Math- Coding theory

MATH 422 Winter 2016 Homework 8

Due at the beginning of class Wednesday March 16

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 6= 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⊥ 6=< h(x) >, where h(x) is the check polynomial for C. (c) d(C) = d(C⊥) but C 6= 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 coun- terexample.

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