Determine all binary cyclic codes of length 4 for each code

Question

1) Determine all binary cyclic codes of length 4. For each code, provide: generator polynomial, check polynomial, generator matrix, parity check matrix, dimension, and minimal distance.

2) Determine all q-ary cyclic codes of length 2, where q = p^k for some prime p and k ≥ 1. (There will be 2 cases: p = 2 and p ≠ 2).

3) For each statement, give an example of a cyclic code C for which the statement is true. Justify each answer.

(a) C = C^⊥.

(b) C such that C^⊥ ≠< h(x) >, where h(x) is the check polynomial for C.

(c) d(C) = d(C^⊥) but C ≠ C^⊥.

4) Find generator polynomials for the following binary Hamming codes: Ham(2, 2), Ham(3, 2), and Ham(4, 2). Justify your answers.

5) For each Hamming code in problem 4, do the following:

(a) Write out a generator matrix, parity check matrix, and check polynomial. Justify your answers.

(b) Give a generator polynomial for the dual code C^⊥ and find the minimum distance of C^⊥.

6) (a) Is every Hamming code equivalent to a cyclic code? Prove it, or provide a counterexample.

(b) Is every cyclic code equivalent to a Hamming code? Prove it, or provide a counterexample.