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Perfect and Related Codes
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)
p2.
Perfect and Related Codes

[1] Some bounds for codes

1. The number of word of length n , weight t
n!
( )
t !(n  t )!
n
t

2. Theorem 3.1.1
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
(n0 )  (1n )  (n2 )  ....  (nt )
p3.
Perfect and Related Codes

3. Theorem 3.1.3 Hamming bound(upper bound)

C: a code of length n, distance d = 2t+1 or 2t+2
2n
C  n
(0 )  (1n )  (n2 )  ....  (nt )

Eg 3.1.4 Give an upper bound of the size of a linear code C of
length n=6 and distance d=3
|C |
26
 6  6

 0

 

  1

64
7
So | C | 9 but the size of a linear code C must be a power of 2 so | C | 8
p4.
Perfect and Related Codes

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)
p5.
Perfect and Related Codes

6. Theorem 3.1.13 Gilbert-Varshamov condition

There exists a linear code of length n, dimension k and distance d
if
(n0 1 )  (1n 1 )  .....  (dn 12 )  2n k
(<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
2n 1
C  n-1
n-1
(0 )  (1n-1 )  (n-1
)

....

(
2
d-2 )
(<pf> choose k such that
then |C| = 2k =
2nk 1 
2n 1
 RHS
2n k 1
(n0 1 )  (1n 1 )  .....  (dn 12 )  2n k
)
p6.
Perfect and Related Codes


Eg 3.1.15 Does there exist a linear code of length n=9, dimension k=2,
and distance d=5? Yes, because
 8  8  8  8
            93  128  292
 0   1   2   3

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?
291
 2.75 but |C| is a power of 2 so |C| ≧ 4
G-V lower bound: |C| ≧
93
29
512
Hamming upper bound: |C| ≦

 11.13
9 9 9
   
0 1  2
1  9  36
but |C| is a power of 2 so |C| ≦ 8
p7.
Perfect and Related Codes


Eg 3.1.17 Does there exists a (15, 7, 5) linear code?
Check G-V condition
 n  1  n  1
 n  1  14 14 14 14

  
  ...  
            
0
1
d

2

 


 0 1 2 3
 1  14  91 364  470  2157  2nk  256
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)
p8.
Perfect and Related Codes

[2]. Perfect Codes

1. Definition:

A code C of length n and odd distance d = 2t+1 is called
perfect code if
2n
C  n
(the Hamming bound)
n
n
n
(0 )  (1 )  (2 )  ....  (t )

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
p9.
Perfect and Related Codes

[3]. Hamming Codes
r
 1. Definition: Hamming code of length 2 -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 
110 


1000111 
101 
0100110



parity check matrix H  011 ,the generator matrix G= 
0010101
100 


0001011



010
001
p10.
Perfect and Related Codes


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)
n
n
n
2r 1
2
2
2
2
2r 1r
 n


2
C
n
n
n
n
n
r
(0 )  (1 )  (2 )  ....  (t ) (0 )  (1 ) 1  n 1  2  1
p11.
Perfect and Related Codes

[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* =  H
0

Why?
H
G H  G b 
0
*
*
j  where j is the nx1 col of all ones.
1
j
 GH Gj  b  0 0  0

1
where Gj+b=0
p12.
Perfect and Related Codes

Eg 3.4.1
C: length 5
10010 
G=  01001 ,


00111
C*: length 6
10010 0 
G*= 01001 0 ,


00111 1
10 
01
 
H= 11  , GH=0
 
10 
01
10 1 
01 1


11 1 
H*= 
 , G*H*=0
10
1


01 1


00 1
p13.
Perfect and Related Codes

[5]. The extended Golay code(C24)

1. Definition
the linear code C24 with generator matrix G= [I, B]
I: 12x12 identity matrix
1 1 0 1 1 1 0 0 0 1 0 1
B:

B
B 1
 j
1

0
1

1

j T  1

0  0

0
0

1
0

1
0 1 1 1 0 0 0 1 0 1 1

1 1 1 0 0 0 1 0 1 1 1
1 1 0 0 0 1 0 1 1 0 1

1 0 0 0 1 0 1 1 0 1 1

0 0 0 1 0 1 1 0 1 1 1
0 0 1 0 1 1 0 1 1 1 1

0 1 0 1 1 0 1 1 1 0 1
1 0 1 1 0 1 1 1 0 0 1

0 1 1 0 1 1 1 0 0 0 1
1 1 0 1 1 1 0 0 0 1 1

1 1 1 1 1 1 1 1 1 1 0
p14.
Perfect and Related Codes

2. Important facts of the extended Golary Code C24

Length n =24, dimension k= 12 , 212=4096 codewords

Parity check matrix
B 
H    (Algorithm 2.5.7)
I 

Another parity check matrix
I 
I 
H    , GH=[I, B]    I 2  B 2  I  BB T  I  I  0
B 
B 




Another generator matrix [B, I]
C24 self-dual; C24  C⊥
24
The distance of C is 8
C24 is a three-error-correcting code
p15.
Perfect and Related Codes

[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
p16.
Perfect and Related Codes

2. Eg. 3.6.2
decode w = 101111101111, 010010010010
s  wH  101111101111  001111101110
= 100000000001
= = > wt ( s )  3 (rule 2)
= = > u  [s, 0]  100000000001, 00000000000
= = > v  w  u  001111101110, 010010010010
p17.
Perfect and Related Codes

[7]. The Golay code

1. C23

Removing a digit from every word in C24
B : 12 x 11 matrix by deleting the last column of B
^


G: 12x23 matrix, G=[I12, B ]

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
p18.
Perfect and Related Codes

[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)= K 2m

RM(r, m) = {(x , x  y ) | x  RM (r , m  1), y  RM (r  1, m  1)}, 0  r  m
2. Eg 3.8.1
R M (0, 0)  {0, 1}
R M (0, 1)  {00, 11}
R M (1, 1)  K 2  {00, 01, 10, 11}
R M (0, 2)  {0000, 1111}
R M (2, 2)  K 4
R M (1, 2)  {(x , x  y ) | x  {00, 01, 10, 11}, y  {00, 11}}
p19.
Perfect and Related Codes

3. Generator matrix G of RM(r, m)
i) 0< r< m
ii) r= 0,
iii)r= m,

G (r , m  1) G (r , m  1) 
G (r , m )  

0
G (r  1, m  1) 


G (0, m )  [11....1]
G (m  1, m ) 
G (m , m )  

0........01


4. Eg 3.8.4 Find G(1,3)
11 11 


G (1, 2)  01 01 


00 11


G (1, 2)
G (1, 3)  
 0
G (1, 2) 

G (0, 2) 

1111

0101
 G (1, 3)  
0011

0000
1111 

0101 
0011 

1111

p20.
Perfect and Related Codes

5. The properties of RM(r,m)
(1) lengt h n= 2m
(2) dist ance d= 2m-r
r
(3) dimension k=  ( mi )
i= 0
(4) RM(r-1,m) is cont ained in RM(r,m), r> 0
(5) dual code RM(m-1-r, m), r< m
p21.
Perfect and Related Codes

[9]. Fast decoding for RM(1,m)

1. The Kronecker product of matrices


A x B = [aijB]
Eg 3.9.1
1 1
H  
I2 

1
-1


1 1 0

1 -1 0
I2  H  
0 0 1

 0 0 1

Definition:
1 0 


0 1
0
1


0
0
H  I2  
1
1


-1
0

H mi  I 2m i  H  I
0 

1 0 1 
0 -1 0 

1 0 -1 

0
1
2i 1
p22.
Perfect and Related Codes

2. Algorithm 3.9.4 decoding the RM(1,m) code
_
1) Replace 0 by -1 in w forming w
_
2) Cimput e w1  w H m1 and wi  wi 1H mi , for i  2, 3, ..., m
3) Find t he posit ion j of t he largest component of wm
4) v( j )  t he binary represent at ion of j(low order digit s first )
if t he jt h component of wm is
posit ive= > t he presummed message is (1, v( j ))
negat ive= > t he presummed message is (0, v( j ))
codeword v = presumed message m x G(1,m)
p23.
Perfect and Related Codes

Eg 3.9.5 m=3, w=10101011
_
1) w  (1, 1, 1, 1, 1, 1, 1, 1)
2)compute :
_
w1  w H 31  (0, 2, 0, 2, 0, 2, 2, 0)
_
w2  w H 32  (0, 4, 0, 0, 2, 2, 2, 2)
_
w 3  w H 33  (2, 6, 2, 2, 2, 2, 2, 2)
3) t he largest component of w 3 is 6 in posit ion 1= > j  1
4) v(1)  100 and 6> 0 = > t he presumed message m = 1100
codeword
1
0
v  1 1 0 0 
0

0
1
1
0
0
1
0
1
0
1
1
1
0
1
0
0
1
1
1
0
1
1
0
1
1
1
1
  1 0 1 0 1 0 1 0
1

1
p24.