# Coding Theory

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

1. CT 4.2.21: d.
2. CT 4.3.4.
3. CT 4.3.6: a.
4. CT 4.3.8 (4.3.11 in the old edition).
5. CT 4.3.9: e (4.3.12: e in the old edition). Find the remainders when each x^i is divided by g.
6. CT 4.4.6: f,g.
7. 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.
8. CT 4.5.5: g; this problem concerns the Golay code.