Prove that for any [n, k] code Cover Zp:

(i) all cosets have the same size;

(ii) C + x = C + Y if Y E C + x, and (C + x) n (C + y) = c/J if Y (j. C + x;

(iii) Every word of Z; is a member of so me coset;

(iv) x, y are in the same coset if and only if their difference is in C;

(v) there are pn-k distinct cosets.