IGNOU MMTE 5 SOLVED ASSIGNMENT

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MMTE 5 2025 - English

Course Code: MMTE-005

Assignment Code: MMTE-005/TMA/2025

Maximum Marks: 100

Note: In this assignment, the notations, symbols, definitions and conventions will be as in the prescribed book ‘Fundamentals of Error-Correcting Codes’ by Huffman and Pless. Also, ‘the book’ will always mean the prescribed book.

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 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.

a) Which of the following binary codes are linear?

i)  

ii)  

Justify your answer.

b) Find the minimum distance for each of the codes.

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

3) 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 .

4) a) If , show that

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

where is the vector in 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

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

b) 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  is a vector of even weight?)

c) Show that, if x ∈ ,

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 € )

d) 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 ∈ .

ii) Show that

(Hint: Start with the relation

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

5) Let a be a root of x² + 1 = 0 in .

a) Check whether α is a primitive element of . If it is not a primitive element in  find a primitive element γ in in terms of a.

b) Make a table similiar to Table 5.1 on page 184 for with the primitive element y

c) Factorise x⁸ over .

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

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

b) Over , (1+x) | (xⁿ - 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).

i) What is the dimension of ?

ii) Let w be subspace of containing all the vectors of even weight. Prove that W has dimension n - 1. (Hint: Consider the map w: → given by

Prove that & is the vector space of all vectors in with even weight. 

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

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

7) a) 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) = x⁵ - x⁴ + x³ + x² - 1

Figure 1: Encoder for convolutional code.

is the generator polynomial of C and

y(x) = x⁷ - x⁶ - x⁴ - x³

is the received word, find the transmitted codeword.

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

Find the weight enumerator W_C(x, y) of C.

c) Find the generating idempotents of duadic codes of length n = 23 over .

(Hint: Mimic example 6.1.7.)

be the Z₄-linear code. Find the Gray image of C.

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

c) Find the convolutional code for the message 11011. The convolutional encoder is given in

Fig. 1.

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