Which of the following binary codes are linear?
i)
ii)
Justify your answer.
A binary code \(\mathcal{C}\) is linear if it is a subspace of \(\mathbb{F}_2^n\), meaning it satisfies the following conditions: 1. It contains the zero codeword (all zeros). 2. It is closed under vector addition (modulo-2 addition). Now, let's check both given codes for these properties. --- _____ ________ ______ ____ _________ ________ ___ _____ ________ ___ _________ _________.
________ ____ ___ _____ ______ __________ _____ ________ ___.
_____ _________ _____ _________ _______ ______ ____ _________ ________ _________.
_________ ______ ____ ____ ______ ________ ___ ______ __________ _________ __________.
_________ _________ _________ ____ _______ ____ __________ _____ ____ _____ ________ ____.
__________ _________ ___ _________ _________ _______ _________ _____ __________ _________ _________ ___.
____ _________ _______ ______ ________ ____ _____ _______.
______ __________ ______ ______ __________.
_____ _________ ___ ___ __________ ___.
________ _____ _______ _____ ____ _________ _____ _____ _____.
____ ________ ___ ______ _______ ______ _____ ___ _______.
___ ___ ____ ________ ______ ___ _______ _______ ________ ______.
___ __________ ______ ________ ____ _____ ____.
__________ _______ ______ ____ ___ ________ _____ ______ ____ ______ _______.
___ ________ _________ ____ _______ ________ _______ _________ ___.
____ ________ ________ ___ _______.
___ __________ _________ _________ _______.
__________ ____ _________ _______ ________ ________.
_______ __________ ______ ____ _________.
_______ _______ _____ _________ _____ _____ ___ _______ ________ ________ _________.
__________ ______ _________ __________ _________ ____ __________.
_______ _______ _____ ____ ____ _______ ______.
__________ ______.
Get Full Answer on WhatsApp
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.
Show that, if x ∈ ,
wt (x) = x.x (mod 3)
Deduce that, if C is a ternary self orthogonal code, the weight of each codeword is divisible by 3.
(Hint: Observe that x2 = 1 for all x ≠0 € )
For each of the linear codes, find the degree, a generator matrix and a parity check matrix.
The 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 perfect code of length 2m + 1, m ∈ .
ii) Show that
(Hint: Start with the relation
iii) Deduce that every repetiition code of odd length is perfect.
Let C1 and C2 be cyclic codes over , with generator polynomials g₁ (x) and g2(x), respectively. Prove that C1
C2 if and only if g2(x) | g1(x).
Over , (1+x) | (xn - 1). Let C be the binary cyclic code (1+x) of length n. Let C1 be any binary cyclic code of length n with generator polynomial g1(x).
i) What is the dimension of ?
ii) Let w be subspace of containing all the vectors of even weight. Prove that W has dimension n - 1. (Hint: Consider the map w:
→
given by
Prove that & is the vector space of all vectors in with even weight.
iv) If C1 has only even weight codewords, what is the relationship between (1+x) and g1(x)?
v) If C1 has some odd weight codewords, what is the relationship between 1 + x and g1(x)?
Let a be a root of x² + 1 = 0 in .
a) Check whether α is a primitive element of . If it is not a primitive element in
find a primitive element γ in
in terms of a.
b) Make a table similiar to Table 5.1 on page 184 for with the primitive element y
c) Factorise x8 over .
d) Find all the possible generator polynomials of a [8, 6] cyclic code.