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Designs, Codes and Cryptography

, Volume 39, Issue 3, pp 387–396 | Cite as

Bounds on Codes Derived by Counting Components in Varshamov Graphs

  • Katie M. O’Brien
  • Patrick Fitzpatrick
Article

Abstract

We are interested in improving the Varshamov bound for finite values of length n and minimum distance d. We employ a counting lemma to this end which we find particularly useful in relation to Varshamov graphs. Since a Varshamov graph consists of components corresponding to low weight vectors in the cosets of a code it is a useful tool when trying to improve the estimates involved in the Varshamov bound. We consider how the graph can be iteratively constructed and using our observations are able to achieve a reduction in the over-counting which occurs. This tightens the lower bound for any choice of parameters n, k, d or q and is not dependent on information such as the weight distribution of a code.

Keywords

Varshamov bound Varshamov graph Greedy codes 

AMS Classfication

94B65 94B25 

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Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Department of Engineering MathematicsUniversity of BristolEngland
  2. 2.Department of MathematicsUniversity College CorkIreland

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