Explain what a simple communication channel is, with the help of a diagram.
Define the dual code-of a code. Find the dual code of a code C generated by the matrix
G <img src='./qimages/12508-1b.jpg'> over F2. Also find the generator matrix of the dual code of C.
Find the 2-cyclotomic cosets modulo 31.

2.(a) Let r be an integer with 0 r m. Let denote the r^th order RM code of length 2^m. Prove that m)l and m)l for m.
Generate a field with 16 elements with the polynomial x^4 x 1.

3.(a) Find the generating idempotent for a cyclic code C of length 7 over F2 with generator polynomial 1 x x^3.
Let C be a cyclic code in Rn and let be a non-zero idempotent in C. Prove that iff is the unity of C. 'There is a unique self-dual code of length 7 over F2.' Is this statement true Give reasons for you answer.

4.(a) Give an example of a BCH code, with justification.
Define a low density parity check code, and give an example. Find the convolutional code for the message 1011. The convolutional encoder is given below <img src='./qimages/12508-4c.jpg'>

5.(a) Find the weight distribution and weight enumerator of the code C generated by the matrix <img src='./qimages/12508-5a.jpg'>
Let p be an odd prime and let a be in Zp with a f 0 (mod p). If a is a square, then prove that the multiplicative order of a is a divisor Give the criteria for the existence of duadic codes of length n over F2 and F3. Also, find n 10, such that duadic codes of length n exist over F2, F3. Justify your answer.

6.(a) Show that the Z4-linear codes with generator matrices <img src='./qimages/12508-6a.jpg'>

are monomially equivalent.
Let C be a self-orthogonal Z4-linear code with c E C. Prove that

wtr,(c) 0(mod
wtE(c) 0 (mod
Let C be the narrow-sense binary BCH code with designed distance 0 which has defining set

T 12}.

Using the primitive 15th root of unity a^4 a the generator polynomial of C is 1 x^4 x^6 x^7 x^8.
If 1 x x^5 x^6 x^9 x^10 is received, find the transmitted code word.