Find the minimum distance for each of the codes.
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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).
For each of the linear codes, find the degree, a generator matrix and a parity check matrix.
Draw the Tanner graph of the code C with parity check matrix
Which of the following binary codes are linear?
i)
ii)
Justify your answer.
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.)
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?)
Find the minimum distance for each of the codes.