a) Let 
and 
be cyclic codes over with generator polynomials
and
respectively. Prove that
if and only if
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1) Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample. (
i) If the weight of each element in the generating matrix of a linear code is at least r, the mininum distance of the code is at least r.
ii) There is no linear self orthogonal code of odd length.
iii) There is no 3-cyclotomic coset modulo 121 of size 25.
iv) There is no duadic code of length 15 over F
v) There is no LDPC code with parameters n = 16, c = 3 and r = 5.
b) Draw the Tanner graph of the code L with parity check matrix
c) Find the convolutional code for the message 11011. The convolutional encoder is given in Fig. 1.
8) a) Let
C=be the-linear code. Find the Gray image of
c) Find the generating idempotents of duadic codes of length n = 23 over (Hint: Mimic example 6.1.7.)
d) The aim of this exercise is to show that every binary repetition code of odd length is perfect
i) Find the value of t and d for a perfect code of length 2m+1, m
ii) Show that
(Hint: Start with the relation
iii) Deduce that every repetiition code of odd length is perfect.
Let C1 and C2 be two binary codes with generator matrices
respectively.
a) Find the minimum distance of both the codes.
b) Find the generator matrix of the c
Let 
be a binary code with a generator matrix each of whose rows has even weight. Show that, every codeword of has even weight ( Hint: Why is it enough to prove that sum of vectors of even weight in is a vector of even weight? )