Design Research Group

Lecture notes on the Web

This list gives you access to lecture notes in design theory, finite geometry and related areas of discrete mathematics on the Web. I have given a brief annotation and table of contents for each set of notes.

Note on formats: HTML files should be handled by your browser. Others require special software to display or print them. On my system, these are xdvi (for DVI), ghostview (for PostScript), and acroread (Acrobat reader for PDF). All these are freely available and can be wired into your browser.

Please email me with further information for this section. I will include readers' reviews if anyone is prepared to submit them!

Contents of this page:
There are several sets of lecture notes from the Socrates Intensive Courses in Finite Geometries and Applications.
Ian Anderson and Iiro Honkala, A short course on combinatorial designs, available from
The basics in less than forty pages.
Format: PostScript
  1. Systems of distinct representatives
  2. 2-designs
  3. t-designs and Steiner systems
  4. Codes and designs

R. A. Bailey, Association schemes and partially balanced designs, available from
Notes of a course currently in progress. Written from a statistician's point of view, these notes are complementary to the treatments by Bannai and Ito or by Brouwer, Cohen and Neumaier: they have much to say about methods of calculation and about highly imprimitive association schemes, for example. Also includes an annotated reading list, and course problem sheets. The notes are adapted from a forthcoming book.
Format: PDF
  1. Definitions of association scheme
  2. Adjacency matrices
  3. Some special association schemes
  4. The Bose-Mesner algebra
  5. Character tables
  6. Techniques
  7. Strongly regular graphs
  8. Block designs
  9. Partially balanced block designs
  10. A little statistics
  11. Efficiency
  12. Cyclic designs
  13. Families of partitions
  14. Orthogonal block structures

A. Betten, H. Fripertinger and A. Kerber, Algebraic combinatorics via finite group actions, available from
A very complete survey of enumeration under group action; in interactive HTML, so you can try out the concepts for yourself. Many exercises. But you have to re-configure your X-windows to get the symbols to print correctly.
Format: HTML
  1. Actions (Actions of groups; Bilateral classes, symmetry classes of mappings; Finite symmetric groups; Complete monomial groups; Enumeration of symmetry classes; The involution principle; Special symmetry classes)
  2. Weights (Enumeration by weight; Cycle indicator polynomials; Sums of cycle indicators, recursive methods; A generalization; The Decomposition Theorem; Species)
  3. Marks
  4. Constructions (Orbit evaluation; Transversals of symmetry classes; Orbits of centralizers; Recursion and orderly generation; Generating orbit representatives; Symmetry adapted bases)
  5. Index

Arrigo Bonisoli, On collineation groups of finite planes, (Socrates course notes), available from
The heart of Dembowski's book Finite Geometries, published in 1968, concerned collineations of finite projective planes. This material has never been satisfactorily brought up to date. These notes do so in part, including a lot of recent material on collineation groups fixing an oval. The addendum, available from,
contain exercises and further explanations.
Format: PDF
  1. Introduction
  2. The basics
  3. Some classics
  4. Machinery
  5. Primitive ovals in projective planes of odd order
  6. An excursion into graphs
  7. The uniqueness of the dual Lüneburg plane of order 64
  8. Recent results, open problems
Contents of addendum by K. Thas:
  1. Notation, definitions, and some exercises
  2. Ovals, ovoids, and a theorem concerning the fixed element structure of a collineation of a finite projective plane
  3. Translation nets, translation planes, Moufang planes, and ( p,L)-transitivity
  4. Some easy remarks on the proofs of some propositions
  5. On Proposition 4.3
  6. On Section 5
  7. Moufang sets and collineations of projective planes
  8. Collineations of finite projective planes and the Petersen graph
  9. References

Matthew Brown, (Hyper)ovals and ovoids in projective spaces, (Socrates course notes), available from
An oval or ovoid in a finite projective space of odd characteristic is "classical" (a conic or elliptic quadric), but there are other examples in characteristic 2, and the classification problems are still open. This survey gives proofs of the classical results, and up-to-date information in characteristic 2 including the recent "Adelaide ovals".
Format: PDF
  1. Introduction
  2. (Hyper)ovals in PG(2,q)
  3. Ovoids in PG(3,q)
  4. References

Francis Buekenhout, History and prehistory of polar spaces and of generalized quadrangles, (Socrates course notes), available from
The story of one of the most important topics in finite geometry, told by one who contributed more than anyone else except Jacques Tits to the story.
Format: PDF
  1. Introduction
  2. Projective spaces
  3. Duality and polarity
  4. The classical groups, Jordan 1870 and Dickson 1901
  5. Tits 1956
  6. Veldkamp 1959
  7. Tits 1959, Feit-Higman 1964
  8. Tits 1961-1974
  9. Buekenhout-Shult 1974
  10. To be done
  11. References

Chris K. Caldwell, Graph theory tutorials, available from
Created with WebTutor; you have to register, and you have to do the exercises before you are allowed to proceed. Aimed at beginning graph theorists.
Format: HTML (CGI)
  1. Introduction to graph theory
  2. Euler circuits and paths
  3. Coloring problems
(More tutorials are promised)
Peter J. Cameron, Finite geometry and coding theory, (Socrates course notes), available from
Mainly about quadratic forms over GF(2) and their role in finite geometry, classical and quantum error-correction, extraspecial groups, etc. Includes exercises.
Format: PostScript
  1. Codes
  2. Symplectic and quadratic forms
  3. Reed-Muller codes
  4. Self-dual codes
  5. Bent functions
  6. Kerdock codes
  7. Some resolved designs
  8. Extraspecial 2-groups
  9. Quantum computing
  10. Quantum codes
  11. Z4 codes
  12. Bibliography

Peter J. Cameron, Classical groups, available from
Notes of a lecture course, roughly following Taylor's book: generation and simplicity of classical groups, and some of their geometry. Includes exercises.
Format: DVI, PostScript, PDF
  1. Fields and vector spaces
  2. Linear and projective groups
  3. Polarities and forms
  4. Symplectic groups
  5. Unitary groups
  6. Orthogonal groups
  7. The Klein correspondence and triality
  8. Further topics
  9. A short bibliography on classical groups

Peter J. Cameron, Projective and polar spaces, available from
Second edition (with revisions and corrections) of QMW Maths Notes 13, first published in 1991. Describes the geometry of projective and polar spaces, with extras such as Clifford algebras, Mathieu groups, and diagram geometry. Includes exercises.
Format: PDF
  1. Projective spaces
  2. Projective planes
  3. Coordinatisation of projective spaces
  4. Various topics
  5. Buekenhout geometries
  6. Polar spaces
  7. Axioms for polar spaces
  8. The Klein quadric and triality
  9. The geometry of the Mathieu groups
  10. Exterior powers and Clifford algebras
  11. References and index

Peter J. Cameron, Polynomial aspects of codes, matroids and permutation groups, available from
These notes include background on codes, matroids and permutation groups, and polynomials associated with them (weight enumerator, Tutte polynomial and cycle index), and describe the links between these objects. Their second purpose is to describe codes over Z4 and the associated matroids and permutation groups.
Format: PDF
  1. Codes
  2. Codes over Z4
  3. Matroids
  4. Matroids and codes
  5. Permutation groups
  6. Cycle index
  7. Codes and permutation groups
  8. IBIS groups

Bill Cherowitzo, Combinatorial structures, available from
A combinatorics course, slanted to topics of interest to design theorists and finite geometers. In well-designed HTML, with homework problems.
Format: HTML
  1. Latin squares
  2. Introduction to finite fields
  3. Hadamard matrices
  4. Block designs
  5. Finite geometries

Queen Mary Combinatorics Study Group Papers, available from
An occasional series of expository papers about topics discussed in the Study Group.
Format: DVI, PostScript, PDF
  1. Fibonacci notes, by Peter Cameron and Dima Fon-Dear-Flaass
  2. Notes on quantum error correction, by Harriet Pollatsek and Keldon Drudge
  3. Problems from the First Anglo-Hungarian Meeting on Groups and Geometries
  4. Five lectures on generalized permutation representations, by Thomas Müller
  5. Borcherds' proof of the moonshine conjecture, after V. Nikulin
  6. Partially ordered sets, by Thomas Britz and Peter Cameron
  7. Causal set glossary and bibliography, by Rafael D. Sorkin
  8. A Markov chain for Steiner triple systems, working notes by Peter Cameron
  9. Primitive lambda-roots, by Peter Cameron and Donald Preece
  10. Decoding the Mathieu group M12, by Robert F. Bailey

Stephen Donkin, Linear algebra, available from
An elementary course on linear algebra. Includes both the concrete approach via matrices and the abstract approach via vector spaces.
Format: PDF
  1. Matrices
  2. Inversion, elementary row operations
  3. Determinants
  4. Vector spaces
  5. Linear maps
  6. The rank of a matrix

J. Eisfeld and L. Storme, (Partial) t-spreads and minimal t-covers in finite projective spaces, available from
A survey of this topic, including extendability of partial spreads to spreads and largest/smallest cardinality of partial spreads/covers of projective spaces.
Format: PostScript
  1. t-spreads of projective spaces
  2. t-covers and partial t-spreads in PG(N,q), where t+1 does not divide N+1
  3. Extendability of spreads in PG(3,q)
  4. Partial spreads in PG(N,q), where t+1 divides N+1
  5. Minimal t-covers in PG(N,q), where t+1 divides N+1
  6. General covering and blocking problems in projective spaces
  7. References

Andrew Granville, Arithmetic properties of Binomial Coefficients, available from
From Lucas' Theorem to recent results on congruences of binomial coefficients modulo prime powers. This is a dynamic survey which is expected to develop.
Format: HTML
  1. Introduction
  2. Elementary Number Theory
  3. Generalization of Lucas' Theorem
  4. Congruences for sums of binomial coefficients
  5. Computing binomial coefficients modulo prime powers
  6. Recognizing the primes
  7. Pascal's triangle via cellular automata
  8. Studying binomial coefficients through their generating function
  9. Bernoulli numbers and polynomials
  10. Theorems of Morley and Emma Lehmer and their generalizations
  11. Some useful p-adic numbers
  12. Congruences modulo higher powers of primes
  13. Concluding remarks
  14. References

Willem H. Haemers, Matrix techniques for strongly regular graphs and related geometries, (Socrates course notes), available from
After surveying strongly regular graphs and their generalisations (distance-regular graphs and association schemes), these notes discuss matrix techniques (partitions and interlacing), with an application to the uniqueness of a particular strongly regular graph. The addendum, available from
gives supplementary results, mostly more geometric in character.
Format: PDF
  1. Strongly regular graphs
  2. Association schemes
  3. Matrix tools
  4. The (81, 20, 1, 6) strongly regular graph
  5. References
Contents of addendum by S. Cauchie and E. Kuijken:
  1. Eigenvalues of a regular graph
  2. The friendship property; polarities in projective planes
  3. The line graph of a graph
  4. (Partial) linear spaces and their point and line graphs
  5. Pseudo-geometric graphs
  6. Spreads
  7. References

J. W. P. Hirschfeld, Semi-linear groups over finite fields, (Socrates course notes), available from
Discusses the types of polarities of projective spaces and the semi-linear groups they define.
Format: PostScript
  1. Polarities
  2. Groups on the line
  3. Orders and isomorphisms among the semi-linear groups
  4. Bibliography

W. D. Joyner, The mathematics of Rubik's cube, available from
An introduction to the discrete mathematics and group theory underlying Rubik's cube and other permutation puzzles. A fine example of non-trivial mathematics arising from "diversions". Many worked examples and exercises.
Format: DVI
  1. Logic and sets
  2. Functions, matrices, relations and counting
  3. Permutations
  4. Permutation puzzles
  5. Groups, I
  6. Graphs and "God's Algorithm"
  7. Symmetry groups of the Platonic solids
  8. Groups, II
  9. The Rubik's cube and the word problem
  10. The 2 × 2 and 3 × 3 cube groups
  11. Other Rubik-like puzzle groups
  12. Interesting subgroups of the cube group
  13. Crossing the Rubicon
  14. Appendix: some solution strategies

M. Klin, Ch. Rücker, G. Rücker, G. Tinhofer, Algebraic Combinatorics in Mathematical Chemistry. Methods and Algorithms, I, Permutation Groups and Coherent (Cellular) Algebras, available from
This valuable exposition and survey brings together ideas about graph isomorphism, cellular algebras, permutation groups, and mathematical chemistry.
Format: PostScript
  1. Introduction
  2. The subject of algebraic combinatorics
  3. Problems related to the perception of the symmetry of chemical graphs
  4. Fundamentals of permutation group theory
  5. Centralizer algebras of permutation groups
  6. Cellular algebras
  7. Galois correspondence between permutation groups and cellular algebras
  8. S-rings over cyclic groups
  9. Automorphism groups of certain chemical graphs
  10. Concluding remarks

O. H. King, Classical groups, (Socrates course notes), available from
A general account of classical groups including Aschbacher's Theorem.
Format: PostScript
  1. Forms and groups
  2. Isomorphisms between classical groups
  3. Aschbacher's Theorem
  4. Bibliography

T. W. Müller, Five lectures on generalized permutation representations, available from
Lectures given by the author in an algebra summer school in Brazil, and repeated in the Queen Mary Combinatorics Study Group. They describe techniques for counting representations of an arbitrary finitely generated group in a wreath product H wr Sn (or a variant on this), with applications to such topics as Quillen complexes and subgroup growth.
Format: PostScript
  1. Some combinatorial aspects of permutation representations
  2. Generalizing permutation representations
  3. Some examples and a formula for the exterior function
  4. Explicit formulae for abelian groups and computations in Quillen complexes
  5. Asymptotics of Hom(GH wr Sn) and subgroup growth

László Lovász, Lecture notes, available from
Various sets of notes by Lovász on topics in discrete mathematics. Packed with insight, clear explanation, and many exercises. Recommended.
Format: PostScript
  1. Semidefinite optimization
  2. Topological methods in combinatorics
  3. Complexity of algorithms
  4. (with K. Vesztergombi) Discrete Mathematics

F. Mazzocca, Nuclei in projective planes, available from
A nucleus of a set S of q+1 points in a projective plane of order q is a point all lines through which meet S in just one point. These notes tell what is known about nuclei, and generalisations to affine planes and to "multiple nuclei".
Format: PostScript
  1. Definition and examples
  2. Some results
  3. The Segre-Korchmáros lemma and its applications
  4. Sets with the maximal number of nuclei
  5. Quasi-odd sets
  6. Generalizations
  7. Further generalizations
  8. References

Steven R. Pagano, Matroids and signed graphs, available from
A student's-eye view of these topics - clearly introduced.
Format: HTML
  1. Introduction and some notes
  2. What is a matroid?
  3. Some common examples of matroids
  4. Circuits, bases, rank, closure
  5. Duality
  6. Minors
  7. Representability
  8. Connectivity
  9. What the heck is a signed graph?
  10. What are you trying to find out?
  11. Have you found anything interesting yet?
  12. Any good references for all this stuff?

John Preskill, Quantum information theory and quantum computation, available from
A careful exposition of this important topic, for the Caltech course PH229. Includes new material on quantum error correction of interest to design theorists. Includes exercises. Recommended.
Format: PostScript
  1. Introduction and overview
  2. Foundations of quantum theory I: States and ensembles
  3. Foundations of quantum theory II: Measurements and ensembles
  4. Quantum entanglement
  5. Quantum information theory
  6. Quantum computation
  7. Quantum error correction

D. R. Stinson, Combinatorial designs with selected applications, available from
A detailed account of BIBDs, with many examples and up-to-date applications.
Format: PostScript
  1. Basic definitions and properties of BIBDs
  2. Symmetric BIBDs
  3. Resolvable BIBDs
  4. Steiner triple systems
  5. Orthogonal Latin squares
  6. Authentication codes
  7. Threshold schemes
  8. Visual cryptography
  9. Group testing
  10. Two-point sampling
  11. References

J. A. Thas and H. Van Maldeghem, Embeddings of geometries in finite projective spaces, available from
A timely survey of polar spaces, generalised polygons, and partial geometries, directed towards discussing their embeddings in projective spaces.
Format: PDF
  1. Definitions
  2. Some important finite point-line geometries
  3. Embeddings of generalized quadrangles
  4. Embeddings of polar spaces
  5. Embeddings of partial geometries
  6. Embeddings of the flag geometries of projective planes
  7. Embeddings of generalized hexagons
  8. Polarized, flat and lax embeddings of generalized polygons
  9. Open cases and conjectures
  10. References

H. Wilf, East side, west side, available from
A set of notes accompanying a Maple package which can count, list, choose a random member of ..., combinatorial objects of various types (subsets, partitions or permutations).
Format: PDF
  1. Introduction
  2. About programming in Maple
  3. Sets and subsets
  4. Permutations and their cycles
  5. Set partitions
  6. Integer partitions
  7. ... and all around the town
  8. The EastWest Maple package
  9. Program notes

Other sources for lecture notes on the Web include

Peter J. Cameron
20 May 2003

Design Theory Pages at Queen Mary: | Resources | Lecture Notes | Design Research Group | Maths Research Centre
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