Some Notes on the Binary GV Bound for Linear Codes Sixth International Workshop on Optimal Codes and Related Topics June , 2009, Varna, BULGARIA Dejan Spasov, Marjan Gusev

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

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

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?

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

Some Intuition 1.Every d -1 columns of are linearly independent 2.Let and let 3.This is OK if 4.But the Varshamov’s estimate will count twice 1 1 n n j j i i

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

Proof Outline Use only odd number of columns

Further Results The code can be extended to a code provided

Comparison: Elia’s result

Comparison: A. Barg et al

Comparison: Jiang & Vardy

For d/n=const For d/n->0

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