Coding Theory and Algebraic Combinatorics

Abstract

This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.

33 pages; handbook chapter, to appear in: "Selected Topics in Information and Coding Theory", ed. by I. Woungang et al., World Scientific, Singapore, 2010

References in corpus (3)

• New upper bounds for kissing numbers from semidefinite programming
• A Census Of Highly Symmetric Combinatorial Designs
• An extension of Delsarte's method. The kissing problem in three and four dimensions

Cited by in corpus (9)

• Commutative association schemes
• Steiner t-designs for large t
• New Combinatorial Construction Techniques for Low-Density Parity-Check Codes and Systematic Repeat-Accumulate Codes
• Computational complexity of reconstruction and isomorphism testing for designs and line graphs
• Authentication and Secrecy Codes for Equiprobable Source Probability Distributions
• Fuzzy linear codes based on nested linear codes
• Perfect Secrecy Systems Immune to Spoofing Attacks
• Efficient Two-Stage Group Testing Algorithms for Genetic Screening
• Ultimate linear block and convolutional codes