Answer the following questions explaining each step in each part.
Problem 1: Construction of a finite field.
(a) Check that the polynomial x3 + x + 1 is irreducible and primitive over F2.
(b) Let α be the class of x in F2/(x3 + x + 1). Give an exponential-vectorial table.
Problem 2: Define a primitive RS code over the finite field in the previous part which can correct two errors.
(a) What minimum distance must we take? Give its length and dimension.
(b) Give its generator polynomial.
(c) Obtain a generator matrix based on the generator polynomial.
(d) Give a parity check matrix H whose first row contains all the powers of α.
(e) What is the result of multiplying the first row of G by the first row of H? Explain all the steps.
(f) What is G.HT?
Problem 3: Consider the same RS code as in the previous problem.
(a) Encode systematically, using the generator polynomial, the first block of information of the bit stream
1111100011111000111111111111100010101010101010101010100011111000
Give your result also as a bit stream.
(b) What is the syndrome of the encoded word? Explain all steps you use to obtain it.
Problem 4: Consider the same RS code as in the previous problems.
(a) We receive the bit stream 001100000000000010100. What is the corresponding symbol stream?
(b) Compute all the syndromes of the received word.
(c) How many symbol errors does the received word have?
(d) What is the error locator polynomial?
(e) Give all the error positions.
(f) Give all error values.
(g) Give the corrected bit stream.
(h) How many bit errors did the received word have?
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