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Determine all binary cyclic codes of length 5.
Note: To find all cyclic codes of length n, find all ideals in B[x]/x^2+1Note: If 1 is an Ideal (I) then R = I.
Example:n=2R=B[x]/x^2+1, x^2=1
R={o,1,x,1+x}
Ideals <0> = 0<1> = Rx = (0, x, x^2...)= (1,...= R<1+x> = (1+x, 0, x+x^2, 1+x^2)x+x^2 = x+11+x^2 = 1+1 = 0 so,
<1+x> = (1+x, 0)
Therefore:
Ideal Code Words<0> 00<1> 00011011x = <1><1+x> 0011
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