perfect and related codes. p2. outline [1] some bounds for codes [2] perfect codes [3] hamming...

Perfect and Related Codes

p2.p2.

OUTLINE

[1] Some bounds for codes

[2] Perfect codes

[3] Hamming codes

[4] Extended codes

[5] The extended Golay code

[6] Decoding the extended Golay code

[7] The Golay code

[8] Reed-Muller codes

[9] Fast decoding RM(1,m)

p3.p3.


[1] Some bounds for codes

1. The number of word of length n , weight t

2. Theorem 3.1.1

!

( )!( )!

nt

nt n t

n n n n0 1 2 t

If 0 t n and if v is a word of length n, then the number

of words of length n of distance at most t is

( ) ( ) ( ) .... ( )

p4.p4.


3. Theorem 3.1.3 Hamming bound(upper bound) C: a code of length n, distance d = 2t+1 or 2t+2

Eg 3.1.4 Give an upper bound of the size of a linear code

C of length n=6 and distance d=3

So but the size of a linear code C must be a power of 2 so

n n n n

0 1 2 t

2C

( ) ( ) ( ) .... ( )

n

7

64

1

6

0

62

||6

C

9|| C 8|| C

p5.p5.


4. Theorem 3.1.7 Singleton bound(upper bound) For any (n, k, d) linear codes, d-1≦ n-k (i.e. k ≦ n-d+1 or |C| ≦2n-d+1 )

<pf> the parity check matrix H of an (n,k,d) linear code is an n by n-k matrix such that every d-1 rows of H are independent. Since the rows have length n-k, we can never have more than n-k independent row vectors. Hence d-1≦ n-k.

5. Theorem 3.1.8

For a (n, k, d) linear code C, the following are equivalent:

d = n-k+1

Every n-k rows of parity check matrix are linearly independent

Every k columns of the generator matrix are linearly independent

C is Maximum Distance Separable(MDS) (definition: if d=n-k+1)

p6.p6.

6. Theorem 3.1.13 Gilbert-Varshamov condition There exists a linear code of length n, dimension k and distance d

if

(<pf> design a parity check matrix under this condition. See Ex3.1.22)

7. Corollary 3.1.14 Gilbert-Varshamov bound(lower bound) If n≠1 and d ≠1, there exists a linear code C of length n and distance at

least d with

(<pf> choose k such that

then |C| = 2k = )


1 1 1

0 1 2( ) ( ) ..... ( ) 2n n n n kd

1

n-1 n-1 n-1 n-10 1 2 d-2

2C

( ) ( ) ( ) .... ( )

n

12 kn 1 1 1

0 1 2( ) ( ) ..... ( ) 2n n n n kd

RHSkn

n

1

1

2

2

p7.p7.

Eg 3.1.15 Does there exist a linear code of length n=9,

dimension k=2, and distance d=5? Yes, because

Eg 3.1.16 What is a lower and an upper bound on the size or

the dimension, k, of a linear code with n=9 and d=5?

G-V lower bound: |C| ≧ but |C| is a power of 2 so |C| ≧

4

Hamming upper bound: |C| ≦

but |C| is a power of 2 so |C| ≦ 8


292128933

8

2

8

1

8

0

8

75.293

2 19

13.113691

512

2

9

1

9

0

929

p8.p8.

Eg 3.1.17 Does there exists a (15, 7, 5) linear code?

Check G-V condition

G-V condition does not hold, so G-V bound does not tell us

Whether or not such a code exists.

But actually such a code does exist. (See BCH code later)


2562247036491141

3

14

2

14

1

14

0

14

2

1...

1

1

0

1

715

kn

d

nnn

p9.p9.


[2]. Perfect Codes 1. Definition:

A code C of length n and odd distance d = 2t+1 is called perfect code if

2. Theorem 3.2.8 If C is perfect code of length code of length n and

distance d = 2t+1, then C will correct all error pattern of weight less than or equal t

n n n n

0 1 2 t

2C (the Hamming bound)

( ) ( ) ( ) .... ( )

n

p10.p10.


[3]. Hamming Codes 1. Definition: Hamming code of length 2r-1

A code of length n = 2r-1, r ≧2, having parity check matrix H whose rows of all nonzero vectors of length r

Eg 3.3.1 the Hamming code of length 7(r=3)

111

1101000111

1010100110

parity check matrix 011 ,the generator matrix G=0010101

1000001011

010

001

H

p11.p11.


H contains r rows of weight one, so its r columns are linearly independent.

Thus a Hamming code has dimension k=2r-1-r and contains 2k codewords.

It is a perfect single error-correcting code (d=3)

1. Any two rows of H are lin. indep so d ≧3 2. 100…0, 010…0, and 110…0 are lin. dep. so d ≦3 and thus

d=3 3. Attains the Hamming bound (t=1)

2 12 1

n n n n n n0 1 2 t 0 1

2 2 2 22 C

( ) ( ) ( ) .... ( ) ( ) ( ) 1 1 2 1

r

rn n n

rrn

p12.p12.

[4]Extended Codes 1. Definition: A code C* of length n+1 obtained from C of length n

Construction:

Let G: kxn and choose G*=[G, b] : kx(n+1) where the last col b of G* is appended so that each row of G* has even weight.

Let H: nx(n-k) then H* = where j is the nx1 col of all ones.

Why?

where Gj+b=0


10

jH

00010

**

bGjGH

jHbGHG

p13.p13.

Eg 3.4.1

C: length 5

C*: length 6


10

10010 01

G= 01001 , H= , GH=011

00111 10

01

10 1

01 110010 0

11 1G*= 01001 0 , H*= , G*H*=0

10 100111 1

01 1

00 1

p14.p14.


[5]. The extended Golay code(C24) 1. Definition

the linear code C24 with generator matrix G= [I, B] I: 12x12 identity matrix

B:

011111111111

110001110110

100011101101

100111011010

101110110100

111101101000

111011010001

110110100011

101101000111

111010001110

110100011101

101000111011

01

j

jBB

T

p15.p15.


2. Important facts of the extended Golary Code C24

Length n =24, dimension k= 12 , 212=4096 codewords Parity check matrix

Another parity check matrix

Another generator matrix [B, I] C24 self-dual; The distance of C is 8 C24 is a three-error-correcting code

(Algorithm 2.5.7)

BH

I

2 2 , GH=[I, B] 0TI I

H I B I BB I IB B

⊥24 24C C

p16.p16.


[6]. Decoding the extended Golay code

1. Algorithm 3.6.1 IMLD for Code C24

w: received word 1) s = wH 2) if wt(s)≦3 then u = [ s,0 ](error pattern) 3) if wt(s+bi) ≦2 for some row bi of B then u=[s+bi, ei] 4) compute the second syndrome sB 5) if wt(sB)≦3 then u=[0, sB] 6) if wt(sB+bi)≦2 for some row bi of B then u=[ei, sB+bi] 7) if u is not yet determined then request retransmission

p17.p17.


2. Eg. 3.6.2

decode w = 101111101111, 010010010010

101111101111 001111101110

= 100000000001

==> wt( ) 3 (rule 2)

==> [ ,0] 100000000001,00000000000

==> 001111101110,010010010010

s wH

s

u s

v w u

p18.p18.


[7]. The Golay code

1. C23

Removing a digit from every word in C24

: 12 x 11 matrix by deleting the last column of B

G: 12x23 matrix, G=[I12, ]

Length n=23, dimension k=12, 212=4096 words, distance d=7

Perfect code, three-error-correcting code

Algorithm 3.7.1 (decoding)

Form w0 or w1, which has odd weight

Decode wi(I is 0 or 1) using Algorithm 3.6.1 to a codeword c in C24

Removing the last digit from c

^B

^B

p19.p19.


[8]. Reed-Muller codes

1. r-th order, length 2m, 0≦r≦m, RM(r, m)

RM(0, m)={00…0, 11…1}, RM(m, m)=

RM(r, m) =

2. Eg 3.8.1

{( , ) | ( , 1), ( 1, 1)},0x x y x RM r m y RM r m r m

2mK

2

4

(0,0) {0,1}

(0,1) {00,11} (1,1) {00,01,10,11}

(0,2) {0000,1111} (2,2)

(1,2) {( , ) | {00,01,10,11}, {00,11}}

RM

RM RM K

RM RM K

RM x x y x y

p20.p20.


3. Generator matrix G of RM(r, m)

4. Eg 3.8.4 Find G(1,3)

( , 1) ( , 1)i) 0<r<m ( , )

0 ( 1, 1)

ii) r=0, (0, ) [11....1]

( 1, )iii)r=m, ( , )

0........01

G r m G r mG r m

G r m

G m

G m mG mm

11 11

(1,2) 01 01

00 11

1111 1111

(1,2) (1,2) 0101 0101(1,3) (1,3)

0011 0011 0 (0,2)

0000 1111

G

G GG G

G

p21.p21.


5. The properties of RM(r,m)

m

m-r

r

i=0

(1) length n=2

(2) distance d=2

(3) dimension k= ( )

(4) RM(r-1,m) is contained in RM(r,m), r>0

(5) dual code RM(m-1-r, m), r<m

mi

p22.p22.


[9]. Fast decoding for RM(1,m) 1. The Kronecker product of matrices

A x B = [aijB]

Eg 3.9.1

Definition:

2

2 2

1 1 1 0 I

1 -1 0 1

1 1 0 0 1 0 1 0

1 -1 0 0 0 1 0 1

0 0 1 1 1 0 -1 0

0 0 1 -1 0 1 0 -1

H

I H H I

12 2m i i

imH I H I

p23.p23.


2. Algorithm 3.9.4 decoding the RM(1,m) code

codeword v = presumed message m x G(1,m)

_

_ 11 1

1) Replace 0 by -1 in forming

2) Cimpute and , for 2,3,...,

3) Find the position of the largest component of

4) ( ) the binary representation of j(low order digits first

im mi i

m

w w

w wH w w H i m

j w

v j )

if the th component of is

positive=>the presummed message is (1, ( ))

negative=>the presummed message is (0, ( ))

mj w

v j

v j

p24.p24.


Eg 3.9.5 m=3, w=10101011

codeword

_

_ 11 3

_ 22 3

_ 33 3

3

1) (1, 1,1, 1,1, 1,1,1)

2) :

(0,2,0,2,0,2,2,0)

(0,4,0,0,2,2, 2,2)

(2,6, 2,2, 2,2,2, 2)

3) the largest component of is 6 in position

w

compute

w wH

w wH

w wH

w

1=> 1

4) (1) 100 and 6>0 => the presumed message m = 1100

j

v

01010101

11110000

11001100

10101010

11111111

0011

v

perfect and related codes. p2. outline [1] some bounds for codes [2] perfect codes [3] hamming...

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