Find the generating idempotents of duadic codes of length n = 23 over F3. (Hint: Mimic example 6.1.7.)
To find the generating idempotents of duadic codes of length \( n = 23 \) over \( F_3 \), we can follow the approach outlined in example 6.1.7. Duadic codes are special linear block codes over a finite _____ _________ _________ ___ _________ ________.
____ _____ ___ __________ ____ _______.
______ __________ ______ ____ ____ ______ ________ ___.
________ _________ __________ ______ _____ ______ _____ _______.
__________ ___ ___ ____ ___ ____ _______ _________ ____.
_________ ____ ___ _______ ________ _________.
_____ _________ ___ _________ ____.
_____ ____ ________ ___ _______ __________ _____ ______ __________.
_______ __________ _________ ______ __________ ______ ________ ________ __________ ______ _________ ______.
_____ _______ ____ ___ ___ _____ _________ __________ ____.
___ _________ ________ ________ _____ ____ _______ _______.
__________ _____ _________ __________ ___ ___ ________ _______.
_______ _______ __________ ________ _____ ___ ___ ______.
______ _______ ___ __________ _____ _____ _______ ______ ____ ________ _________.
__________ ___ _________ _________ ______ ________ ___ ____ _____ ______.
________ ____ ________ _______ __________.
__________ _________ ___ ___ __________ ______.
________ _________ ___ __________ ________ _________ ____ ______ ___.
______ ________ _________ _______ ____ __________ _____ _________ _____.
________ _______ ______ ____ _________ __________.
______ _____ ________ ___ __________.
___ _____ _____ ___ _________ ___ _____ ___.
__________ ___ _______ ___ ______ ____ __________ ___ _______ ______ ________ __________.
______ ______ ___ _____ _____.
_____ _____ ______.
Get Full Answer on WhatsApp
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.
If C1 has some odd weight codewords, what is the relationship between 1+x and g1(x)?
If C1 has only even weight codewords, what is the relationship between (1+x) and g1(x)?
Show that, if x ∈ Fn3
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. (3)
(Hint: Observe that x
2 = 1 for all x 6= 0 ∈ F3)
Find the convolutional code for the message 11011. The convolutional encoder is given in Fig. 1.
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