Prove that the linear fractional transformation maps the circle
into itself. Also prove
that f(z) is conformal in .
Consider the linear fractional transformation [ \phi(z)=\frac{2z-1}{2-z}. ] First, we show that the circle (C:\ |z|=1) is mapped into itself. Let (|z|=1). Then (\bar z = 1/z). Compute [ |\phi(z)|^2=\phi(z)\overline{\phi(z)} =\frac{2z-1}{2-z}\cdot \frac{2\bar ________ __________ ____ ____ _______ _________ __________.
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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.
Let C be a cyclic code of length eight over with generator polynomial
. Find the generator matrix of C, the generator polynomial of C and the parity check matrix of C.
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 ?
For each of the linear codes, find the degree, a generator matrix and a parity check matrix.
Find the parity check matrix of the code . Decode the following vectors
i)
ii)
iii)
iv)
Let and
be two binary codes with generator matrices
respectively.
i) Find the minimum distance of both the codes.
Table 1: Table for F16.
| 0000 | 0 | 1000 | α³ | 1011 | α⁷ | 1110 | α¹¹ |
|---|---|---|---|---|---|---|---|
| 0001 | 1 | 0011 | α⁴ | 0101 | α⁸ | 1111 | α¹² |
| 0010 | α | 0110 | α⁵ | 1010 | α⁹ | 1101 | α¹³ |
| 0100 | α² | 1100 | α⁶ | 0111 | α¹⁰ | 1001 | α¹⁴ |