IV054 Coding, Cryptography and Cryptographic Protocols
2012 - Exercises III.
1. Consider the following binary linear [8, 5]-code C generated with
G =






1 1 1 1 0 0 0 0
1 0 0 0 1 0 0 0
0 1 0 0 0 1 0 0
0 0 1 0 0 0 1 0
1 1 1 0 0 0 0 1






.
(a) Prove that C is a cyclic code.
(b) Find the generator polynomial of C.
2. Which of the following binary codes are cyclic? Explain your reasoning.
(a) C1 = {000, 001, 100, 101}
(b) C2 = {000, 001, 010, 100}
(c) C3 = {0, 1}
(d) C4 = {0000, 0101, 1010, 1111}
3. Compute a generator polynomial and a parity check polynomial of a binary cyclic code of length 12
and dimension 5. Encode the word 00100.
4. Provide the generator polynomial of the smallest binary cyclic code containing codeword 0001001.
5. Consider a binary cyclic code C with a generator polynomial g(x). Show that g(1) = 0 if and only
if weight of each word in C is even.
6. How many quinary cyclic codes of length seven are there? Give a generator polynomial for each of
them.
7. Let C be a cyclic code over Fq of length 7 such that 1110000 is an element of C. Show that C is a
trivial code (ie. Fn
q or {0n
}) if q is not a power of 3.