Book contents
- Frontmatter
- Contents
- Introduction
- 1 A brief introduction to design theory
- 2 Strongly regular graphs
- 3 Quasi-symmetric designs
- 4 Strongly regular graphs with no triangles
- 5 Polarities of designs
- 6 Extension of graphs
- 7 Codes
- 8 Cyclic codes
- 9 Threshold decoding
- 10 Reed-Muller codes
- 11 Self-orthogonal codes and designs
- 12 Quadratic residue codes
- 13 Symmetry codes over GF(3)
- 14 Nearly perfect binary codes and uniformly packed codes
- 15 Association schemes
- References
- Index
15 - Association schemes
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- Contents
- Introduction
- 1 A brief introduction to design theory
- 2 Strongly regular graphs
- 3 Quasi-symmetric designs
- 4 Strongly regular graphs with no triangles
- 5 Polarities of designs
- 6 Extension of graphs
- 7 Codes
- 8 Cyclic codes
- 9 Threshold decoding
- 10 Reed-Muller codes
- 11 Self-orthogonal codes and designs
- 12 Quadratic residue codes
- 13 Symmetry codes over GF(3)
- 14 Nearly perfect binary codes and uniformly packed codes
- 15 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 [21].
Association schemes were introduced by Bose and Shimamoto [13] 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, …, Γn, satisfying the conditions
(i) given p ϵ X, the number ni of q ϵ X with {p, q} ϵΓi depends only on i;
(ii) given p, q ϵ X with {p, q} ϵ Γk the number aijk of r ϵ X with {p, r} ϵ Γi, {q, r} ϵ Γj 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.
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- Chapter
- Information
- Graph Theory, Coding Theory and Block Designs , pp. 96 - 106Publisher: Cambridge University PressPrint publication year: 1975