IV054 Coding, Cryptography and Cryptographic Protocols
2009 ­ Exercises III.
1. Consider the binary cyclic code C of length 7 with the generating polynomial
g(x) = x3
+ x + 1.
a) Find the generating matrix G and the parity check matrix H.
b) Decide whether the code C is perfect or not.
c) Encode the message 1001.
2. Find a generator polynomial for the smallest binary cyclic code containing codewords
00101000 and 01001000.
3. Let g(x) = gkxk
+    + g1x + g0 = 0 be a generator polynomial of some cyclic
code C. Show that g0 = 0.
4. For k  {0, 1, . . . , 5} let nk be the number of different cyclic codes over GF(31)
which have length 5 and dimension k. Find n0, n1, . . . , n5.
5. How many ternary cyclic codes of length 6 are there? Give the generator polynomial
for each such code and one generator matrix for each dimension.
6. Which of the following codes are cyclic?
a) {000, 111, 222}  F3
3
b) {000, 100, 010, 001}  F3
q
c) {x0x1...xn-1  Fn
q | n-1
i=0 xi = 0}
d) {x0x1...xn-1  Fn
8 | n-1
i=0 x2
i = 0}
e) {x0x1...xn-1  Fn
2 | n-1
i=0 (x2
i + xi) = 0}
7. Show that the dual code of a cyclic code is cyclic.