Solve your IGNOU Doubts

Question:

Let C₁ and C₂ be cyclic codes over $\mathbb{F}_{q}$ , with generator polynomials g₁ (x) and g₂(x), respectively. Prove that C₁ $\subseteq$ C₂ if and only if g₂(x) | g¹(x).

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Question:

Let a be a root of x² + 1 = 0 in $\mathbb{F}_{9}$ .

a) Check whether α is a primitive element of $\mathbb{F}_{9}$ . If it is not a primitive element in $\mathbb{F}_{9}$ find a primitive element γ in $\mathbb{F}_{9}$ in terms of a.

b) Make a table similiar to Table 5.1 on page 184 for $\mathbb{F}_{9}$ with the primitive element y

c) Factorise x⁸ over $\mathbb{F}_{3}$ .

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

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Question:

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 ∈ $\mathbb{N}$ .

ii) Show that

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

(Hint: Start with the relation

$2^{2m+1}=\sum_{i=0}^{2m+1}\binom{2m+1}{i})$

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

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Question:

Show that, if x ∈ $\mathbb{F}_2^n$ ,

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.

(Hint: Observe that x2 = 1 for all x ≠0 € $\mathbb{F}_{3}$ )

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Question:

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

(Hint: Why is it enough to prove that sum of vectors of even weight in $\mathbb{F}_2^n$ is a vector of even weight?)

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Question:

If $x,y\in\mathbb{F}_2^n$ , show that

wt(x+y)=wt(x)+wt (y) -2wt (xy)

where $x\cap y$ is the vector in $\mathbb{F}_2^n$ which has 1s precisely at those positions where x and y have 1s.

(Hint: Let x = (x₁,x₂,...,x_n) and y = (y₁, y₂,..., y_n). Suppose

$n_1=|\{i\mid x_i=y_i=1\}|,\quad n_2=|\{i\mid x_i=1,y_i=0\}|,\quad n_3=|\{i\mid x_i=0,y_i=1\}|$

Observe that wt (x) = n₁ + n₂ and wt (y) = n₁+n₃.)

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Question:

Let C1 and C2 be two binary codes with generator matrices

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

respectively.

a) Find the minimum distance of both the codes.

b) Find the generator matrix of the 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 .

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Question:

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

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Question:

Find the minimum distance for each of the codes.

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Question:

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.

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Question:

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.

There is no 3-cyclotomic coset modulo 121 of size 25.

iv) There is no duadic code of length 15 over F₂.

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

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Question:

A library wants to improve its service facilities in terms of the waiting time of its borrowers. The library has two counters at present and borrowers arrive according to Poisson distribution with arrival rate 1 every 6 minutes and service time follows exponential distribution with a mean of 10 minutes. The library has relaxed its membership rules and a substantial increase in the number of borrowers is expected. Find the number of additional counters to be provided if the arrival rate is expected to be twice the present value and the average waiting time of the borrower must be limited to half the present value.

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Question:

Ships arrive at a port at the rate of one in every 4 hours with exponential distribution of inter- arrival times. The time a ship occupies a berth for unloading has exponential distribution with an average of 10 hours. If the average delay of ships waiting for berths is to be kept below 14 hours, how many berths should be provided at the port?

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Question:

For a multi-channel queuing system with λ = 12/hours, u=5/hours, c=3, p_o, = 0.056, calculate

i) The average time a customer is in the system

ii) The average number of customers in the system

iii) Whether any time would be saved for customers if the three-channel system with the service rate of 5 per hour is replaced by a single-channel system with an average service rate of 15 per hour?

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Question:

A company has three factories that supply to three markets. The transportation costs from each factory to each market are given in the table. Capacities of the factories and market requirements are shown. Find the minimum transportation cost.

	M₁	M₂	M₃	a_i
F₁	2	1	3	20
F₂	1	2	3	30
F₃	2	1	2	10
b_j	10	10	20	40/60

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Question:

The owner of a readymade garments store sells two types of shirts: Zee-shirts and Button-down shirts. He makes a profit of Rs. 5 and Rs. 10 per shirt on Zee-shirts and Button-down shirt, respectively. He has two tailors, A and B at his disposal to stitch the shirts. Tailors A and B can devote at the most 7 hours and 15 hours per day, respectively. Both these shirts are to be stitched by both the tailors. Tailors A and B spend 2 hours and 5 hours, respectively in stitching one Zee-shirts, and 4 hours and 3 hours, respectively in stitching a Button-down shirt. How many shirts of both types should be stitched in order to maximize daily profit?

a) Formulate and solve this problem as an LP problem.

b) If the optimal solution is not integer-valued, use Gomory technique to derive the optimal integer solution.

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Question:

Do the stability analysis of the following model formulated to study the effect of toxicant on one competing species where the environment toxicant concentration is being taken to change w.r.t. time.

$\begin{align*}\frac{dN_1}{dt}&=r_1 N_1-\alpha_1 N_1 N_2-d_1 C_0 N_1\\\frac{dN_2}{dt}&=r_2 N_2-\alpha_2 N_1 N_2\\\frac{dC_0}{dt}&=k_1 P-g_1 C_0-m_1 C_0\\\frac{dP}{dt}&=Q-hP-kPN_1+gC_0N_2\end{align*}$

along with the initial conditions.

$N_1(0)=N_{10},\quad N_2(0)=N_{20},\quad C(0)=0,\quad P(0)=P_0>0$

Here,

N₁ (t)=Density of prey population

N₂(t)=Density of predator population

C_o(t) = Concentration of the toxicant in the individual of the prey population

P=Constant environmental toxicant concentration.

a,, a, are the predation rates, r,r, are the growth rates or birth rates, d, is the death rate due to C, m, is the depuration rate, Q, h, k, g are positive rate constants.

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Question:

The population dynamics of a species is governed by the discrete model

$x_{n+1}=x_n\exp\left[r\left(1-\frac{x_n}{K}\right)\right]$

where r and k are positive constants. Determine the steady states and discuss the stability of the model. Find the value of r at which first bifurcation occurs. Describe qualitatively the behaviors of the population for $r=2+\varepsilon,\quad\text{where}0<\varepsilon\ll 1.$ . Since a species becomes extinct if x_n ≤1 1 for any n> 1, show using iterations, that irrespective of the size of r> 1 the species could become extinct if the carrying capacity k $k<r\exp[1+e^{r-1}-2r]$

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Question:

Do the stability analysis of the following competing species system of equations with diffusion and advection

$\begin{align*}\frac{\partial N_1}{\partial t}&=a_1 N_1-b_1 N_1 N_2+D_1\frac{\partial^2 N_1}{\partial x^2}-V_1\frac{\partial N_1}{\partial x}\\\frac{\partial N_2}{\partial t}&=-d_1 N_2+c_1 N_1 N_2+D_2\frac{\partial^2 N_2}{\partial x^2}-V_2\frac{\partial N_2}{\partial x},\quad 0\leq x\leq L\end{align*}$

where Vand V, are advection velocities in x direction of the two populations with densities N, and N, respectively. a, is the growth rate, b, is the predation rate, d, is the death rate, C, is the conversion rate. D, and D, are diffusion coefficients. The initial and boundary conditions are:

$N_i(x,0)=f_i(x)>0,\quad 0\leq x\leq L,\quad i=1,2,\dots$

$N_i=\overline{N}\text{at}x=0\text{and}x=L\quad\forall t,\quad i=1,2,\dots$

where $\overline{N_{i}$ are the equilibrium solutions of the given system of equations. Interpret the solution obtained and also write the limitations of the model.