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.
To solve the problem of finding the generating idempotents of duadic codes of length \( n = 23 \) over \( \mathbb{F}_3 \), we first need to understand the context of duadic codes and idempotents. Idempotents in coding ______ ___ __________ _________ __________ ________ _______ __________ _________.
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_____ ________ _______ _________ _____ ______ _______ _________.
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_____ __________ _________ _________ ____ ______ __________ ________.
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For each of the linear codes, find the degree, a generator matrix and a parity check matrix.
Let C be the [5, 2] ternary code generated by
Find the weight enumerator WC(x, y) 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) = 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.
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)?
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 binary codes are linear?
i)
ii)
Justify your answer.
Find the convolutional code for the message 11011. The convolutional encoder is given in
Fig. 1.