CYCLIC CODE
Submitted By:
Farida Mohamed Ahmed
18P1474
Hagar Ahmed Mohamed
18P6623
Mariam Hany Wadie
18P5465
Nada Ayman Abd El Qader
Rahma Ahmed Hesham
18P3028
18P9374
Submitted To:
Dr. Fatma Newagy
Eng. Moustafa Mohamed
1
Main Concept
Agenda
• Cyclic Code properties
• Encoding
• Decoding
Encoder & Decoder Circuit Design
Advantages & Disadvantages
Standards & Applications
Cyclic Code
 Subclass of linear block code
Properties:
 Linearity property
The Sum of two codewords gives another codeword
→ Ex: {101 , 110,011} → 101 + 110 = 011
→ 101 + 011 = 110
→ 110 + 011 = 101
Cyclic Shifting Property
Cyclic shifting of any codeword right or left by any number of bits gives another codeword
Ex: {000,011,101,110}
 shifting 011 right by 1 bit gives 101
 shifting 101 right by 1 bit gives 110
 shifting 110 right by 1 bit gives 011
 shifting 000 right by 1 bit gives 000
Cyclic codes
Encoding
Systematic
Division
method
Generator
matrix
method
Non-systematic
Multiplication
method
Generator
matrix
method
Decoding
Cyclic encoders
Circuit (shift
register)
Syndrome
decoding
(error pattern)
Cyclic
Decoders
Circuit (shift
register)
Non-systematic Codeword
 In Non-Systematic codeword
→ codeword ≠ [msg, parity]
 Else msg and parity bits are not in sequence.
Cyclic code for non-systematic codeword
(Multiplication Method)

How to get codeword?
c(x) = m(x)*g(x) , m(x)  message polynomial
, g(x)  generator polynomial
, c(x)  codeword polynomial

Note:
•
How to get polynomials from bits
Ex: msg polynomial from msg bits, msg = 1011
 1st (least significant) bit corresponds to x0
 2nd bit corresponds to x1
 3rd bit corresponds to x2
 4th (most significant) bit corresponds to x3
‫ ؞‬polynomial of the msg = 1(x ) + 0(x ) + 1(x ) + 1(x ) = x + x + 1
3
2
1
0
3
1
Cyclic Code for non-systematic codeword
(Multiplication Method)
 Ex: construct Non-Systematic codeword (7,4) using cyclic codes , generator polynomial
𝑔(𝑥) = 𝑥 3 + 𝑥 2 + 1 , msg = 1010
 msg polynomial = 𝑥 3 + 𝑥
 c(x) = m(x)*g(x) = 𝑥 3 + 𝑥 ∗ 𝑥 3 + 𝑥2 + 1 = 𝑥 6 + 𝑥 5 + 𝑥 3 + 𝑥 4 + 𝑥 3 + 𝑥
= 𝑥6 + 𝑥5 + 𝑥4 + 𝑥
(note: we use modulo 2 sum x3 + x3 = 0)
 Then get codeword bits from codeword polynomial
Coeff. of 𝑥 0 = 0
Coeff. of 𝑥 4 = 1
Coeff. of 𝑥 1 = 1
Coeff. of 𝑥 5 = 1
Coeff. of 𝑥 2 = 0
Coeff. of 𝑥 6 = 1 → codeword bits = 1110010
Coeff. of 𝑥 3 = 0
Cyclic Code for non-systematic codeword
(Generator Matrix Method)
 How to get codeword?!
c = m*g
, m  message matrix
, g generator matrix
, c codeword matrix
Cyclic Code for non-systematic codeword
(Generator Matrix Method)
 Getting Generator matrix from generator polynomial:
•
G is of size nxk
•
Rows of generator matrix is given by {Ri = xk-1 g(x)} , i= 1, 2…,k
•
EX: construct generator matrix (7,4), generator polynomial = x3 + x + 1
 i=1 , R1 = x3(x3 + x + 1) = x6 + x4 + x3 , R1 matrix = [ 1011000]
 i=2 , R2 = x2(x3 + x + 1) = x5 + x3 + x2 , R1 matrix = [ 0101100]
 i=3 , R3 = x1(x3 + x + 1) = x4 + x2 + x1 , R1 matrix = [ 0010110]
 i=4 , R1 = x0(x3 + x + 1) = x3 + x + 1, R1 matrix = [ 0001011]
Cyclic code for non-systematic codeword
(Generator Matrix Method)
R1
• G=
R2
R3
R4
=
1
0
1
1
0
0
0
0
1
0
1
1
0
0
0
0
1
0
1
1
0
0
0
0
1
0
1
1
Cyclic code for non-systematic codeword
(Generator Matrix Method)
 EX: construct Non-Systematic codeword (7,4) using cyclic codes, generator polynomial
g(x) = x3 + x + 1, msg = 1011
G=
1
0
1
1
0
0
0
0
1
0
1
1
0
0
0
0
1
0
1
1
0
0
0
0
1
0
1
1
C =m * G = [1011]
=1000101
1
0
1
1
0
0
0
0
1
0
1
1
0
0
0
0
1
0
1
1
0
0
0
0
1
0
1
1
Cyclic code for systematic codeword
(Division Method)

In Systematic codeword
codeword = [msg, parity]

How to get codeword?!
c(x) = xn-k m(x) + p(x) , p(x) = rem [xn−k m(x)/g(x)]
, m(x)  message polynomial
, g(x)  generator polynomial
, c(x)  codeword polynomial
Cyclic code for systematic codeword
(Division Method)
 Ex: construct Systematic codeword (7,4) using cyclic codes, generator polynomial
g(x) = x3 + x2 + 1, msg = 1010
 msg polynomial = x3 + x
 c(x) = xn-k m(x) + p(x) , p(x) = rem [xn−k m(x)/g(x)] , n= 7 & k =4
 xn−k m(x)/g(x) = x3 (x3 + x ) / (x3 + x2 + 1 ) = ( x6 + x4) / (x3 + x2 + 1 )
x3 + x2 + 1
‫ ؞‬p(x) = rem [xn−k m(x)/g(x)] = 1
c(x) = xn-k m(x) + p(x) = x6 + x4 +1
x3 + x2 + 1
x6 + x4
x6 + x5 + x3
x5 + x4 + x3
x5 + x4 + x2
x3 + x2
x3 + x2 +1
1
Cyclic code for systematic codeword
(Division Method)
 Then get codeword bits from codeword polynomial
Coeff. of x0 = 1
Coeff. of x1 = 0
Coeff. of x2 = 0
Coeff. of x3 = 0
Coeff. of x4 = 1
Coeff. of x5 = 0
Coeff. of x6 = 1  codeword bits = 1010 001
msg Parity
bits
Cyclic code for systematic codeword
(Generator Matrix Method)

In Systematic codeword
codeword = [msg , parity]

How to get codeword?!
c= m* g
, m  message matrix
, g generator matrix
, c codeword matrix
Cyclic code for systematic codeword
(Generator Matrix Method)
 Getting Generator matrix from generator polynomial:
•
G = [I : P], G is of size nxk
•
Identity matrix is of size kxk
•
To get parity matrix → 1st row : rem [xn−1 /g(x)]
→ 2nd row : rem [xn−2 /g(x)]
..
.
→ kth row: rem [xn−k /g(x)]
Cyclic code for systematic codeword
(Generator Matrix Method)
 Generator Matrix:
•
Ex: construct generator matrix (7,4), generator polynomial = x3 + x + 1
To get parity matrix → 1st row : rem [x6 / (x3 + x + 1)]
 1st row bits = 101
x3 + x + 1
x3 + x + 1 x6
x6 + x4 + x3
x4 + x3
x4 + x2 + x
x3 + x2 + x
x3 + x +1
x2 + 1
Cyclic code for systematic codeword
(Generator Matrix Method)
 2nd row : rem [x5 / (x3 + x + 1)]
2nd row bits = 111
 3rd row : rem [x4 / (x3 + x + 1)]]
3rd row bits = 110
x2 + 1
x3 + x + 1 x5
x5 + x3 + x2
x3 + x2
x3 + x + 1
x2 + x + 1
x
x3 + x + 1 x4
x4 + x2 + x
x2 + x
Cyclic code for systematic codeword
(Generator Matrix Method)
→ 4th row : rem [x3 / (x3 + x + 1)]] = x+1
1
x3 + x + 1 x3
x3 + x + 1
x+ 1
4th row bits = 011
G = [I : P] =
1
0
0
0
1
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
0
0
0
0
1
0
1
1
Cyclic code for systematic codeword
(Generator Matrix Method)
 Ex: construct Systematic codeword (7,4) using cyclic codes, generator
polynomial g(x) = x3 + x + 1, msg = 1011
G=
1
0
0
0
1
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
0
0
0
0
1
0
1
1
C = m * G = 1011
1
0
0
0
1
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
0
0
0
0
1
0
1
1
= 1011000
Decoding of cyclic codes
a. Syndrome Decoding:
 It’s used in cyclic codes in order to decode the sequence and to decode and
correct it.
 Syndrome polynomial is obtained as
S(x)= Reminder of
R 𝑥
𝑔 𝑥
 If S(x) =0, then No errors
 If S(x)≠ 0, there are some errors
Decoding of cyclic codes
a. Syndrome Decoding:
 Ex: Consider received sequence is R=1011000 for (7,4) code dimension and
generator polynomial g(x)=𝑥 3 + 𝑥 + 1
Find received sequence is valid or not?
𝑅 𝑥 = 𝑥6 + 𝑥4 + 𝑥3
𝑔 𝑥 = 𝑥3 + 𝑥 + 1
𝑥3
𝑥3 + 𝑥 + 1 𝑥6 + 𝑥4 + 𝑥3
𝑥6 + 𝑥4 + 𝑥3
0
0 0
S(x)=0 → No Error
Cyclic encoder Circuit design
 Codewords could be generated using n-k flipflops
Ex: For generator polynomial = 1 + x + x3 = 1 + x +0(x2) + x3
1
1
FF
FF
FF
P1
LSB
1
0
P2
P3
MSB
Input
message
Cyclic encoder Circuit Design
 Construct codeword (7,4) using cyclic codes, generator polynomial
g(x) = x3 + x + 1, msg = 1011
‫ ؞‬Codeword = [msg |p] = [1011000]
FF
FF
FF
P1
LSB
P2
P3 Input
MSB message
Input
P1 = inputi-1 xor P3i-1
P2 = inputi-1 xor P1i-1 xor P3i-1
P3 = P2i-1
1
0
0
0
After 1st clk cycle
0
1
1
0
After 2nd clk cycle
1
0
1
1
After 3rd clk cycle
1
0
0
1
0
0
0
After 4th clk cycle
P1
LSB
P2
P3
MSB
Cyclic Decoder Circuit Design
b. Syndrome Calculator:
 Ex: Consider a cyclic code with code dimension (7,4) and g(x)=𝑥 3 + 𝑥 + 1. If the received
codeword is R=1001000. Determine the syndrome by using a syndrome calculator.
g(x)=𝑥 3 + 𝑥 + 1
Before Shift
𝑔2 = 0
𝑆3
R +
𝑆3
1
𝑆1
+
𝑆2
𝑆3
Syndrome
SW 2 Output
R
1
0
0
1
0
0
0
𝑆1
0
1
0
0
0
0
1
𝑆2
0
0
1
0
1
0
1
S3
0
0
0
1
0
1
0
After Shift
𝑆1′ = R⊕S3
1
0
0
0
0
1
0
WB
so, S=110
S2′ =𝑆1 ⊕S3
0
1
0
1
0
1
1
S3′ = 𝑆2
0
0
1
0
1
0
1
MSB
Advantages of cyclic code
 They are easy to implement.
 Simple and easy error detection and correction methods.
 Easy implementation for encoding and decoding circuit for cyclic codes
using shift registers.
 They can be easily implemented in both hardware or software.
 Small overhead.
Disadvantages of cyclic code
 Low robustness during overloads.
 Error correction is slightly more complicated than error detection; due to the
complexity of the combination logic circuit used for error correction.
Some standard cyclic codes
 The hamming code:
Single error-correcting codes can be used in cyclic form.
 The BCH code:
Extension of hamming for n-errors correcting codes.
 CRC code:
Cyclic redundancy check code which is a shortened cyclic error-detecting code used
in automatic repeat request systems.
 Reed Solomon code:
A non-binary cyclic code.
Reed Solomon Codes Applications
 Reed-Solomon codes have been widely used in mass storage systems to correct the
burst errors caused by media defects.
 Special types of Reed-Solomon codes have been used to overcome the unreliable
nature of data transmission over erasure channels.
 Several bar-code systems use Reed-Solomon codes to allow correct reading even if a
portion of a bar code is damaged.
 Reed-Solomon codes were used to encode pictures sent by the Voyager spacecraft.
Other applications of cyclic code
 Satellite & wireless communication :
The information sent digitally in satellite & wireless communication uses cyclic code
in the encoding and decoding process.
 Compact discs(CD):
Since CD errors are burst errors, cyclic codes can be used in CDs to correct these
burst errors.
 Error correction:
Used to correct double errors and burst errors.
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