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 mininum distance of the code is at least r.

ii) There is no linear self orthogonal code of odd length.

There is no 3-cyclotomic coset modulo 121 of size 25.

iv) There is no duadic code of length 15 over F₂.

v) There is no LDPC code with parameters n = 16, c = 3 and r = 5.

i) True. If the weight of each element in the generating matrix of a linear code is at least r, then the minimum distance of the code is at least r. This is because the minimum distance of a code is the smallest Hamming distance between any two distinct codewords. Since the weight of each codeword is at least r, the ________ ______ _________ ___ _____ __________.
___ _______ _____ ______ ________ ______ _______ ___ _________ ____ ______ _________.
________ _______ _____ ________ _________ _________ __________ _______ _____ __________ __________ ___.
_________ _______ _______ _______ ____ ____ _______ ____ _________ _______ ____.
___ ___ _________ _______ ____ _________ _________.
_____ ________ __________ ________ _______.
_______ __________ ___ ______ ________ ____ ____ __________ _______ ________ _______ ______.
_____ __________ ____ ___ ___ __________ ____ ________ ______ _______ ________.
________ _________ ____ _______ _____ _____ _________ ______.
_______ __________ ____ __________ ________ _________ _________ ________.
______ _________ ____ __________ ____ ____ ________ _________ __________ ______.
______ __________ ____ ________ _________ _________ ______.
________ _______ _____ _____ ______ _____.
_______ ______ _________ ____ _____ ___ _____ __________ ______.
____ _____ _______ __________ _____ _________ _________.
____ _______ _____ _____ ______ ___ ___ __________ _______ _______ ____ _______.
__________ ________ _________ _________ __________ ________ ___ ___ __________ ___ ______ ________.
______ ______ _______ ______ _______.
____ _____ ________ ______ ____ ____ __________ _______ ______.
_____ __________ ________ ___ ____.
________ ___ _____ _______ ____ ________ _______ ________ ________.
____ ________ ______ ________ ________.
___ ______ ___ __________ _______ _________ _____ ___ ___.
_______ _______ ________ _____ _______ ______ _______ _________ ________.
__________ ____ _____ _____ ___ __________.
_____ ____ ________ _________ ___ ______ __________ _____.
_____ _______ ______ _______ ___ ______ ______.
____ ______ __________ ________ ___.
_________ ______ ____ _________ ______ _____ ___ ______ ___ ___ _____.
___ ___ __________ ___ ____.
________ ___ __________ _______ ____ _______ __________ ___ ___ ________ __________.
____ ____ ________ _____ _____.
__________ _____ _____ ______ ________ ________ ____.
______ _____ ______ _______ _________ ____ _________.
_______ _______ _________ _____ ________ ___ __________.
______ _________ _________ ______ ___ ________ ______ ____ _____.
__________ __________ ___ _____ _______ ________ _____ _________ ___.
_______ _________ _______ _______ ______.
Get Full Answer on WhatsApp