4 research outputs found
Asymptotic Improvement of the Gilbert-Varshamov Bound on the Size of Binary Codes
Given positive integers and , let denote the maximum size
of a binary code of length and minimum distance . The well-known
Gilbert-Varshamov bound asserts that , where
is the volume of a Hamming sphere of
radius . We show that, in fact, there exists a positive constant such
that whenever . The result follows by recasting the Gilbert- Varshamov bound into a
graph-theoretic framework and using the fact that the corresponding graph is
locally sparse. Generalizations and extensions of this result are briefly
discussed.Comment: 10 pages, 3 figures; to appear in the IEEE Transactions on
Information Theory, submitted August 12, 2003, revised March 28, 200
A Generalized Gilbert-Varshamov Bound Derived via Analysis of a Code-Search Algorithm
This correspondence derives a generalization of the Gilbert- Varshamov bound that is applicable to block codes whose codewords must be drawn from irregular sets; the bound improves by a factor of four a similar result recently published by Kolesnik and Krachkovsky. It is derived by analyzing a code search algorithm we refer to as the "Altruistic Algorithm". This algorithm iteratively deletes potential codewords so that at each iteration the "worst" candidate is removed; the bound is derived by demonstrating that, as the algorithm proceeds, the average volume of a sphere of a given radius approaches the maximum such volume and so a bound previously expressed in terms of the maximum volume can in fact be expressed in terms of the average volume. Examples of applications where the bound is relevant include error-correcting (d, k)- constrained codes and binary codes for code division multiple access