Prove that, in a linear code, the minimum distance is the same as the minimum weight.

State and prove the sphere packing bound.

Find all the primitive elements in F11.

Find all the code-words of the code with generator matrix

0 0 1 1
0 1 0 0 1
0 0 1 1

How many errors can detect How many can it correct

Construct a field with 8 elements.

Let C be narrow-sense binary BCH code of designed distance 0 which has defining set

T 12}.

Let a^4 where a is primitive 15th root of unity, and generator polynomial of C is

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.

Define cyclic code and give an example.

Prove that a BCH code of designed distance o has minimum weight at least o.

Let C be a cyclic code over Fq with generating idempotent Prove that the generator polynomial of C is
=gcd x^n computed in Fq

Let C be any self-dual 12, ternary code. Prove that the weight enumerator of C is

WC y^12 264x^6y^6 440x^9y^3 24x^12

Construct the generating idempotents of the duadic codes of length 11 over F3.

Let C be the Z4 -linear code of length 3 with generator matrix G is

0 1
0 1

List the 16 code-words in C.

List the 16 code-words in the Gray image of C.

Define a convolutional code and give an example.

If a polynomial generator matrix of an convolutional code C is basic and reduced, prove that G is canonical.

Write the Message Passing Decoding Algorithm.