IV054 Coding, Cryptography and Cryptographic Protocols
2008 ­ Exercises III.
1. Suppose that a cyclic shift of each row of a generator matrix G of a linear code
C belongs to C. Show that C is a cyclic code.
2. Determine d and find generator polynomials and generator matrices for
(a) all binary cyclic codes in R4;
(b) all ternary cyclic codes in R5.
3. (a) How many binary cyclic codes of length 7 are there ?
(b) Find a binary cyclic code of length 7 which contains exactly 32 codewords
or show that such a code does not exist.
4. Let C be a binary cyclic code and g(x) its generator polynomial. Show that
C = C
if and only if xn-k
g(x)g(x-1
) = xn
- 1.
5. (a) Which Hamming codes are maximum distance separable ?
(b) Let C be a q-ary [n, k]-code that is maximum distance separable. What is
the number of words with minimum weight d = n - k + 1 in C ?
6. Show that C
is equivalent to the cyclic code h(x) where h(x) is the check
polynomial of a cyclic code C.