Find the constant c such that can be extended to be analytic at
, when
is fixed.
Let (P(z)=z^{n}+z^{n-1}+\cdots+z-n=\sum_{k=1}^{n}z^{k}-n). Then (f(z)=\dfrac{1}{P(z)}+\dfrac{c}{z-1}). At (z=1), (P(1)=n-n=0), so the first term ________ ________ _______ ______ _____ ________.
____ ____ ___ ____ ____.
_________ _____ ________ _________ ________ ______ _____ ____ ________ _____.
____ _______ _____ _______ __________ _______ ____.
_______ ____ _____ ____ _____ ______ ____ ______ ____ ________ ________ ___.
__________ _____ _______ __________ ______ ________ __________ __________ ___ ______ _______.
__________ ________ ______ _____ _________.
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) For the binary, (6,3) linear code C with generator matrix
prepare a standard array for decoding. Use it to decode the vectors (1, 1, 1, 0, 1, 1), and (1, 1, 0, 1, 1, 1). \hfill (7)
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)
Find the radius of convergence of the following series.
i)
ii)
he 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 binary repetition code of length . \hfill (2)
ii) Show that
(Hint: Start with the relation
iii) Deduce that every repetition code of odd length is perfect.
Show that there is only one linear fractional transformation that maps three given distinct points z1, z2 and z3 in the extended z plane onto three specified distinct points w1, w2 and w3 in the extended w plane.
) Let C be the ternary [8,3] narrow-sense BCH code of designed distance , which has defining set
. Use the primitive root 8th root of unity you chose in 4a) to avoid recomputing the the table of powers. If
is the generator polynomial of C and
is the received word, find the transmitted codeword. Use the following table in 1.
For each of the linear codes, find the degree, a generator matrix and a parity check matrix.