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.
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Communicated by P. Wild
This work is taken from the author’s thesis [10]
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O’Brien, K.M., Fitzpatrick, P. Bounds on Codes Derived by Counting Components in Varshamov Graphs. Des Codes Crypt 39, 387–396 (2006). https://doi.org/10.1007/s10623-005-6118-6
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DOI: https://doi.org/10.1007/s10623-005-6118-6