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.
iii) 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. The minimum distance of a linear code is the minimum weight among all non-zero codewords. If the weight of each element in the generating matrix is at least r, then any non-zero codeword will have a weight of at least r, ensuring that the minimum distance is at least r.
ii) True. A linear self-orthogonal _______ _____ ________ ________ _________ ___ ___ __________ _________ ____ ______ _____.
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Let C be the [5, 2] ternary code generated by
Find the weight enumerator WC (x, y) of C
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 ∈ N.
ii) Show that
(Hint: Start with the relation
2 2m+1
Which of the following binary codes are linear?
i) C = {(0,0,0,0),(1,0,1,0),(0,1,1,0),(1,1,1,0)}
ii) C = {(0,0,0),(1,1,0),(1,0,1),(0,1,1)}
Justify your answer. (3)
b) Find the minimum distance for each of the codes. (4)
c) For each of the linear codes, find the degree, a generator matrix and a parity check matrix. (3)
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. (3) ( Hint: Why is it enough to prove that sum of vectors of even weight in F n 2 is a vector of even weight? )
Let α be a root of x2 +1 = 0 in F9.
a) Check whether α is a primitive element of F9. If it is not a primitive element in F9 find a
primitive element γ in F9 in terms of α. (4)
b) Make a table similiar to Table 5.1 on page 184 for F9 with the primitive element γ
c) Factorise x
8 −1 over F3. (6)
d) Find all the possible generator polynomials of a [8,6] cyclic code.
Let C = {0000,1113,2222,3331,0202,1313,2020,3131,0022,1131,2200,3313,0220,1333,2002,3111} be the Z4-linear code. Find the Gray image of C.
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) = x 5 −x 4 +x 3 +x 2 −1
is the generator polynomial of C and
y(x) = x 7 −x 6 −x 4 −x 3
is the received word, find the transmitted codeword.