Question

a)  If x,y\in F_{2,}^{n} show that

wt(x+y)=wt(y)-2wt(x\cap y)

where x\cap y is the vector in F_{2}^{n} which has 1s precisely at those positions where x and y have 1s. 

(Hint: Let x=(x_1,x_2,.....,x-n)\, and\, y=(y_1,y_2,....,y_n). SupposeImage ignouassignments-ignouacademy-com--p-that-42424Observe that wt(x)=n_1+n_2 \: and \: wt (y)=n_1+n_3.

08 Mar 2024
Answer :
Word Count : 723
To solve the problem, we'll analyze the weights of the vectors \( x \), \( y \), and \( x + y \) in the vector space \( \mathbb{F}_2^n \). ### Definitions: 1. Weight (wt): The weight of a vector is the number of its non-zero (i.e., 1) entries. 2. Addition in \( \mathbb{F}_2^n \): The sum \( x + y \) is computed component-wise modulo 2. This means: - \( (x + y)_i = 1 \) if \( x_i \neq y_i \). - \( (x + y)_i = 0 \) if \( x_i = y_i \). ### Step-by-Step Solution: 1. Partition the Indices: - Let \( n_1 \) be the number of positions where both \( x \) and ________ __________ _____ ________ ________ __________ ________ ____.
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Related Questions

b) Draw the Tanner graph of the code L with parity check matrix \begin{bmatrix} 1 &0 &0 &0 &1 &0 &0 &1 & 0 &1 \\0 &1 &0 &0 &1 &1 &0 &0 &1 &0 \\0 &0 &1 &0 &0 &1 &1 &0 &0 &1 \\0 &0 &0 &1 &0 &0 &1 &1 & 1 &0 \end{bmatrix}.

c) Find the convolutional code for the message 11011. The convolutional encoder is given in Fig. 1.

Over F_{2,}\left ( 1+x \right )|\left ( x^{n} -1\right ). Let L be the binary cyclic code \left ( 1+x \right ) of lenth n. l_{1} be any binary cyclic code of length n with generator polynomial gg_{1} \left ( x \right ).

i) What is the dimension of l?

ii) Let w be subspace of F_{2}^{n} containing all the vectors of even weight. Prove that W has dimension n−1. (Hint: Consider the map w F_{2}^{n} →F_{2}^{} given by 

w((a_{1}a_{2}....a_{n}))=a_{1}+a_{2}+...+a_{n. )

iii) Prove that l is the vector space of all vectors in F_{2}^{n} with even weight. 

iv) If l_{1} has only even weight codewords, what is the relationship between  (1+x) and g_{1}(x)?

v) If l_{1} has some odd weight codewords, what is the relationship between  1+x and g_{1}(x)?

7) a) Le l be the ternary \left [ 8,3 \right ] narrow-senseBCH code of designed distance \delta =5, which has defining set T=\left \{ 1,2,3,4,5,6 \right \}.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

 

Image ignouassignments-ignouacademy-com--p-vector-12752

                   Figure 1: Encoder for convolutional code.

is the generator polynomial of  Image ignouassignments-ignouacademy-com--p-wherenbspnbspis-31526 and

y(x)=x^{7}-x^{6}-x^{4}-x^{3}

is the received word, find the transmitted codeword.

a)  If x,y\in F_{2,}^{n} show that

wt(x+y)=wt(y)-2wt(x\cap y)

where x\cap y is the vector in F_{2}^{n} which has 1s precisely at those positions where x and y have 1s. 

(Hint: Let x=(x_1,x_2,.....,x-n)\, and\, y=(y_1,y_2,....,y_n). SupposeImage ignouassignments-ignouacademy-com--p-anbspnbspnbspshow-28090Observe that wt(x)=n_1+n_2 \: and \: wt (y)=n_1+n_3.

b) Let l be the  \left [ 5,2 \right ] ternary code generated by

                                                     G=\begin{pmatrix} 1 &1 &0 &0 &1 \\0 &1 &0 &1 &1 \end{pmatrix}.

Find the weight enumerato Wl (x,y) of l.

a) Let Image ignouassignments-ignouacademy-com--p-vector-41631 and Image ignouassignments-ignouacademy-com--p-anbspnbspnbspshow-28971 be cyclic codes overF_{q} with generator polynomials g_{1}(x) and g_{2(x),}respectively. Prove that Image ignouassignments-ignouacademy-com--p-anbspnbspnbspshow-66710

\subseteq Image ignouassignments-ignouacademy-com--p-vector-65330 if and only if g_{2(x)} |g_{1}(x).

 

Show that, ifx\epsilon F_{2}^{N,}

 wt(x)\equiv x.x (mod 3)

Deduce that, if Image ignouassignments-ignouacademy-com--p-anbspnbspnbspshow-59429is a ternary self orthogonal code, the weight of each codeword is divisible by 3.

(Hint: Observe that x^{2}=1for all x Image ignouassignments-ignouacademy-com--p-vector-650460\epsilon F_{3} )

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