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Bounds on Codes Derived by Counting Components in Varshamov Graphs

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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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References

  1. A Barg S Guritman J Simonis (2000) ArticleTitleStrengthening the Gilbert–Varshamov bound Linear Algebra Appl 307 119–129 Occurrence Handle10.1016/S0024-3795(99)00271-2 Occurrence Handle2001g:94016

    Article  MathSciNet  Google Scholar 

  2. Bierbrauer J (2004) Introduction to coding theory. Chapman and Hall

  3. RA Brualdi V Pless (1993) ArticleTitleGreedy codes J Combinat Theory, Series A 64 10–30 Occurrence Handle95a:94010

    MathSciNet  Google Scholar 

  4. Brouwer AE (1998) Bounds on the size of linear codes. In: Pless V, Huffman W (eds), Handbook of coding theory Elsevier Science Online version of the tables: http://www.win.tue.nl/math/dw/voorlincod.html

  5. J Conway NJA Sloane (1986) ArticleTitleLexicographic codes: error correcting codes from game theory IEEE Trans Inform Theory 32 IssueID3 337–347 Occurrence Handle10.1109/TIT.1986.1057187 Occurrence Handle87f:94049

    Article  MathSciNet  Google Scholar 

  6. Edel Y (1996). Eine Verallgemeinerung von BCH-codes. PhD thesis, Univ. Heidelberg

  7. Y Edel J Bierbrauer (1997) ArticleTitleLengthening and the Gilbert-Varshamov bound IEEE Trans Inform Theory 43 991–992 Occurrence Handle10.1109/18.568708 Occurrence Handle98a:94033

    Article  MathSciNet  Google Scholar 

  8. EN Gilbert (1952) ArticleTitleA comparison of signalling alphabets Bell Syst Tech J. 31 504–522

    Google Scholar 

  9. AA Hashim (1978) ArticleTitleImprovement on Varshamov-Gilbert lower bound on minimum Hamming distance of linear codes Proc Inst Eng. 125 104–106 Occurrence Handle80g:94054

    MathSciNet  Google Scholar 

  10. O’Brien K (2003). Constructions and properties of greedy codes. PhD Thesis, Department of Mathematics, University College Cork

  11. V Pless WC Huffman (Eds) (1998) Handbook of coding theory Elsevier Science Amsterdam

    Google Scholar 

  12. RR Varshamov (1957) ArticleTitleEstimate of the number of signals in error correcting codes Dokl Acad Nauk SSSR 117 739–741 Occurrence Handle0081.36905

    MATH  Google Scholar 

  13. AJ Zanten Particlevan (1997) ArticleTitleLexicographic order and linearity Des Codes Cryptogr 10 85–97 Occurrence Handle10.1023/A:1008244404559 Occurrence Handle0868.94054 Occurrence Handle97k:94057

    Article  MATH  MathSciNet  Google Scholar 

  14. van Zanten AJ, Monroe L, Lukito A Dimensions of Standard Lexicodes, Report 96-66, Technische Universiteit Delft

Download references

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Correspondence to Katie M. O’Brien.

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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

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