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Binary Cyclic Codes

Determine all binery cyclic codes of length 5.

Note: To find all cyclic codes of length n, find all ideals in B[x]/x^2+1
Note: If 1 is an Ideal (I) then R = I.

Example:
n=2
R=B[x]/x^2+1, x^2=1

R={o,1,x,1+x}

Ideals <0> = 0
<1> = R
x = (0, x, x^2...)
= (1,...
= R
<1+x> = (1+x, 0, x+x^2, 1+x^2)
x+x^2 = x+1
1+x^2 = 1+1 = 0 so,

<1+x> = (1+x, 0)

Therefore:

Ideal Code Words
<0> 00
<1> 00
01
10
11
x = <1>
<1+x> 00
11

Solution Summary

Binary Cyclic Codes are emphasized.

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