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