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.
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.
i) True. If the weight of each element in the generating matrix of a linear code is at least r, then the minimum distance of the code is at least r. This is because the minimum distance of a code is the smallest Hamming distance between any two distinct codewords. Since the weight of each codeword is at least r, the Hamming distance between any two distinct codewords must be at least r.
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Find the convolutional code for the message 11011. The convolutional encoder is given in
Fig. 1.
Let C be a binary code with a generator matrix each of whose rows has even weight. Show that, every codeword of C has even weight.
(Hint: Why is it enough to prove that sum of vectors of even weight in is a vector of even weight?)
Let C1 and C2 be cyclic codes over , with generator polynomials g₁ (x) and g2(x), respectively. Prove that C1
C2 if and only if g2(x) | g1(x).
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)?
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.)
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 € )
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.