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IGNOU MMTE 5 SOLVED ASSIGNMENT

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Rs. 123

IGNOU MMTE 5 SOLVED ASSIGNMENT

~~Rs. 200~~
Rs. 123

Last Date of Submission of IGNOU MMTE-05 (MSCMACS) 2026 Assignment is for January 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).
Semester Wise
January 2026 Session: 30th March, 2026 (for June 2026 Term End Exam).
July 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).

Title Name	IGNOU MMTE 5 SOLVED ASSIGNMENT
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	2026
Session
Language	English Medium
Assignment Code	MMTE-05/Assignmentt-1//2026
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2026. Latest MMTE 05 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MMTE-05 (MSCMACS) 2026 Assignment is for January 2026 Session: 30th September, 2026 (for December 2026 Term End Exam). Semester Wise January 2026 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).

~~Rs. 200~~
Rs. 123

Questions Included in this Help Book

Ques 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 minimum 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 $\mathbf{F}_2$ .
v) There is no LDPC code with parameters $n = 16$ , $c = 3$ and $r = 5$ .

Ques 2.

Which of the following binary codes are linear?

i) $\mathcal{C}=\{(0,0,0,0),(1,0,1,0),(0,1,1,0),(1,1,1,0)\}$

ii) $\mathcal{C}=\{(0,0,0),(1,1,0),(1,0,1),(0,1,1)\}$

Justify your answer.

Ques 3.

Find the minimum distance for each of the codes.

Ques 4.

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

Ques 5.

For a linear code C with generator matrix

$$G = \begin{bmatrix} 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 \end{bmatrix}$
find the parity check matrix. Check that any two columns of the parity check matrix are linearly independent and there are three columns that are linearly dependent. What is the minimum distance of C ?

Ques 6.

Find the parity check matrix of the code $\overline{\mathcal{H}}_3$ . Decode the following vectors
i) $(1\ 1\ 1\ 0\ 0\ 0\ 0\ 1)$
ii) $(1\ 0\ 1\ 1\ 0\ 0\ 1\ 0)$
iii) $(0\ 0\ 1\ 1\ 0\ 1\ 0\ 0)$
iv) $(0\ 1\ 1\ 1\ 1\ 0\ 0\ 1)$

Ques 7.

Find the parity check matrix of the code $\mathcal{H}_{2,5}$ .

Ques 8.

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ 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}, \quad G_2 = \begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{bmatrix},$

respectively.

i) Find the minimum distance of both the codes.

Table 1: Table for F₁₆.

0000	0	1000	α³	1011	α⁷	1110	α¹¹
0001	1	0011	α⁴	0101	α⁸	1111	α¹²
0010	α	0110	α⁵	1010	α⁹	1101	α¹³
0100	α²	1100	α⁶	0111	α¹⁰	1001	α¹⁴

ii) Find the generator matrix of the code

$\mathcal{C}=\{(\mathbf{u}|\mathbf{u}+\mathbf{v})\mid\mathbf{u}\in\mathcal{C}_1,\mathbf{v}\in\mathcal{C}_2\}$

obtained from C1 and C2 by (u|u+v) construction. Also, find the minimum distance of C .

Ques 9.

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ 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}, \quad G_2 = \begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{bmatrix},$

respectively.

i) Find the minimum distance of both the codes.

Table 1: Table for F₁₆.

0000	0	1000	α³	1011	α⁷	1110	α¹¹
0001	1	0011	α⁴	0101	α⁸	1111	α¹²
0010	α	0110	α⁵	1010	α⁹	1101	α¹³
0100	α²	1100	α⁶	0111	α¹⁰	1001	α¹⁴

ii) Find the generator matrix of the code

$\mathcal{C}=\{(\mathbf{u}|\mathbf{u}+\mathbf{v})\mid\mathbf{u}\in\mathcal{C}_1,\mathbf{v}\in\mathcal{C}_2\}$

obtained from C1 and C2 by (u|u+v) construction. Also, find the minimum distance of C .

Ques 10.

) For the binary, (6,3) linear code C with generator matrix $G = \begin{bmatrix} 1 & 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 1 \end{bmatrix},$
prepare a standard array for decoding. Use it to decode the vectors (1, 1, 1, 0, 1, 1), and (1, 1, 0, 1, 1, 1). \hfill (7)

Ques 11.

Prepare a syndrome table for C in part a) and decode the vectors (1, 1, 1, 1, 0, 1) and (0, 1, 0, 1, 1, 1). \hfill (5)

Ques 12.

he 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 binary repetition code of length $2m + 1, m \in \mathbf{N}$ . \hfill (2)
ii) Show that

$$\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}{i}\mathbf{)}$
iii) Deduce that every repetition code of odd length is perfect.

Ques 13.

) Let C be the ternary [8,3] narrow-sense BCH code of designed distance $\delta = 5$ , 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. Use the following table in 1.

Ques 14.

Let C be the [5, 2] binary code generated by
$$G = \begin{pmatrix} 1 & 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 1 \end{pmatrix}.$
Find the weight enumerator C of C . Use McWilliams identity to find the weight enumerator of C. Verify your answer by finding the generator matrix of C and finding the weight distribution of C.

Ques 15.

Let C be a cyclic code of length eight over $\mathbf{F}_3$ with generator polynomial $\overline{2} + x + x^3 + x^4$ . Find the generator matrix of C, the generator polynomial of C and the parity check matrix of C.

Ques 16.

Factor x⁵ - 1 over $\mathbf{F}_2$ . Give the generator polynomials of all cyclic codes of length five over $\mathbf{F}_2$ .

Ques 17.

Factor x⁸ - 1 over $\mathbf{F}_5$ . Give the generator polynomials of all cyclic codes of length eight over $\mathbf{F}_5$ .

Ques 18.

Determine whether each of the following statement is true or false. Justify your answer with a short proof or a counter example.
i) If $z = a + ib$ , where a and b are integers, then $|1 + z + z^2 + \dots + z^n| \geq |z|^n$ if a > 0.
ii) If f(z) and $\overline{f(\bar{z})}$ are analytic functions in a domain, then f is necessarily a constant.
iii) A real-valued function u(x, y) is harmonic in D iff u(x, -y) is harmonic in D.
iv) $\lim_{n \to \infty} (n!)^{1/n} = \infty$ .
v) The inequality $|e^a - e^b| \leq |a - b|$ holds for $a, b \in D = \{w : \text{Re } w \leq 0\}$ .
vi) If $f(z) = \sum_{n=0}^{\infty} a_n(z - a)^n$ has the property that $\sum_{n=0}^{\infty} f^{(n)}(a)$ converges, then f is necessarily an entire function.
vii) If a power series $\sum_{n=0}^{\infty} a_n z^n$ converges for |z| < 1 and if $b_n \in \mathbb{C}$ is such that |b_n| < n² |a_n| for all $n \geq 0$ , then $\sum_{n=0}^{\infty} b_n z^n$ converges for |z| < 1.
viii) If f is entire and $f(z) = f(-z)$ for all z, then there exists an entire function g such that $f(z) = g(z^2)$ for all $z \in \mathbb{C}$ .
ix) A mobius transformation which maps the upper half plane $\{z : \text{Im } z > 0\}$ onto itself and fixing $0, \infty$ and no other points, must be of the form $Tz = \alpha z$ for some $\alpha > 0$ and $\alpha \neq 1$ .
x) If f is entire and $\text{Re } f(z)$ is bounded as $|z| \to \infty$ , then f is constant.

Ques 19.

If $f = u + iv$ is entire such that $u_x + v_y = 0$ in $\mathbb{C}$ then show that f has the form $f(z) = az + b$ where $a, \mathbf{b}$ are constants with $\operatorname{Re} a = 0$ .

Ques 20.

Consider $f(z) = z^2 - z$ and the closed circular region $R = \{z : |z| \leq 1\}$ . Find points in R where |f(z)| has its maximum and minimum values.

Ques 21.

Find the points where the function $f(z) = \frac{\operatorname{Log}(z + 4)}{z^2 + i}$ is not analytic.

Ques 22.

) Evaluate the following integrals:
i) $I = \int_{0}^{2\pi} f(e^{i\theta}) \cos^2(\theta/2) \, d\theta$ .

ii) $I = \int_{0}^{2\pi} f(e^{i\theta}) \sin^2 \theta/2 \, d\theta$ .

Ques 23.

Find the image of the circle $|z| = r (r \neq 1)$ under the mapping $w = f(z) = \frac{z - i}{z + i}$ . What happens when $r = 1$ ?

Ques 24.

If $p(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1} + z^n (n \geq 1)$ , then show that there exists a real R > 0 such that $2^{-1} |z|^n \leq |p(z)| \leq 2 |z|^n$ for $|z| \geq R$ .
b) Find all solutions to the equation $\sin z = 5$ .

Ques 25.

Find the constant c such that $f(z) = \frac{1}{z^n + z^{n-1} + \dots + z^2 + z - n} + \frac{c}{z - 1}$ can be extended to be analytic at $z = 1$ , when $n \in \mathbb{N}$ is fixed.

Ques 26.

Find all the singularities of the function $f(z) = \exp \left( \frac{z}{\sin z} \right)$ .

Ques 27.

Evaluate $\oint_c \frac{dz}{z^2 + 1}$ where c is the circle $|z| = 4$ .

Ques 28.

Find the maximum modulus of $f(z) = 2z + 5i$ on the closed circular region defined by $|z| \leq 2$ .

Ques 29.

Evaluate $\int_{c} \frac{z^3 + 3}{z(z - i)^2} \, dz$ , where c is the eight like figure shown in Fig. 1.

Ques 30.

Find the radius of convergence of the following series.
i) $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k!} (z - 1 - i)^k$ $\quad \quad$ ii) $\sum_{k=1}^{\infty}\left(\frac{6k+1}{2k+5}\right)^k(z-2i)^k$

Ques 31.

Expand $f(z) = \frac{1}{(z-1)^2 (z-3)}$ in a Laurent series valid for

i) 0 < |z - 1| < 2 and $\quad \quad \quad \quad$ ii) 0 < |z - 3| < 2.

Ques 32.

Find the zeros and singularities of the function $f(z) = \frac{z}{4 \cos^2 z - 1}$ in $|z| \leq 1$ . Also find the residue at the poles

Ques 33.

Prove that the linear fractional transformation $\phi(z) = \frac{2z - 1}{2 - z}$ maps the circle $c : |z| = 1$ into itself. Also prove

that f(z) is conformal in $\bar{D} = \{z : |z| \leq 1\}$ .

Ques 34.

Find the image of the semi-infinite strip x > 0, 0 < y < 1 when $w = i/z$ . Sketch the strip and its image.

Ques 35.

Show that there is only one linear fractional transformation that maps three given distinct points z₁, z₂ and z₃ in the extended z plane onto three specified distinct points w₁, w₂ and w₃ in the extended w plane.

Ques 36.

Evaluate the following integrals

a) $\int_{0}^{\infty} \frac{x^2 + 2}{(x^2 + 1)(x^2 + 4)} \, dx$ .
b) $\int_{-\infty}^{\infty} \frac{\sin^2 2x}{1 + x^2} \, dx$ .

Ques 37.

Ques 38.

Which of the following binary codes are linear?

i) $\mathcal{C}=\{(0,0,0,0),(1,0,1,0),(0,1,1,0),(1,1,1,0)\}$

ii) $\mathcal{C}=\{(0,0,0),(1,1,0),(1,0,1),(0,1,1)\}$

Justify your answer.

Ques 39.

Find the minimum distance for each of the codes.

Ques 40.

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

Ques 41.

For a linear code C with generator matrix

Ques 42.

Ques 43.

Find the parity check matrix of the code $\mathcal{H}_{2,5}$ .

Ques 44.

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ 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}, \quad G_2 = \begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{bmatrix},$

respectively.

i) Find the minimum distance of both the codes.

Table 1: Table for F₁₆.

0000	0	1000	α³	1011	α⁷	1110	α¹¹
0001	1	0011	α⁴	0101	α⁸	1111	α¹²
0010	α	0110	α⁵	1010	α⁹	1101	α¹³
0100	α²	1100	α⁶	0111	α¹⁰	1001	α¹⁴

ii) Find the generator matrix of the code

$\mathcal{C}=\{(\mathbf{u}|\mathbf{u}+\mathbf{v})\mid\mathbf{u}\in\mathcal{C}_1,\mathbf{v}\in\mathcal{C}_2\}$

obtained from C1 and C2 by (u|u+v) construction. Also, find the minimum distance of C .

Ques 45.

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ 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}, \quad G_2 = \begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{bmatrix},$

respectively.

i) Find the minimum distance of both the codes.

Table 1: Table for F₁₆.

0000	0	1000	α³	1011	α⁷	1110	α¹¹
0001	1	0011	α⁴	0101	α⁸	1111	α¹²
0010	α	0110	α⁵	1010	α⁹	1101	α¹³
0100	α²	1100	α⁶	0111	α¹⁰	1001	α¹⁴

ii) Find the generator matrix of the code

$\mathcal{C}=\{(\mathbf{u}|\mathbf{u}+\mathbf{v})\mid\mathbf{u}\in\mathcal{C}_1,\mathbf{v}\in\mathcal{C}_2\}$

obtained from C1 and C2 by (u|u+v) construction. Also, find the minimum distance of C .

Ques 46.

Ques 47.

Prepare a syndrome table for C in part a) and decode the vectors (1, 1, 1, 1, 0, 1) and (0, 1, 0, 1, 1, 1). \hfill (5)

Ques 48.

$$\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}{i}\mathbf{)}$
iii) Deduce that every repetition code of odd length is perfect.

Ques 49.

$$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. Use the following table in 1.

Ques 50.

Ques 51.

Ques 52.

Factor x⁵ - 1 over $\mathbf{F}_2$ . Give the generator polynomials of all cyclic codes of length five over $\mathbf{F}_2$ .

Ques 53.

Factor x⁸ - 1 over $\mathbf{F}_5$ . Give the generator polynomials of all cyclic codes of length eight over $\mathbf{F}_5$ .

Ques 54.

Ques 55.

If $f = u + iv$ is entire such that $u_x + v_y = 0$ in $\mathbb{C}$ then show that f has the form $f(z) = az + b$ where $a, \mathbf{b}$ are constants with $\operatorname{Re} a = 0$ .

Ques 56.

Consider $f(z) = z^2 - z$ and the closed circular region $R = \{z : |z| \leq 1\}$ . Find points in R where |f(z)| has its maximum and minimum values.

Ques 57.

Find the points where the function $f(z) = \frac{\operatorname{Log}(z + 4)}{z^2 + i}$ is not analytic.

Ques 58.

) Evaluate the following integrals:
i) $I = \int_{0}^{2\pi} f(e^{i\theta}) \cos^2(\theta/2) \, d\theta$ .

ii) $I = \int_{0}^{2\pi} f(e^{i\theta}) \sin^2 \theta/2 \, d\theta$ .

Ques 59.

Find the image of the circle $|z| = r (r \neq 1)$ under the mapping $w = f(z) = \frac{z - i}{z + i}$ . What happens when $r = 1$ ?

Ques 60.

Ques 61.

Find the constant c such that $f(z) = \frac{1}{z^n + z^{n-1} + \dots + z^2 + z - n} + \frac{c}{z - 1}$ can be extended to be analytic at $z = 1$ , when $n \in \mathbb{N}$ is fixed.

Ques 62.

Find all the singularities of the function $f(z) = \exp \left( \frac{z}{\sin z} \right)$ .

Ques 63.

Evaluate $\oint_c \frac{dz}{z^2 + 1}$ where c is the circle $|z| = 4$ .

Ques 64.

Find the maximum modulus of $f(z) = 2z + 5i$ on the closed circular region defined by $|z| \leq 2$ .

Ques 65.

Evaluate $\int_{c} \frac{z^3 + 3}{z(z - i)^2} \, dz$ , where c is the eight like figure shown in Fig. 1.

Ques 66.

Find the radius of convergence of the following series.
i) $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k!} (z - 1 - i)^k$ $\quad \quad$ ii) $\sum_{k=1}^{\infty}\left(\frac{6k+1}{2k+5}\right)^k(z-2i)^k$

Ques 67.

Expand $f(z) = \frac{1}{(z-1)^2 (z-3)}$ in a Laurent series valid for

i) 0 < |z - 1| < 2 and $\quad \quad \quad \quad$ ii) 0 < |z - 3| < 2.

Ques 68.

Find the zeros and singularities of the function $f(z) = \frac{z}{4 \cos^2 z - 1}$ in $|z| \leq 1$ . Also find the residue at the poles

Ques 69.

Prove that the linear fractional transformation $\phi(z) = \frac{2z - 1}{2 - z}$ maps the circle $c : |z| = 1$ into itself. Also prove

that f(z) is conformal in $\bar{D} = \{z : |z| \leq 1\}$ .

Ques 70.

Find the image of the semi-infinite strip x > 0, 0 < y < 1 when $w = i/z$ . Sketch the strip and its image.

Ques 71.

Ques 72.

Evaluate the following integrals

a) $\int_{0}^{\infty} \frac{x^2 + 2}{(x^2 + 1)(x^2 + 4)} \, dx$ .
b) $\int_{-\infty}^{\infty} \frac{\sin^2 2x}{1 + x^2} \, dx$ .

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