1. Show that if g(X) generates a cyclic code of odd minimum distance d, then (X+1)g(X)...

 

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1. Show that if g(X) generates a cyclic code of odd minimum distance d, then (X+1)g(X) generates a cyclic code of minimum distance at least d+1. 2. The polynomial g(X)=1+X+X3 generates a cyclic Hamming code of length 7. Find its dimension and minimum distance. 3. Find the dimension and minimum distance of the code of length 7 generated by the polynomial (X + 1)g(X) where g(X)=1+X+X. 1. Show that if g(X) generates a cyclic code of odd minimum distance d, then (X+1)g(X) generates a cyclic code of minimum distance at least d+1. 2. The polynomial g(X)=1+X+X3 generates a cyclic Hamming code of length 7. Find its dimension and minimum distance. 3. Find the dimension and minimum distance of the code of length 7 generated by the polynomial (X + 1)g(X) where g(X)=1+X+X.