Find the convolutional code for the message 11011. The convolutional encoder is given in
Fig. 1.
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Draw the Tanner graph of the code C with parity check matrix
Show that, if x ∈ ,
wt (x) = x.x (mod 3)
Deduce that, if C is a ternary self orthogonal code, the weight of each codeword is divisible by 3.
(Hint: Observe that x2 = 1 for all x ≠0 € )
If , show that
wt(x+y)=wt(x)+wt (y) -2wt (xy)
where is the vector in
which has 1s precisely at those positions where x and y have 1s.
(Hint: Let x = (x1,x2,...,xn) and y = (y1, y2,..., yn). Suppose
Observe that wt (x) = n1 + n2 and wt (y) = n1+n3.)
Which of the following binary codes are linear?
i)
ii)
Justify your answer.
Find the minimum distance for each of the codes.
Over , (1+x) | (xn - 1). Let C be the binary cyclic code (1+x) of length n. Let C1 be any binary cyclic code of length n with generator polynomial g1(x).
i) What is the dimension of ?
ii) Let w be subspace of containing all the vectors of even weight. Prove that W has dimension n - 1. (Hint: Consider the map w:
→
given by
Prove that & is the vector space of all vectors in with even weight.
iv) If C1 has only even weight codewords, what is the relationship between (1+x) and g1(x)?
v) If C1 has some odd weight codewords, what is the relationship between 1 + x and g1(x)?
Find the generating idempotents of duadic codes of length n = 23 over .
(Hint: Mimic example 6.1.7.)
be the Z4-linear code. Find the Gray image of C.