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Some Notes on the Binary GV Bound for Linear Codes

This document summarizes key findings on the binary GV bound within linear codes, presented at the Sixth International Workshop on Optimal Codes and Related Topics in Varna, Bulgaria, June 2009. Highlights include the greedy algorithm approach, the Varshamov estimate, and significant results about code parameters. The paper outlines proof techniques and explores comparisons with related works. It concludes with a discussion on extending codes, providing improved parameters, and potential for generalization within the realm of coding theory.

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Some Notes on the Binary GV Bound for Linear Codes

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  1. Some Notes on the Binary GV Bound for Linear Codes Sixth International Workshop on Optimal Codes and Related Topics June 16 - 22, 2009, Varna, BULGARIA Dejan Spasov,MarjanGusev

  2. Agenda • Intro • The greedy algorithm • The Varshamov estimate • Main result(s) • Proof outline • Comparison with other results

  3. The Greedy Algorithm • Given d and m; Initialize H • For each • add x to H , if the x is NOT linear combination of d-2 columns of H x

  4. The Varshamov’s Estimate • The greedy code will have parameters AT LEAST as good as the code parameters that satisfy • Example: Let m=32 • The greedy [ 8752, 8720, 5 ] does exist • Varshamov - [ 2954, 2922, 5 ] • Can we find a better estimate?

  5. Main Result • The code can be extended to a code provided • The existence of can be confirmed by the GV bound or recursively until

  6. Some Intuition • Every d -1 columns of are linearly independent • Let and let • This is OK if • But the Varshamov’s estimate will count twice 1 n 2 1 1 2 2 j i

  7. Proof Outline • - all vectors that are linear combination of d-2 columns from H • Find • As long as • Keep adding vectors • - Varshamov bound

  8. Proof Outline Use only odd number of columns

  9. Further Results • The code can be extended to a code provided

  10. Comparison: Elia’s result 1 0 0 0 0

  11. Comparison: A. Barg et al. 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

  12. Comparison: Jiang & Vardy

  13. Comparison: Jiang & Vardy For d/n=const For d/n->0

  14. Conclusion • The greedy [ 8752, 8720, 5 ] does exist • Varshamov- [ 2954, 2922, 5 ] • The Improvement - [ 3100, 3100-32, 5 ] • The asymptotical R≥1-H(δ) ? • Generalization

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