Define 'linear and give an example with justification.

Prove that a binary repetition code of odd length is a perfect code.

Let a Make a table that gives the powers of a as a linear combination of a and a^2. Use it to write as a
linear combination of a and a^2.

Compute the 3-cyclotomic cosets modulo 7.

Find the generator matrix and the parity check matrix for the binary cyclic code of length 7 with generator polynomial x 1).

Let C be a cyclic code over Fq with generating idempotent Prove that the generator polynomial of C is
computed in Fq[x].

Let C be a narrow-sense binary BCH code of designed distance 0 which has the defining set T 12}. Using the primitive 15th root of unity a,where a^4 1 the generator polynomial
of C is 1 x^4 x^6 x^7 x^8. Suppose C is used to transmit a code word and x^6 x^10 x^12 is received. Find the transmitted code word.
You may use the following table:

<img src='./qimages/10182-3a.jpg'>

Construct the generating idempotents of the duadic codes of length 7 over F4.

Let C be the binary code generated by

<img src='./qimages/10182-4a.jpg'>

Find the weight distribution of C.

Find the weight distribution of C1 by using the Mac-Williams identity.

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

<img src='./qimages/10182-4b.jpg'>

List the 16 code words in C.

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

Let C be the convolutional code with generator matrix 1 D]. Prove that the code C has free distance 3.

Explain the Tanner graph of a code.

Let C be the binary code with parity-check matrix.

<img src='./qimages/10182-5b.jpg'>

Give the Tanner graph of this code.

6. Which of the following statements are True, and which are False? Justify your answers.

l(mod 10).

If C is an k]-code with parity-check matrix then any two words y in C have the same syndrome only if x=y.

If x and y are two code words in an LDPC code, with the minimum distance between them being less than then x and y will differ in only one component.

The dimension of a code C is the same as the dimension of the dual of C.

The minimum distance of any code is its error-coding capability.