1) (a) Give an example (with justification) of a linear code which can detect:

i) up to 1 error in any codeword.

ii) up to 2 errors in any codeword.

iii) up to 3 errors in any codeword.

(b) Give an example (with justification) of a linear code which can correct:

i) up to 1 error in any codeword.

ii) up to 2 errors in any codeword.

iii) up to 3 errors in any codeword.

2) (a) Recall that F = F₃[x]/ < x² + 1 > is a finite field of order 3². Find a primitive element and write out the cyclotomic cosets along with their minimal polynomials.

(b) Use F to construct a BCH code C of length n = 8 and design distance 5. Give a generator polynomial and generator matrix for C. What is the dimension of C?

(c) Give a check polynomial for C and find a parity check matrix H.

(d) What is the minimum distance of C? Justify your answer. How many errors can C correct in any codeword?

(e) Is C a perfect code? If so, prove it. If not, give an example of a received vector which C cannot correct.

3) (a) Recall that F = F₂[x]/ < x³ + x² + 1 > is a finite field of order 2³. Find a primitive element and write out the cyclotomic cosets along with their minimal polynomials.

(b) Use F to construct a BCH code C of length n = 7 and design distance 5. Give a generator polynomial and generator matrix for C. What is the dimension of C?

(c) Give a check polynomial for C and find a parity check matrix H.

(d) What is the minimum distance of C? Justify your answer. How many errors can C correct in any codeword?

(e) Is C a perfect code? If so, prove it. If not, give an example of a received vector which C cannot correct.

4) Give an example of a BCH code which is equivalent to a Hamming code. Justify your answer.

5.) (a) Is every binary Hamming code equivalent to a BCH code? Prove it, or provide a counterexample.

(b) Is every binary BCH code equivalent to a Hamming code? Prove it, or provide a counterexample.

(c) Is every perfect binary BCH code equivalent to a Hamming code? Prove it, or provide a counterexample.