Which of the following binary codes are linear?
i)
ii)
Justify your answer.
A binary code is linear if it contains the all-zero codeword and is closed under addition modulo 2. For (i), the code is (\mathcal{C}={(0,0,0,0),(1,0,1,0),(0,1,1,0),(1,1,1,0)}). The all-zero vector ____ _________ _________ ___ ____ _____ ________ ______ ________.
__________ _________ _________ __________ ___ ______ _____ __________ ________.
______ ______ _________ _____ ______ __________ ______ ___.
______ _________ _________ _____ _________ ____ ______ __________.
_______ _________ __________ _________ _____ _______ ___.
___ _____ ______ ____ __________ _________ __________ _____ _____.
_____ _____ ____ ___ _________ _________ _____.
_______ _________ ___ ___ __________.
__________ __________ _______ ________ __________ ____ _____ ___ _________ ______.
________ __________ _______ _______ ______ ____ ______ ______ ________ ________ _____ _________.
_________ _______ __________ _________ _________ _____ ____ _________ ________ ________ __________.
__________ ____ _________ ____ ________ ________.
____ ________ _____ ___ ____ __________ _____.
________ ________ ________.
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Find all the singularities of the function .
Prove that the linear fractional transformation maps the circle
into itself. Also prove
that f(z) is conformal in .
Find the parity check matrix of the code .
For a linear code C with generator matrix
find the parity check matrix. Check that any two columns of the parity check matrix are linearly independent and there are three columns that are linearly dependent. What is the minimum distance of C ?
Determine whether each of the following statement is true or false. Justify your answer with a short proof or a counter example.
i) If , where a and b are integers, then
if a > 0.
ii) If f(z) and are analytic functions in a domain, then f is necessarily a constant.
iii) A real-valued function u(x, y) is harmonic in D iff u(x, -y) is harmonic in D.
iv) .
v) The inequality holds for
.
vi) If has the property that
converges, then f is necessarily an entire function.
vii) If a power series converges for |z| < 1 and if
is such that |bn| < n2 |an| for all
, then
converges for |z| < 1.
viii) If f is entire and for all z, then there exists an entire function g such that
for all
.
ix) A mobius transformation which maps the upper half plane onto itself and fixing
and no other points, must be of the form
for some
and
.
x) If f is entire and is bounded as
, then f is constant.
If is entire such that
in
then show that f has the form
where
are constants with
.
Evaluate , where c is the eight like figure shown in Fig. 1.
