Factor x⁵ - 1 over . Give the generator polynomials of all cyclic codes of length five over . 

Let C be a cyclic code of length eight over  with generator polynomial . Find the generator matrix of C, the generator polynomial of C and the parity check matrix of C. 

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

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. 

) Let C be the ternary [8,3] narrow-sense BCH code of designed distance , which has defining set . Use the primitive root 8th root of unity you chose in 4a) to avoid recomputing the the table of powers. If

is the generator polynomial of C and

is the received word, find the transmitted codeword. Use the following table in 1. 

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 . \hfill (2)
ii) Show that

(Hint: Start with the relation

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

 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)

) For the binary, (6,3) linear code C with generator matrix 
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)

Let  and  be two binary codes with generator matrices

respectively.

i) Find the minimum distance of both the codes.

Table 1: Table for F₁₆.

ii) Find the generator matrix of the code

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

Let  and  be two binary codes with generator matrices

respectively.

i) Find the minimum distance of both the codes.

Table 1: Table for F₁₆.

ii) Find the generator matrix of the code

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

Find the parity check matrix of the code .

 Find the parity check matrix of the code . Decode the following vectors
i) 
ii) 
iii) 
iv) 

For a linear code C with generator matrix

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 ? 

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

 Find the minimum distance for each of the codes. 

Which of the following binary codes are linear?

i)   

ii)   

Justify your answer. 

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 .
v) There is no LDPC code with parameters ,  and .

 Find the optimum solution to the following transportation problem:

 A sales manager wishes to assign four sales territories to four salespersons. The salespersons differ in their ability and skills and consequently the sales expected in each territory are different. The estimates of sales per month for each sales-person in different territories are given below:

Find the optimal assignment of the four salespersons to the four different territories and the maximum monthly sales.

Obtain all the basic solutions to the following system of linear equations:

Which of the solutions are feasible? Justify.

At present a company is purchasing an item ‘x’ from outside suppliers. The assumption of unit is 10000 units/year. The cost of the item is ₹ 5 per unit and the ordering cost is estimated to be ₹ 100 per order. The cost of carrying inventory is 25%. If the consumption rate is uniform, determine the economic order quantity.