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III. Cyclic Codes
Description of Cyclic Codes
Cyclic Shift:
 v=(v0,v1,v2,…, vn-1)
 Cyclic shift of v: v(1)=(vn-1,v0,v1,…,vn-2)
It means cyclically shifting the components of v
one place to the right
 v(i)=(vn-i, vn-i+1,…, vn-1,v0,v1,…,vn-i-1): cyclically
shifting v i places to the right
Definition of Cyclic Codes:
An (n,k) linear code C is called a cyclic code if every
cyclic shift of a codeword in C is also a codeword in C
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Code Polynomials
 Each codeword corresponds to a polynomial of
degree n-1 or less:
 For a codeword v=(v0,v1,v2,…, vn-1) the
corresponding code polynomial is:
v(X)= v0+v1X+v2X2+…+vn-1Xn-1
 Code polynomial for v(i):
v(i)(X)=vn-i+vn-i+1X+…+vn-1Xi-1+v0Xi+v1Xi+1+…+vn-i-1Xn-1
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Example: (7,4) cyclic code
Information Message
Codeword
Code Polynomial
(0 0 0 0)
(0 0 0 0 0 0 0)
0
(1 0 0 0)
(1 1 0 1 0 0 0)
1+X+X3
(0 1 0 0)
(0 1 1 0 1 0 0)
X+X2+X4
(1 1 0 0)
(1 0 1 1 1 0 0)
1+X2+X3+X4
(0 0 1 0)
(0 0 1 1 0 1 0)
X2+X3+X5
(1 0 1 0)
(1 1 1 0 0 1 0)
1+X+X2+X5
(0 1 1 0)
(0 1 0 1 1 1 0)
X+X3+X4+X5
(1 1 1 0)
(1 0 0 0 1 1 0)
1+X4+X5
(0 0 0 1)
(0 0 0 1 1 0 1)
X3+X4+X6
(1 0 0 1)
(1 1 0 0 1 0 1)
1+X+X4+X6
(0 1 0 1)
(0 1 1 1 0 0 1)
X+X2+X3+X6
(1 1 0 1)
(1 0 1 0 0 0 1)
1+X2+X6
(0 0 1 1)
(0 0 1 0 1 1 1)
X2+X4+X5+X6
(1 0 1 1)
(1 1 1 1 1 1 1)
1+X+X2+X3+X4+X5+X6
(0 1 1 1)
(0 1 0 0 0 1 1)
X+X5+X6
(1 1 1 1)
(1 0 0 1 0 1 1)
1+X3+X5+X6
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Algebraic Relation between v(X) and v(i)(X)
Xiv(X)=v0Xi+v1Xi+1+…+vn-i-1Xn-1+…+vn-1Xn+i-1
Add (vn-i+vn-i+1X+…+vn-1Xi-1) twice
Xiv(X)= vn-i+vn-i+1X+…+vn-1Xi-1+
v0Xi+v1Xi+1+…+vn-i-1Xn-1+…+vn-1Xn+i-1+
vn-i+vn-i+1X+…+vn-1Xi-1
Xiv(X)= vn-i+vn-i+1X+…+vn-1Xi-1+v0Xi+v1Xi+1+…+vn-i-1Xn-1+
vn-iXn+vn-i+1Xn+1+…+vn-1Xn+i-1+vn-i+vn-i+1X+…+vn-1Xi-1
=v(i)(X)+(Xn+1)(vn-i+vn-i+1X+…+vn-1Xi-1)
Xiv(X)=v(i)(X)+(Xn+1)q(X)
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v(i)(X) is the remainder of
dividing Xiv(X) by (Xn+1)
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Theorem 1
The nonzero code polynomial of minimum
degree in a cyclic code is unique
Proof:
 Let g(X)=g0+ g1X+…+gr-1Xr-1+Xr be a non-zero code polynomial
of minimal degree in C
 If g(X) is not unique, there a exists a code polynomial g’(X)
such that: g’(X)=g’0+ g’1X+…+g’r-1Xr-1+Xr
 For C to be linear g(X)+g’(X) is…… a Codeword. The
corresponding code polynomial is:
(g0+ g’0)+(g1+ g’1)X+…+(gr-1+ g’r-1)Xr-1 with degree<r
 THIS CONTRADICTS THE DEFINITION THAT g(X) IS THE
NON-ZERO CODE POLYNOMIAL OF MINIMUM DEGREE
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Theorem 2
Let g(X)=g0+ g1X+…+gr-1Xr-1+Xr be a non-zero
code polynomial of minimal degree in an (n,k)
cyclic code C. Then g0 must be equal to 1
Proof:
 Suppose g0=0
 g(X)=g1X+…+gr-1Xr-1+Xr
 If we cyclically shift g(X) 1 place to the left we get another code
polynomial g’(X)=g1+g2X…+Xr-2+Xr-1 with degree<r
 THIS CONTRADICTS THE DEFINITION THAT g(X) IS THE
NON-ZERO CODE POLYNOMIAL OF MINIMUM DEGREE
 Therefore g0=1
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From Theorems 1 & 2
The non-zero code polynomial of minimal
degree in an (n,k) cyclic code C has the form:
g(X)=1+ g1X+g2X2+…+gr-1Xr-1+Xr
In the table for the (7,4) cyclic code in slide 4
g(X)=1+X+X3
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Cyclic Shifts of the Minimal Degree Polynomial g(X)
The polynomials Xg(X), X2g(X),…,Xn-r-1g(X)
are cyclic shifts of g(X) where:
Xg(X)
= g(1)(X)
X2g(X) = g(2)(X)
They are code
:
:
polynomials
Xn-r-1g(X) = g(n-r-1)(X)
Given that Xg(X), X2g(X),…, Xn-r-1g(X) are code polynomials in a
linear code
v(X)= [u0+u1X+u2X2+…+un-r-1Xn-r-1]g(X) is also a code polynomial in C
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Theorem 3
Let g(X)=1+ g1X+g2X2+…+gr-1Xr-1+Xr be the minimal degree
polynomial in an (n,k) cyclic code C. A binary polynomial of degree
n-1 or less is a code polynomial if and only if it is a multiple of g(X)
Proof:


Let v(X) be a binary polynomial of degree n-1 or less such that v(X) is
a multiple of g(x). Then:
v(X)= [a0+a1X+a2X2+…+an-r-1Xn-r-1]g(X).
v(X) is a linear combination of code polynomials,
g(X), Xg(X), …Xn-r-1g(X)
v(X) is a code polynomial in C
Let v(X) be a code polynomial in C
v(X)=a(X)g(X)+b(X) where the degree of b(X)<r
b(X)=v(x)+a(X)g(X)
v(X) is a code polynomial, a(X)g(X) are code polynomials
b(X) is also a code polynomial of degree<r
g(X) is the minimal degree nonzero polynomial
b(X)=0
v(X) is a multiple of g(X)
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The Number of Code Polynomials
From Theorem 3:
A code polynomial must be a multiple of g(X) & Any multiple of
g(X) is a code polynomial
The number of multiples of g(X) of degree<n-1 are: n-r
The number of code polynomials are 2n-r
k=n-r where k is the number of information bits in the code
word, r is the number of parity check bits in the codeword
g(X)=1+g1X+g2X2+…+gn-k-1Xn-k-1+Xn-k
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Theorem 4
In an (n,k) cyclic code, there exists one and only one
code polynomial of degree n-k
g(X)=1+g1X+g2X2+…+gn-k-1Xn-k-1+Xn-k
1. Every code polynomial is a multiple of g(X)
2. Every binary polynomial of degree n-1 or less that is
multiple of g(X) is a code polynomial
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Generator Polynomial
From Theorem 4:
Every code polynomial v(X) in an (n,k) cyclic code could
be expressed as:
v(X)= u(X)g(X) =(u0+ u1X+…+uk-1Xk-1 )g(X)
If (u0, u1,…,uk-1) is the information message,
v(X) represents the corresponding codeword
An (n,k) cyclic code is completely specified by its non
zero minimal degree code polynomial g(X)
g(X) is called the generator polynomial
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Theorem 5
The generator polynomial g(X) of an (n,k) cyclic code is
a factor of Xn+1
Proof:
Multiply g(X) by Xk
Xkg(X) has a degree n. By dividing Xkg(X) by Xn+1
Xkg(X)=(Xn+1)+g(k)(X)
g(k)(X) is a cyclic shift of g(X) and therefore is a code
polynomial
g(k)(X) is a multiple of g(X) g(k)(X)=a(X)g(X)
Xkg(X)=(Xn+1)+a(X)g(X)
(Xn+1)=[XK+a(X)]g(X)
Therefore g(X) is a factor of Xn+1
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Theorem 6
If g(X) is a polynomial of degree n-k and is a factor of
Xn+1, then g(X) generates an (n,k) cyclic code
Proof:
 It is possible to generate 2k polynomials from linear
combinations of the polynomials g(X), Xg(X), …,Xk-1g(X)
v(X)=(u0+ u1X+…+uk-1Xk-1)g(X)WE HAVE A LINEAR BLOCK
CODE
 IS THIS CODE CYCLIC?
Xv(X)=vn-1(Xn+1)+v(1)(X)
Since BOTH Xv(X) and (Xn+1) are divisible by g(X), v(1)(X)
must be also be divisible by g(X)
Therefore v(1)(X) must be a linear combination of g(X), Xg(X),
…,Xk-1g(X)
Therefore v(1)(X) must be a code polynomial in the (n,k) linear
block code
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Example
(X7+1)=(1+X)(1+X+X3)(1+X2+X3)
There are two factors of degree 3
Each factor could be used to generate a (7,4)
cyclic code
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