Cooperatives

Water Resources

Rural Urban Linkages in India

Describe major institutions of Indian rural society.

Explain major approaches of rural development.

Discuss in brief about agrarian movements in Pre-independent India.

Discuss the impact of emerging knowledge on the behaviour of rural society.

Describe the concept, aims and objectives of rural development.

What do you mean by rural society? Describe important characteristics of Indian rural society.

Let α be a root of x² +1 = 0 in F9.
a) Check whether α is a primitive element of F9. If it is not a primitive element in F9 find a
primitive element γ in F9 in terms of α. (4)
b) Make a table similiar to Table 5.1 on page 184 for F9 with the primitive element γ
c) Factorise x
8 −1 over F3. (6)
d) Find all the possible generator polynomials of a [8,6] cyclic code.

Deduce that every repetiition code of odd length is perfect.

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 ∈ N.

ii) Show that

(Hint: Start with the relation

2 ^2m+1

Show that, if x ∈ Fⁿ₃
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. (3)
(Hint: Observe that x
2 = 1 for all x 6= 0 ∈ F3)
 

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. (3) ( Hint: Why is it enough to prove that sum of vectors of even weight in F n 2 is a vector of even weight? )

If x, y ∈ F
n
2
, show that
wt(x+y) = wt(x) +wt(y)−2wt(x∩y)
where x∩y is the vector in F
n
2 which has 1s precisely at those positions where x and y
have 1s. (2)
(Hint: Let x = (x1, x2,..., xn) and y = (y1, y2,..., yn). Suppose
n1 = |{i | xi = yi = 1}|,n2 = |{i | xi = 1, yi = 0}|,n3 = |{i | xi = 0, yi = 1}|
Observe that wt(x) = n1 +n2 and wt(y) = n1 +n3. )

Let C1 and C2 be two binary codes with generator matrices

,

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 .

Which of the following binary codes are linear?
i) C = {(0,0,0,0),(1,0,1,0),(0,1,1,0),(1,1,1,0)}
ii) C = {(0,0,0),(1,1,0),(1,0,1),(0,1,1)}
Justify your answer. (3)
b) Find the minimum distance for each of the codes. (4)
c) For each of the linear codes, find the degree, a generator matrix and a parity check matrix. (3)

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 F2.
v) There is no LDPC code with parameters n = 16, c = 3 and r = 5.

Compute the DFT of the vector (−1, 3, 1, −1).

Let {(−1, −5), (0, −4), (1, −1)} and {(−1, 14), (0, 7), (1, 4)} be the point-value representations of two polynomials See Answer →