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ignou MMTE 5 solved assignment 2024

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Last Date of Submission of IGNOU MMTE-05 (MSCMACS) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Semester Wise
January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam).
July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).

Title Name	ignou MMTE 5 solved 2024
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MSCMACS
Course Name	M.Sc. Mathematics with Applications in Computer Science
Subject Code	MMTE 5
Subject Name	Coding Theory
Year	2024
Session	-
Language	English Medium
Assignment Code	MMTE-05/Assignmentt-1//2024
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2024. Latest MMTE 05 2024 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MMTE-05 (MSCMACS) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam). Semester Wise January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam). July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Assignment Code	MMTE 5/2024

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Questions Included in this Help Book

Ques 1.

1) 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 F $F_{2.}$

v) There is no LDPC code with parameters n = 16, c = 3 and r = 5.

Ques 2.

a) Which of the following binary codes are linear?

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.

Ques 3.

Let C1 and C2 be two binary codes with generator matrices

$G_1=\begin{bmatrix} 1 &0 & 0 & 1\\ 0 & 1 & 0&0 \\0 & 0 &1 &1 \end{bmatrix},$ $G_2=\begin{bmatrix} 1 & 0 & 0&1 \\ 0 & 1 & 1 & 0 \end{bmatrix},$

respectively.

a) Find the minimum distance of both the codes.

b) Find the generator matrix of the c

Ques 4.

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 $x=(x_1,x_2,.....,x-n)\, and\, y=(y_1,y_2,....,y_n). Suppose$ Observe that $wt(x)=n_1+n_2 \: and \: wt (y)=n_1+n_3.$

Ques 5.

Let be a binary code with a generator matrix each of whose rows has even weight. Show that, every codeword of 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? )

Ques 6.

Show that, if $x\epsilon F_{2}^{N,}$

$wt(x)\equiv x.x$ (mod 3)

Deduce that, if is a ternary self orthogonal code, the weight of each codeword is divisible by 3.

(Hint: Observe that $x^{2}=1for all x$ $0\epsilon F_{3} )$

Ques 7.

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.

Ques 8.

Let α be a root of $x^{2}+1=0$ in $F_{9.}$

a) Check whether α is a primitive element of $F_{9.}$ If it is not a primitive element in $F_{9.}$ find a primitive element γ in $F_{9.}$ in terms of α. b) Make a table similiar to Table 5.1 on page 184 for $F_{9.}$ with the primitive element γ

c) Factorise $X^{8}- 1$ over $F_{3.}$

d) Find all the possible generator polynomials of a [8,6] cyclic code.

Ques 9.

a) Let and be cyclic codes over $F_{q}$ with generator polynomials $g_{1}(x)$ and $g_{2(x),}$ respectively. Prove that

$\subseteq$ if and only if $g_{2(x)} |g_{1}(x).$

Ques 10.

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 g $g_{1}$ $\left ( x \right ).$

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 $g_{1}(x)?$

Ques 11.

7) a) Le l be the ternary $\left [ 8,3 \right ]$ narrow-sense $BCH$ 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

$g(x)=x^{5}-x^{4}+x^{3}+x^{2}-1$

Figure 1: Encoder for convolutional code.

is the generator polynomial of and

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

is the received word, find the transmitted codeword.

Ques 12.

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 $Wl (x,y) of l.$

Ques 13.

c) Find the generating idempotents of duadic codes of length n = 23 over $F_{3.}$ (Hint: Mimic example 6.1.7.)

Ques 14.

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 $C.$

Ques 15.

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