State whether the following statements are True or False. Give short proof or a counter example in support of your answer.
 a) The length of chromosomes to determine the maximum value of the set:
 S ={X 0 ≤x ≤ 4096}is 12.
 b) In the Hopfield network, the neurons belonging to the same layer receive input from
the neurons of the previous layer and send their value only to the neurons of the next
layer.
 c) In a single layer neural network, if x ,0
ω > then the output is −1.
 d) The fuzzy relation (R) given below, is an equivalence relation.

e) The Self Organizing Map (SOM) is a supervised learning technique.

Find the convolutional code for the message 11011. The convolutional encoder is given in Fig. 1.

Draw the Tanner graph of the code C with parity check matrix

Let C = {0000,1113,2222,3331,0202,1313,2020,3131,0022,1131,2200,3313,0220,1333,2002,3111} be the Z4-linear code. Find the Gray image of C.

Find the generating idempotents of duadic codes of length n = 23 over F3. (Hint: Mimic example 6.1.7.)

Let C be the [5, 2] ternary code generated by

Find the weight enumerator WC (x, y) of C

Let C be the ternary [8,3] narrow-sense BCH code of designed distance δ = 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.

If C1 has some odd weight codewords, what is the relationship between 1+x and g1(x)?

If C1 has only even weight codewords, what is the relationship between (1+x) and g1(x)?

Prove that C is the vector space of all vectors in F n 2 with even weight.

Let w be subspace of F n 2 containing all the vectors of even weight. Prove that W has dimension n−1. (Hint: Consider the map w: F n 2 → F2 given by w((a1,a2,...,an)) = a1 +a2 +···+an.)

What is the dimension of C ?

Over F2, (1+x) | (x n −1). Let C be the binary cyclic code (1+x) of length n. Let C1 be any binary cyclic code of length n with generator polynomial g1(x).

Let C1 and C2 be cyclic codes over Fq with generator polynomials g1(x) and g2(x), respectively. Prove that C1 ⊆ C2 if and only if g2(x) | g1(x)

Prove that
l¹⊂
l² If: T :(l² ||.||)(l¹,||.||₂)(l^1,||.||₂) is a compact operator, show that: T:,||.||||.||₂ is also compact.

Find a bounded linear functional f on 3 l such that 3 e(f ) 3 = and || f|| = .3

Let Y A : X0 ⊆ X → be a closed operator where X and Y are Banach spaces. Define x x Ax , x X . A = + ∈ 0 Then show that the norm A ⋅ is complete.

Define 3 3 A : C → C by A z( , z , z ) iz( , e z , z ) 2 3

Check whether the identity map on an infinite dimensional space is compact.

Let X and Y be Banach spaces and :F X → Y be a linear map which is continuous and open. Will F always be closed? Will F be always surjective? Give reasons for your answer.