Some Notes on the Binary GV Bound for Linear Codes
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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, Marjan Gusev
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Agenda• Intro
• The greedy algorithm• The Varshamov estimate
• Main result(s)• Proof outline
• Comparison with other results
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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
2mx F
H x
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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?
, 2 2mV n d
, ,n k d
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Main Result• The code can be extended to a
code provided
• The existence of can be confirmed by the GV bound or recursively until
, ,n k d
1, ,n l k l d
min 2,
12
, 2 2d l
n k
ii i
lV n d i
i
, ,n k d
1m d
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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
x x
1i j d
H 1 n2
x
1 2 j
1 2 i
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Proof Outline
• - all vectors that are linear combination of d-2 columns from H
• Find
• As long as • Keep adding vectors
• - Varshamov bound
H
, 2H m d
, 2H m d
, 2 , 2H m d V n d
, 2H m d 12m
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Proof Outline
0 0 1 1 1
0
0
H
12m
m
Use only odd number of columns
min 2,
1
12
, 2 2 , 2d l
m
ii i
lH m d V n d i
i
l
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Further Results• The code can be extended to a
code provided , ,n k d
1, ,n l k l d
min 2,
2 312
, 3 2d l
md i d
ii i
lC V n d C
i
2 2max 1 , 2
2
22
22 0
maxd i d id i i p d
z p d
d i
d iz d p j
C C p
p n pC p
j z j
0 0 1 1 1
0
0
H
3d
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Comparison: Elia’s result
H0000
1
12, 3 2n kV n d
23, 3 2n kV n d
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Comparison: A. Barg et al.
H0000
1
0000
0
1
0
0000
0
1
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Comparison: Jiang & Vardy
2log
2 2
min ,
1 12
10 , 12
log , 1 log , 1
1, 1
6
n M
w id d
w i dw ij
V n d
V n d e n d
n w n we n d
w i i j
2log, 1 2n McV n d
n
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Comparison: Jiang & Vardy
2log, 1 2n McV n d
n
min 2,
12
, 2 2d l
n k
ii i
lV n d i
i
For d/n=const
For d/n->0
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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