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 __________ ________ _____ ____ ___ __________ _____ ______ __________.
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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.
Find the generating idempotents of duadic codes of length n = 23 over .
(Hint: Mimic example 6.1.7.)
be the Z4-linear code. Find the Gray image of C.
Find the minimum distance for each of the codes.
Which of the following binary codes are linear?
i)
ii)
Justify your answer.
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)?
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.
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.