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 minimum 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 .
v) There is no LDPC code with parameters ,
and
.
i) True. In a linear code, the minimum distance (d_{\min}) is equal to the minimum weight of any nonzero codeword. Since every codeword is a linear combination of the rows of the generating matrix, if each row has weight at least (r), any nonzero combination will have weight at least (r), so (d_{\min} \ge r). ii) True. A linear ______ _____ _______ ______ ______.
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Evaluate , where c is the eight like figure shown in Fig. 1.
If is entire such that
in
then show that f has the form
where
are constants with
.
Find the radius of convergence of the following series.
i)
ii)
Find the minimum distance for each of the codes.
Prepare a syndrome table for C in part a) and decode the vectors (1, 1, 1, 1, 0, 1) and (0, 1, 0, 1, 1, 1). \hfill (5)
Factor x8 - 1 over . Give the generator polynomials of all cyclic codes of length eight over
.
) Evaluate the following integrals:
i) .
ii) .