2007 - Exercises III.
1. Determine whether the following code are cyclic:
(a) C\ = {x | x E Z3 and w(x) = 0 (mod 3)}
(b) C2 = {x I x E Z3 and X\ + ... + x7 = 0 (mod 3)}
(c) C3 = {x I x E Z% and x = i ˇ 22222, i E Z6}
2. Let C be the smallest binary cyclic code which contains the word 011011.
(a) List the codewords of C.
(b) Determine the polynomial g{x) which generates C.
(c) Use g(x) to encode a message 11.
3. Find two different binary cyclic codes of length 8 which are equivalent or show
that such codes do not exist.
4. (a) Factorize x6
-- 1 E 3[x] into irreducible polynomials.
(b) Let rik be the number of ternary cyclic codes of length 6 and dimension k.
Determine n^ for k E { 0 , 1 , . . . , 6}.
(c) For each cyclic code of dimension 5, find the check polynomial and a parity
check matrix and determine whether it contains the word 120210.
5. Let C\ and C2 be linear cyclic codes with generator polynomials g\[x) and
g2Íx). Show that the following code is linear and cyclic and find its generator
polynomial.
C = {a + c2 I d E d, c2 E C2}
6. Let C be a g-ary cyclic code of length n and let f(x) be its generator polynomial.
Show that all the codewords coC\... cn-\ E C satisfy Co + c\ ˇ ˇ ˇ + cn-\ = 0 in q
if and only if the polynomial x -- 1 is a factor of f(x) in q[x].
7. Let C\ and C2 be cyclic codes of the same length over Zg. Is C a cyclic code?
(a) C = {cic2 I C\ E Ci,c2 E C2 }, where Cic2 denotes the concatenation of the
codewords;
(b) C = Ci U C2
(c) C = C\ n C2
(d) C = d U ci
(e) C = S* \ C\, where S* is the set of all strings over Zq of the same length
as codewords of C\;
(f) C = {c\  c2 I C\ E Ci,c2 E C2 }, where  denotes characterwise addition
modulo q.