GEORGIA INSTITUTE OF TECHNOLOGY
School of Electrical and Computer Engineering
ECE 6606
Coding Theory
Assigned: Tuesday, June 9, 2009
Due: Friday, June 19, 2009
Problem Set #3
Problem 1-5: Solve the questions 23,24, 28, 30 and 33 from Chapter 2 of the textbook.
Problem 6-10: Solve the questions 4 (only parts a, b) ,7, 9, 12 (only parts a, b, e) and 14
(only part a) from Chapter 3 of the textbook.
Problem 11: Prove that the following statements are true or false:
(a) GF (32) is a subfield of GF (256).
(b) GF (16) is a subfield of GF (256).
Problem 12: Prove that if Fis a finite field of order q, then every a∈Fsatisfies aq=a.
Problem 13: Let f(x) be an mth degree polynomial in GF (2)[x]. The reciprocal g(x)
of f(x) is defined as g(x) = xmf(1/x). Prove that if f(x) is primitive then g(x) is
primitive as well.
Problem 14: Let f(x)∈Fq[x] be an irreducible polynomial over a finite filed Fq. Let α
be a root of f(x) in an extension field of Fq. Then for a polynomial h(x)∈Fq[x] we
have h(α) = 0 if and only if f(x) divides h(x).
1
