If C1 has some odd weight codewords, what is the relationship between 1+x and g1(x)?
In coding theory, particularly in the realm of polynomial codes, the weight of a codeword refers to the number of non-zero coefficients in the polynomial representation of that codeword. If C1, a code, contains codewords with odd weight, we can explore the relationship between the polynomial representation of 1+x and g1(x), where g1(x) represents a polynomial ____ ___ ______ ___ _____ ___ _______ ______.
_______ _______ ___ _________ ______ _______.
______ ____ __________ ____ ______ _______.
_______ _______ __________ _____ _____ ___ ____ ____ _______.
__________ ___ _______ _________ __________.
___ __________ ______ ________ _________ ______ _________ ______ __________ __________.
____ _________ __________ ______ _______ __________ __________ ________ ___.
_____ _________ ______ _______ ________ ______ ________ ________.
__________ _______ _________ _______ _____ _____ ______ _________.
________ __________ _____ ________ __________ _____.
___ ________ ________ ______ ___ _________ __________ _________ ____ ________ ____ _________.
___ ___ _________ _______ ______ __________ __________.
___ __________ ____ ________ ________ _______ ________ _________.
________ ___ __________ ___ ____ _________.
________ ____ _____ ________ ___ _______ _________ ______ ___.
_____ _________ ____ __________ _________ ___ ________.
_________ ___ ________ _______ ________ ______ ________ ________ ___ _______ __________.
___ ________ __________ _______ ______ ___ _______.
____ __________ ___ __________ ________ ______ _____ ______ _____ ______ __________ _______.
___ _______ _______ _______ _______ _________ ___.
_________ ________ ___ _________ ____ _______.
________ __________ ____ ________ ______ ____ _________ ____ ________.
__________ _______ ______ _____ _________ ___ _______ _____ ____ _________ ____ _________.
______ __________ ________ _____ ___ _________ ____ ___.
____ ______ ____ ________ _____.
_________ _______ ____ ______ ___ ___ __________.
____ _____ __________ ________ ___ ________ _________ ______ _______ _________ __________.
________ __________ __________ __________ __________ ________ __________.
Get Full Answer on WhatsApp
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 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 .
What is the dimension 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.
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)
Deduce that every repetiition code of odd length is perfect.
Let C be the [5, 2] ternary code generated by
Find the weight enumerator WC (x, y) of C