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).
We are tasked with proving that if \( C_1 \subseteq C_2 \), then the generator polynomial \( g_2(x) \) divides \( g_1(x) \), and vice versa, that is, if \( g_2(x) \) divides \( g_1(x) \), then \( C_1 \subseteq C_2 \). ### Definitions: 1. Cyclic Codes: A cyclic code is a type of linear block code where if a codeword is part of the code, then any cyclic shift of that codeword is also part of the code. 2. Generator Polynomial: The generator polynomial \( g(x) \) of a cyclic code is a polynomial that generates all the codewords of the code through its multiples. We assume ____ ____ _____ _____ ___ __________ ________ ___.
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Let C be a binary code with a generator matrix each of whose rows has even weight. Show that, every codeword of C has even weight.
(Hint: Why is it enough to prove that sum of vectors of even weight in is a vector of even weight?)
For each of the linear codes, find the degree, a generator matrix and a parity check matrix.
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 € )
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 C be the ternary [8,3] narrow-sense BCH code of designed distance δ = 5, which has defining set T = {1,2,3,4,6}. Use the primitive root 8th root of unity you chose in 4a) to avoid recomputing the the table of powers. If
g(x) = x5 - x4 + x3 + x2 - 1
Figure 1: Encoder for convolutional code.
is the generator polynomial of C and
y(x) = x7 - x6 - x4 - x3
is the received word, find the transmitted codeword.
Let C be the [5, 2] ternary code generated by
Find the weight enumerator WC(x, y) of C.
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.