Book contents
- Frontmatter
- Contents
- Preface
- 1 A brief introduction to design theory
- 2 Strongly regular graphs
- 3 Quasi-symmetric designs
- 4 Partial geometries
- 5 Strongly regular graphs with no triangles
- 6 Polarities of designs
- 7 Extensions of graphs
- 8 1-factorisations of K6
- 9 Codes
- 10 Cyclic codes
- 11 Threshold decoding
- 12 Finite geometries and codes
- 13 Self-orthogonal codes, designs and projective planes
- 14 Quadratic residue codes
- 15 Symmetry codes over GF(3)
- 16 Nearly perfect binary codes and uniformly packed codes
- 17 Association schemes
- References
- Index
17 - Association schemes
Published online by Cambridge University Press: 16 March 2010
- Frontmatter
- Contents
- Preface
- 1 A brief introduction to design theory
- 2 Strongly regular graphs
- 3 Quasi-symmetric designs
- 4 Partial geometries
- 5 Strongly regular graphs with no triangles
- 6 Polarities of designs
- 7 Extensions of graphs
- 8 1-factorisations of K6
- 9 Codes
- 10 Cyclic codes
- 11 Threshold decoding
- 12 Finite geometries and codes
- 13 Self-orthogonal codes, designs and projective planes
- 14 Quadratic residue codes
- 15 Symmetry codes over GF(3)
- 16 Nearly perfect binary codes and uniformly packed codes
- 17 Association schemes
- References
- Index
Summary
After a short account of the theory of association schemes, this final chapter contains an outline of part of the thesis of P. Delsarte, in which many of the concepts of classical coding theory and design theory are generalised to classes of association schemes. For proofs, we refer the reader to [39].
Association schemes were introduced by Bose and Shimamoto [18] as a generalisation of strongly regular graphs. An association scheme consists of a set X together with a partition of the set of 2-element subsets of X into n classes Γ1,…, Γnsatisfying the conditions
(i)given p ε X, the number ni. of q ε X with {p, q}εΓ1 depends only on i;
(ii) given p, q ε X with {p, q } ε εΓk, the number of rε X with {p, r} εΓi εΓi {q, r } εΓi depends only on i, j, and k. It is convenient to take a set of n ‘colours’ c1, …, cn, and colour an edge of the complete graph on X with colour ci if it belongs to Γi; so Γi. is the ci.-coloured subgraph. The first condition asserts that each graph Γi. is regular; the second, that the number of triangles with given colouring on a given base depends only on the colouring and not on the base. A complementary pair of strongly regular graphs forms an association scheme with two classes, and conversely.
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- Information
- Graphs, Codes and Designs , pp. 124 - 135Publisher: Cambridge University PressPrint publication year: 1980