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 theory correspond to vectors that, when used in _________ ______ ______ ________ _________.
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Let C1 and C2 be two binary codes with generator matrices
respectively.
a) Find the minimum distance of both the codes.
b) Find the generator matrix of the code
C = {(u|u+v)|u ∈ C1,v ∈ C2}
obtained from C1 and C2 by (u|u+v) construction. Also, find the minimum distance of C .
Draw the Tanner graph of the code C with parity check matrix
Let C be the [5, 2] ternary code generated by
Find the weight enumerator WC(x, y) of C.
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.
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 € )
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?)
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.