Bounds on Codes Based on Graph Theory

TL;DR

This paper presents a unified graph-theoretic framework to derive and extend bounds on the maximum size of error-correcting codes, connecting algebraic graph theory and combinatorics.

Contribution

It introduces a single framework that simplifies deriving known bounds and enables the discovery of new bounds for error-correcting codes using graph properties.

Findings

Unified derivation of Hamming and Singleton bounds

New bounds on code sizes using graph-theoretic techniques

Applications to various coding theory problems

Abstract

Let $A_q(n,d)$ be the maximum order (maximum number of codewords) of a $q$-ary code of length $n$ and Hamming distance at least $d$. And let $A(n,d,w)$ that of a binary code of constant weight $w$. Building on results from algebraic graph theory and Erd\H{o}s-ko-Rado like theorems in extremal combinatorics, we show how several known bounds on $A_q(n,d)$ and $A(n,d,w)$ can be easily obtained in a single framework. For instance, both the Hamming and Singleton bounds can derived as an application of a property relating the clique number and the independence number of vertex transitive graphs. Using the same techniques, we also derive some new bounds and present some additional applications.