The g can constructed from gx by first constructing

Cyclic codes can be realized by the generating matrix, G.
•
The G can constructed from g ( x ) by first constructing the parity
submatrix, P, as
1. 1
st
row of P: Rem ( x
n – 1
/ g ( x ))
2. 2
nd
row of P: Rem ( x
n – 2
/ g ( x ))
3. :
4. k
th
row of P: Rem ( x
n – k
/ g ( x ))

ECE 550 Communication Theory – Channel Coding Hafiz Malik
Example  Decoding of Cyclic Codes
•
For the cyclic code (7, 4) with ,
assume a received vector y = (1 1 0 0 0 0 0)
3 ( ) 1 g x x x   

Because s ( x ) = x , the syndrome vector is s =(0 1
0). This corresponds to the error pattern (0 0 0 0 0
1 0) from Table 15.2. The decoded codeword is
therefore
y = (1 1 0 0 0 0 0) + (0 0 0 0 0 1 0)
= (1 1 0 0 0 1 0)

ECE 550 Communication Theory – Channel Coding Hafiz Malik
Cyclic Code Generation: Example
•
Consider Hamming (7,4,3) code with generating polynomial
g ( x ) = x
3
+ x
2
+ 1
Construct G and H matrixes

ECE 550 Communication Theory – Channel Coding Hafiz Malik
Example 15.5

ECE 550 Communication Theory – Channel Coding Hafiz Malik
Decoding of Cyclic Block Codes
•
For decoding we exploit that fact that c ( x ) is devisable by g ( x ), if an
errors during transmission, then the received polynomial r ( x ) will not
be multiple of g ( x ).
•
If the number of errors in r ( x ) are correctable then,
r( x ) / g( x ) = m( x ) + s( x )
where syndrome polynomial s(x) = Rem (r(x) / g(x) )
has degree n – k – 1 or less.
Let e ( x ) be the error polynomial, then,
r(x) = c(x) + e(x)

s(x) = Rem (e(x)/g(x))

ECE 550 Communication Theory – Channel Coding Hafiz Malik
BCH and Reed-Solomon Codes
•
The BCH (Bose-Chaudhuri-Hocqunghen) codes is class of
random cyclic codes.
•
For m > 0 and t (t < 2
m
-1 ),
there exist a t-error correcting code (n, k) with
n = 2
m
-1, n – k <= mt, and 2t+1<= d
min
<= 2t + 2
•
Easy implementation makes them very popular.
•
Hamming code is a special case of BCH codes.
•
Reed-Solomon codes are a special case of nonbinary BCH codes
–
Applications: DVD, CD-ROM, HDTV, high speed modems, boardband
wireless systems, etc.

ECE 550 Communication Theory – Channel Coding Hafiz Malik
CRC Codes for Error Detection
•
The CRC (Cyclic Redundancy Check) codes are used for error detection
in the received data.
•
The CRC codes are cyclic codes designed to detect errors at the
reviewer.
•
For integrity verification of the transmitted data packet, each packet at
the transmitter is encoded by CRC codes of length n <= 2
m
– 1 with
code generating polynomial
g ( x ) = (1 – x ) g
c
( x )
where g
c
( x ) is generator polynomial of a cyclic Hamming code
•
The most commonly used m = 12, 16, 32
1 :
1 : 16
1 : 12
1 : 8
5 12 16
2 15 16
1 2 3 11 12
2 4 6 7 8
   
   
     
     
x x x CCITT CRC
x x x CRC
x x x x x CRC
x x x x x CRC

ECE 550 Communication Theory – Channel Coding Hafiz Malik
Probability of Codeword Error: HD Decoding
•
The probability of codeword error, P
cw
( e ), is defined as the
probability that a transmitted codeword is decoded in error
•
Under hard-decision decoding, an ( n , k , d
min
) block code can
correct t or fewer errors if d
min
 2 t + 1
•
Thus a received vector may be decoded in error if it contains
more than t errors. Therefore, P
cw
( e ) can be expressed as
•
By the application of union bound, we can write
 
 
1
( ) 1 or more errors in codeword of length
errors in codeword of length
cw
n
j t
P e P t n
P j n
 
 
 
 

 
 
 