Factor x5 - 1 over . Give the generator polynomials of all cyclic codes of length five over
.
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Prove that the linear fractional transformation maps the circle
into itself. Also prove
that f(z) is conformal in .
Find the image of the circle under the mapping
. What happens when
?
Let C be the [5, 2] binary code generated by
Find the weight enumerator C of C . Use McWilliams identity to find the weight enumerator of C. Verify your answer by finding the generator matrix of C and finding the weight distribution of C.
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)
Evaluate where c is the circle
.
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.
Find the parity check matrix of the code .