**Coding Theory**

** Fifth Assignment, Cyclic Codes: due Mon. Apr. 16**

- CT 4.2.21: d.
- CT 4.3.4.
- CT 4.3.6: a.
- CT 4.3.8 (4.3.11 in the old edition).
- CT 4.3.9: e (4.3.12: e in the old edition).
Find the remainders when each
* x^i *
is divided by * g*.
- CT 4.4.6: f,g.
- A study of binary cyclic codes of length 14.

a. Factor *x^12+1 *.

b. How many distinct cyclic codes of length 14 are there?

c. Describe concisely all possible codes by giving for each code:
- the generator polynomial
*g(x) * (You can leave it factored.),
- its dimension,
- the generator polynomial for the dual code.

d. What are the possible dimensions? How many codes are there for each dimension?

e. There are three distinct codes of rate 1/2. For each one find
- the generator polynomial
*g(x) *
- the generator polynomial for the dual code
* h¯(x) *
- the generating matrix based on the generator polynomial,
- the matrix
*A* such that
*[A I]* is a systematic encoding matrix, or equivalently
such that

Transpose(*[I A]*) is a parity check matrix.

- CT 4.5.5: g; this problem concerns the Golay code.

* *