Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
i) If the weight of each element in the generating matrix of a linear code is at least r, the minimum distance of the code is at least r.
ii) There is no linear self orthogonal code of odd length.
iii) There is no 3-cyclotomic coset modulo 121 of size 25.
iv) There is no duadic code of length 15 over .
v) There is no LDPC code with parameters ,
and
.
___ ____ ___ ___ ________ ____ _______ ____ _______.
______ _____ _________ ___ _________ _______ __________.
_______ _____ ___ _______ ________ ____.
__________ _____ _____ ______ ___ __________ __________ _____ ______ _____ _____.
_______ _____ ____ _____ _____ _________.
__________ ___ ________ ______ ___ _______ ___ _______ ___.
_________ ________ ___ __________ __________.
___ _________ ________ ____ ________.
________ ______ _________ __________ ____ ___.
___ ________ __________ _______ ___ _____ _________ ____.
_________ ________ __________ ___ _______.
__________ ___ ________ ____ ______.
____ _______ _____ __________ _______ ________ __________ ______ _________ _____ _____.
_______ ___ ___ ______ ________.
_______ __________ ________ _____ ____ ___ _____ _____ ____ ______.
______ ___ _______ _____ _____ _____ ____ ______ ____ _____ _____.
__________ ____ __________ _____ ___.
_______ ________ ___ _____ _____.
______ _________ _____ ____ _________.
__________ _____ _____ ____ ________ _______ ___ _________ ____ ___ _________.
_________ _________ _________ __________ ____ _____ _____ ________ __________ _______.
_________.
- Solved Assignments
- Study Notes
- Guess Papers
- Question Papers
✓ Unlock up to 60% of the answer
✓ No payment required
For each of the linear codes, find the degree, a generator matrix and a parity check matrix.
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.
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.
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.
Consider and the closed circular region
. Find points in R where |f(z)| has its maximum and minimum values.
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.
Factor x5 - 1 over . Give the generator polynomials of all cyclic codes of length five over
.