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arXiv · 0806.4979

Bounds on Codes Based on Graph Theory

Salim Y. El Rouayheb·C. N. Georghiades·E. Soljanin·A. Sprintson
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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 Erdős-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.

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Salim Y. El Rouayheb, C. N. Georghiades, E. Soljanin, A. Sprintson. 2008-06-30. Bounds on Codes Based on Graph Theory. https://arxiv.org/abs/0806.4979

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