some notes on the binary gv bound for linear codes

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

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 , if the x is NOT linear combination of d-2 columns of H

2mx F

H 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?

, 2 2mV n d

, ,n k d

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

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

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

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

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

Comparison: Elia’s result

H0000

1

12, 3 2n kV n d

23, 3 2n kV n d

Comparison: A. Barg et al.

H0000

1

0000

0

1

0

0000

0

1

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

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

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