What is the dimension of C ?
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Let w be subspace of F n 2 containing all the vectors of even weight. Prove that W has dimension n−1. (Hint: Consider the map w: F n 2 → F2 given by w((a1,a2,...,an)) = a1 +a2 +···+an.)
Draw the Tanner graph of the code C with parity check matrix
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 code
C = {(u|u+v)|u ∈ C1,v ∈ C2}
obtained from C1 and C2 by (u|u+v) construction. Also, find the minimum distance
of C .
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 F2.
v) There is no LDPC code with parameters n = 16, c = 3 and r = 5.
Let α be a root of x2 +1 = 0 in F9.
a) Check whether α is a primitive element of F9. If it is not a primitive element in F9 find a
primitive element γ in F9 in terms of α. (4)
b) Make a table similiar to Table 5.1 on page 184 for F9 with the primitive element γ
c) Factorise x
8 −1 over F3. (6)
d) Find all the possible generator polynomials of a [8,6] cyclic code.
Find the convolutional code for the message 11011. The convolutional encoder is given in Fig. 1.
If C1 has only even weight codewords, what is the relationship between (1+x) and g1(x)?