CODING THEORY AND ALGEBRAIC COMBINATORICS

2008/11/08 by Michael Huber · 1 citation
Computer Science · Engineering · Mathematics · #Algebra over a field #Algebraic combinatorics #Algebraic number #Algebraic structure #Association scheme #Cellular Automata and Applications #Coding (social sciences) #Coding theory #Coding theory and cryptography #Combinatorial design #Combinatorics #Computer science #Enumerative combinatorics #Epistemology #Extremal combinatorics #Focus (optics) #Mathematics #Physics #Pure mathematics #Simple (philosophy) #Theoretical computer science #cs.IT #graph theory and CDMA systems #math.CO #math.IT

paper · pdf · doi:10.1142/9789812837172_0004

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

arxiv created 2008/11/08 · openalex publication_date 2010/02/01 · arxiv updated 2018/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• On Harmonic Weight Enumerators of Binary Codes
• A Mathematical Theory of Communication
• AN EXAMINATION OF THE DIFFERENT POSSIBLE SOLUTIONS OF A PROBLEM IN INCOMPLETE BLOCKS
• On the Nonexistence of Perfect Codes over Finite Fields
• On Linear Associative Algebras Corresponding to Association Schemes of Partially Balanced Designs
• Some Sphere Packings in Higher Space
• A survey of perfect codes
• New upper bounds for kissing numbers from semidefinite programming
• Association schemes and coding theory
• A new combinatorial approach to M24
• The Nonexistence of Certain Finite Projective Planes
• New upper bounds on sphere packings I
• Twenty-three constructions for the Leech lattice
• The Search for a Finite Projective Plane of Order 10
• Sphere Packings, Lattices and Groups

Cited by

• New Combinatorial Construction Techniques for Low-Density Parity-Check Codes and Systematic Repeat-Accumulate Codes
• Mathematical Methods in Computer Science: Essays in Memory of Thomas Beth