Find the convolutional code for the message 11011. The convolutional encoder is given in Fig. 1.
To find the convolutional code for the message "11011" using the given convolutional encoder in Fig. 1, we need to follow the encoding process. Convolutional codes are typically represented by a trellis diagram, where each node in the _________ ________ ___ _______ ______ ________.
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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 ∈ 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)
If C1 has only even weight codewords, what is the relationship between (1+x) and g1(x)?
Over F2, (1+x) | (x n −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).
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.
If C1 has some odd weight codewords, what is the relationship between 1+x and g1(x)?