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III. Cyclic Codes

III. Cyclic Codes. Cyclic Shift: v=( v 0 ,v 1 ,v 2 , … , v n-1 ) Cyclic shift of v: v (1) = ( v n-1 ,v 0 ,v 1 , … ,v n-2 ) It means cyclically shifting the components of v one place to the right

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III. Cyclic Codes

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  1. III. Cyclic Codes

  2. 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 vone 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 Description of Cyclic Codes 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

  3. Code Polynomials • Each codeword corresponds to a polynomial of degree n-1 or less: • For a codeword v=(v0,v1,v2,…, vn-1) thecorresponding 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

  4. Example: (7,4) cyclic code

  5. 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) v(i)(X) is the remainder of dividing Xiv(X) by (Xn+1) Xiv(X)=v(i)(X)+(Xn+1)q(X)

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