Find the maximum modulus of on the closed circular region defined by
.
The function (f(z)=2z+5i) is linear and hence analytic everywhere, so its modulus on the closed disk (|z|\le 2) attains its ____ _______ _________ _________ _____ _______ ___.
____ ___ _________ _________ ______ _________ _______ ______ ___ ____.
_______ ______ ____ __________ ____.
___ ___ _________ __________ ____.
__________ ______ ______ _______ ____.
______ ____ ____ ___ _____ _____ _________ ___ ___.
__________ ____ _____ ___ _______ ________ _________ ____ ________ ________ _______.
________ ___ _____ ___ ______.
______ _________ _________.
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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)
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.
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 ?
) 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.
Factor x8 - 1 over . Give the generator polynomials of all cyclic codes of length eight over
.
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)
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.