Coding Theory and Algebraic Combinatorics

TL;DR

This chapter explores the deep connections between coding theory and algebraic combinatorics, focusing on their interactions with designs, geometries, groups, and packings, highlighting key results, examples, and open problems.

Contribution

It provides a comprehensive overview of the interplay between coding theory and algebraic combinatorics, including recent advances and open challenges.

Findings

Summary of key results in code-combinatorial design interactions

Illustrative examples and constructions in the field

Identification of open problems and future research directions

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.