)s. (I don’t know whether a SOMA(k, 10) exists with 4 < k < 8.) For example, the SOMA(3, 10) below (which can be seen to be the superposition of a Latin square of order 10 with a (10 × 10)/2 square, and has an automorphism group of size 20) has the largest A-measure (33220051/48764357 ≈ 0.6812) of any (10 × 10)/3 semi-Latin square known to me. If three MOLS of order 10 were to exist then their superposition would have A-measure equal to 58/85 ≈ 0.6824.

1 11 16 2 13 14 3 21 23 4 26 29 5 17 20 7 18 22 6 25 28 8 15 27 9 19 30 10 12 24 2 18 19 1 12 17 4 11 15 5 22 25 6 27 30 10 13 21 7 20 23 3 24 26 8 14 29 9 16 28 3 28 29 4 16 20 1 13 18 6 12 14 2 23 24 9 17 26 10 15 22 7 19 25 5 21 27 8 11 30 4 21 25 5 26 30 6 17 19 1 15 20 3 11 13 8 12 28 9 18 27 10 14 23 7 16 24 2 22 29 5 12 15 6 22 24 2 27 28 3 16 18 1 14 19 4 23 30 8 13 26 9 20 29 10 11 25 7 17 21 8 20 24 3 25 27 7 14 26 10 19 28 9 15 21 6 11 29 2 16 17 1 22 30 4 12 13 5 18 23 9 14 22 8 19 21 5 24 29 7 11 27 10 16 26 2 20 25 3 12 30 4 17 18 1 23 28 6 13 15 10 17 27 9 11 23 8 16 22 2 21 30 7 12 29 3 14 15 4 19 24 5 13 28 6 18 20 1 25 26 7 13 30 10 18 29 9 12 25 8 17 23 4 22 28 1 24 27 5 11 14 6 16 21 2 15 26 3 19 20 6 23 26 7 15 28 10 20 30 9 13 24 8 18 25 5 16 19 1 21 29 2 11 12 3 17 22 4 14 27 Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 18 / 28

Uniform semi-Latin squares

Definition

An (n × n)/k semi-Latin square S is uniform if every pair of entries of S, not in the same row or column, intersect in a constant number µ = µ(S)

• f points.

Example

Here is a uniform (3 × 3)/2 semi-Latin square S with µ(S) = 1: 1 4 2 5 3 6 3 5 1 6 2 4 2 6 3 4 1 5

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 19 / 28

On uniform semi-Latin squares

Lemma

If S is a uniform (n × n)/k semi-Latin square then k = µ(S)(n − 1).

Lemma

If S is a uniform semi-Latin square then an s-fold inflation of S is also uniform, and if S and T are both n × n uniform semi-Latin squares (with disjoint symbol sets) then the superposition of S and T is also uniform.

Lemma

An (n × n)/(n − 1) semi-Latin square S is uniform if and only if S is a superposition of n − 1 MOLS of order n.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 20 / 28

The optimality of uniform semi-Latin squares

The statistical importance of uniform semi-Latin squares is due to the following:

Theorem

Suppose that S is an uniform (n × n)/k semi-Latin square. Then S is Schur-optimal; that is, ∆(S) is Schur-optimal in the class of underlying block designs of (n × n)/k semi-Latin squares.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 21 / 28

Proof.

Outline: The dual ∆(S)∗ of ∆(S) is a partially balanced incomplete-block design with respect to the L2-type association scheme, so we may determine the canonical efficiency factors of ∆(S)∗, and hence those

• f ∆(S) (these differ only in the number equal to 1).

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 22 / 28

Proof.

Outline: The dual ∆(S)∗ of ∆(S) is a partially balanced incomplete-block design with respect to the L2-type association scheme, so we may determine the canonical efficiency factors of ∆(S)∗, and hence those

• f ∆(S) (these differ only in the number equal to 1).

The canonical efficiency factors of ∆(S) turn out to be 1 − 1/(n − 1), with multiplicity (n − 1)2, and 1, with multiplicity nk − 1 − (n − 1)2.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 22 / 28

Proof.

Outline: The dual ∆(S)∗ of ∆(S) is a partially balanced incomplete-block design with respect to the L2-type association scheme, so we may determine the canonical efficiency factors of ∆(S)∗, and hence those

• f ∆(S) (these differ only in the number equal to 1).

The canonical efficiency factors of ∆(S) turn out to be 1 − 1/(n − 1), with multiplicity (n − 1)2, and 1, with multiplicity nk − 1 − (n − 1)2. The underlying block design of any (n × n)/k semi-Latin square has at most (n − 1)2 canonical efficiency factors not equal to 1.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 22 / 28

Proof.

Outline: The dual ∆(S)∗ of ∆(S) is a partially balanced incomplete-block design with respect to the L2-type association scheme, so we may determine the canonical efficiency factors of ∆(S)∗, and hence those

• f ∆(S) (these differ only in the number equal to 1).

The canonical efficiency factors of ∆(S) turn out to be 1 − 1/(n − 1), with multiplicity (n − 1)2, and 1, with multiplicity nk − 1 − (n − 1)2. The underlying block design of any (n × n)/k semi-Latin square has at most (n − 1)2 canonical efficiency factors not equal to 1. The result follows.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 22 / 28

The existence of uniform semi-Latin squares

For a given n, the question is: what are the values of µ for which there exist uniform (n × n)/((n − 1)µ) semi-Latin squares?

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 23 / 28

The existence of uniform semi-Latin squares

For a given n, the question is: what are the values of µ for which there exist uniform (n × n)/((n − 1)µ) semi-Latin squares? Since the µ-fold inflation of a uniform semi-Latin square is uniform, the existence of a uniform (n × n)/((n − 1)µ) semi-Latin square for all integers µ > 0 is equivalent to the existence of a set of n − 1 MOLS

• f order n, and such a set exists if n is a prime power (and it is a

major unsolved problem whether such a set exists for some n not a prime power).

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 23 / 28

The existence of uniform semi-Latin squares

For a given n, the question is: what are the values of µ for which there exist uniform (n × n)/((n − 1)µ) semi-Latin squares? Since the µ-fold inflation of a uniform semi-Latin square is uniform, the existence of a uniform (n × n)/((n − 1)µ) semi-Latin square for all integers µ > 0 is equivalent to the existence of a set of n − 1 MOLS

• f order n, and such a set exists if n is a prime power (and it is a

major unsolved problem whether such a set exists for some n not a prime power). There are not even two MOLS of order 6. However, I can show that there exist uniform (6 × 6)/(5µ) semi-Latin squares for all integers µ > 1.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 23 / 28

The existence of uniform semi-Latin squares

For a given n, the question is: what are the values of µ for which there exist uniform (n × n)/((n − 1)µ) semi-Latin squares? Since the µ-fold inflation of a uniform semi-Latin square is uniform, the existence of a uniform (n × n)/((n − 1)µ) semi-Latin square for all integers µ > 0 is equivalent to the existence of a set of n − 1 MOLS

• f order n, and such a set exists if n is a prime power (and it is a

major unsolved problem whether such a set exists for some n not a prime power). There are not even two MOLS of order 6. However, I can show that there exist uniform (6 × 6)/(5µ) semi-Latin squares for all integers µ > 1. It is a celebrated computational result (Lam et al. (1989)) that there is no projective plane of order 10, so the the current smallest open case is whether there exists a uniform (10 × 10)/18 semi-Latin square.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 23 / 28

Uniform semi-Latin squares from 2-transitive groups

Let G be a transitive permutation group of order nk on the set {1, . . . , n}, with n > 1.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 24 / 28

Uniform semi-Latin squares from 2-transitive groups

Let G be a transitive permutation group of order nk on the set {1, . . . , n}, with n > 1. G determines an (n × n)/k semi-Latin square, denoted SLS(G), having symbol set G itself, and with symbol g in the (i, j)-entry if and only if ig = j. (More generally, semi-Latin squares can be constructed using “transitive multisets” of permutations.)

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 24 / 28

Uniform semi-Latin squares from 2-transitive groups

Let G be a transitive permutation group of order nk on the set {1, . . . , n}, with n > 1. G determines an (n × n)/k semi-Latin square, denoted SLS(G), having symbol set G itself, and with symbol g in the (i, j)-entry if and only if ig = j. (More generally, semi-Latin squares can be constructed using “transitive multisets” of permutations.) The study of semi-Latin squares of the form SLS(G) was started in Soicher (submitted).

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 24 / 28

Uniform semi-Latin squares from 2-transitive groups

Let G be a transitive permutation group of order nk on the set {1, . . . , n}, with n > 1. G determines an (n × n)/k semi-Latin square, denoted SLS(G), having symbol set G itself, and with symbol g in the (i, j)-entry if and only if ig = j. (More generally, semi-Latin squares can be constructed using “transitive multisets” of permutations.) The study of semi-Latin squares of the form SLS(G) was started in Soicher (submitted). SLS(G) is uniform if and only if G is 2-transitive. (More generally, uniform semi-Latin squares come from “2-transitive multisets” of permutations.)

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 24 / 28

SLS(PGL2(q))

Using the Classification of Finite Simple Groups, all the finite 2-transitive permutation groups have been classified, and tables of these groups are given in Cameron (1999). For example, consideration of the 2-transitive groups PGL2(q) and PSL2(q), of degree q + 1, where q is a prime power, yields the following result:

Theorem

Let q be a prime power. Then there exists a uniform, and hence Schur-optimal, ((q + 1) × (q + 1))/(q(q − 1)) semi-Latin square S :=SLS(PGL2(q)), which is a superposition of Latin squares each isomorphic to SLS(Cq+1), and in which every pair of distinct symbols

• ccur together in at most two entries. Moreover, if q is odd then S is also

a superposition of two uniform ((q + 1) × (q + 1))/(q(q − 1)/2) semi-Latin squares isomorphic to SLS(PSL2(q)).

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 25 / 28

Refrences

R.A. Bailey, Efficient semi-Latin squares, Statist. Sinica 2 (1992), 413–437. R.A. Bailey, Semi-Latin squares, http://www.maths.qmul.ac.uk/∼rab/sls.html R.A. Bailey and P.J. Cameron, Combinatorics of optimal designs, in Surveys in Combinatorics 2009, S. Huczynska et al. (eds), Cambridge University Press, Cambridge, 2009, pp. 19–73. R.A. Bailey and G. Royle, Optimal semi-Latin squares with side six and block size two, Proc. Roy. Soc. London Ser. A 453 (1997), 1903–1914. P.J. Cameron, Permutation Groups, Cambridge University Press, Cambridge, 1999.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 26 / 28

References

C.-S. Cheng and R.A. Bailey, Optimality of some two-associate-class partially balanced incomplete-block designs, Ann. Statist. 19 (1991), 1667–1671. The DesignTheory.org website, http://designtheory.org The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.4.12, 2008, http://www.gap-system.org

• A. Giovagnoli and H.P. Wynn, Optimum continuous block designs, Proc.
• Roy. Soc. London Ser. A 377 (1981), 405–416.

C.W.H. Lam, L. Thiel and S. Swiercz, The non-existence of finite projective planes of order 10, Canad. J. Math. 41 (1989), 1117–1123. W.J. Martin and B.E. Sagan, A new notion of transitivity for groups and sets of permutations, J. London Math. Soc. 73 (2006), 1–13.

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 27 / 28

References

L.H. Soicher, On the structure and classification of SOMAs: generalizations of mutually orthogonal Latin squares, Electron. J. Combin. 6 (1999), R32, 15 pp. L.H. Soicher, SOMA Update, http://www.maths.qmul.ac.uk/∼leonard/soma/ L.H. Soicher, The DESIGN package for GAP, Version 1.4, 2009, http://designtheory.org/software/gap design/ L.H. Soicher, Uniform semi-Latin squares and their Schur optimality, submitted for publication. Preprint available at: http://www.maths.qmul.ac.uk/∼leonard/optimalsls v3.pdf

Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 28 / 28