Reference no: EM131003859
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 = pk 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.
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