Question

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-ternarynbspnbspnarrowsensenbspcode-63666

                   Figure 1: Encoder for convolutional code.

is the generator polynomial of  Image ignouassignments-ignouacademy-com--p-lnbspbe-35400 and

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

is the received word, find the transmitted codeword.

10 Mar 2024
Answer :
Word Count : 1405
Let’s break this problem into steps. --- ## 1. Understand the problem This is an error-correction coding problem where: - We have a ternary narrow-sense BCH code (over \( GF(3) \)) with a certain designed distance. - Defining set \( T \) is given, probably in the original full problem statement but missing here. - They mention a primitive 8th root of unity chosen in problem 4a (not provided here), so we need to recall or reconstruct that. - We have a generator polynomial \( g(x) \) for a convolutional code ? — wait, reading carefully, it says: “is the generator polynomial of % and … is the received word, find the transmitted codeword.” The convolutional code diagram is just Figure 1, maybe not directly relevant — maybe the problem is actually about a cyclic block code not convolutional. The received word is given in polynomial form but cut off in the text you provided. --- My guess from standard problems: We have a primitive 8th root of unity \( \alpha \) in \( GF(3^m) \) for \( m \) such that \( 8 \mid 3^m - 1 \). For \( GF(3) \), smallest \( m \) such that \( 8 \mid 3^m - 1 \) is \( m=2 \)? Check: \( 3^2 - 1 = 8 \), yes. So \( GF(9) \) contains primitive 8th roots of unity. Let’s say \( \alpha \) is a primitive 8th root of unity in \( GF(9) \) with primitive polynomial \( x^2 + x + 2 \) or \( x^2 + 1 \) over \( GF(3) \) — but \( x^2 + 1 \) is irreducible over \( GF(3) \) since \( -1 \equiv 2 \), no, check: \( x^2 + 1 \) mod 3: \( 0^2+1=1\), \( 1^2+1=2\), \( 2^2+1=5\equiv 2\), never 0, so irreducible, so \( GF(9) \cong GF(3)[x]/(x^2+1) \). Let \( \beta \) be a root: \( \beta^2 + 1 = 0 \), so \( \beta^2 = 2 \). Now \( \beta^4 = 4 \equiv 1 \) mod 3? Wait \( 2^2 = 4 \equiv 1 \) mod 3, yes, ___ ___ ______ ______ _________ _____ __________ ________.
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Related Questions

a) Which of the following binary codes are linear?

Image ignouassignments-ignouacademy-com--p-ternarynbspnbspnarrowsensenbspcode-22144

Justify your answer.

 b) Find the minimum distance for each of the codes.

 c) For each of the linear codes, find the degree, a generator matrix and a parity check matrix.

Let Image ignouassignments-ignouacademy-com--p-ternarynbspnbspnarrowsensenbspcode-82747 be a binary code with a generator matrix each of whose rows has even weight. Show that, every codeword of Image ignouassignments-ignouacademy-com--p-lnbspbe-23135 has even weight ( Hint: Why is it enough to prove that sum of vectors of even weight in F_{2}^{n} is a vector of even weight? ) 

 

8) a) Let 

C=\left \{ 0000,1113,2222,3331,1313,2020,3131,0022,1131,2200,3313,0220,1333,2002,3111 \right \}be the-linear code. Find the Gray image of C.

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)?

d) 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\epsilon

ii) Show that

\sum_{i=0}^{m\binom{2m+1}{i=2^{2m}\sum _{i=0}^{m}\binom{2m+1}{i}=2^{2m}

(Hint: Start with the relation

2^{2m+1}=\sum _{i=0}^{2m+1}\binom{2m+1} )

iii) Deduce that every repetiition code of odd length is perfect.

 

a) Let Image ignouassignments-ignouacademy-com--p-ternarynbspnbspnarrowsensenbspcode-58595 and Image ignouassignments-ignouacademy-com--p-lnbspbe-64564 be cyclic codes overF_{q} with generator polynomials g_{1}(x) and g_{2(x),}respectively. Prove that Image ignouassignments-ignouacademy-com--p-lnbspbe-22714

\subseteq Image ignouassignments-ignouacademy-com--p-ternarynbspnbspnarrowsensenbspcode-62784 if and only if g_{2(x)} |g_{1}(x).

 

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.

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