GEORGIA INSTITUTE OF TECHNOLOGY
School of Electrical and Computer Engineering
ECE 6606
Coding Theory
HWK #5, Due: Friday July 10
Problem 1: Solve the question 10 from Chapter 8 of the textbook.
Problem 2: Consider a binary (15,11) cyclic code with generator polynomial g(x) = x4+
x3+ 1.
(a) Find the error correction capability of this code.
(b) Is it a BCH code? Explain why.
(c) Is it a perfect code? Why?
Problem 3: This problem concerns cyclic BCH codes of length 15.
(a) Compute the exact dimension of the cyclic t-error correcting BCH code of length
15, for t= 1,2,...,7.
(b) For each of the codes in part (a), compute the generator polynomial, assuming
that the primitive root αsatisfies α4+α+ 1 = 0.
Problem 4: Let the vector ˆ
V= [ ˆ
V0,ˆ
V1, . . . , ˆ
Vn−1] be the n-point finite-field DFT of the
vector V= [V0, V1, . . . , Vn−1]. Prove that the DFT of the “phase-shifted” vector
V(s)(i.e., the vector V(s)= [V0, αsV1, α2sV2, . . . , α(n−1)sVn−1]) is equal to ˆ
V(s)=
[ˆ
Vs,ˆ
Vs+1, . . . , ˆ
V((s+n−1))n], where αis the element of order nin the field. Here the
notation ((b))nmeans: bmod n.
Problem 5 (optional to solve): Let F=GF (8), containing a primitive root αsatisfying
α3=α+ 1, and let V= [0,0, α3,0,0, α2,0]. Compute the 7-point finite-field DFT of
this vector.
Problem 6-7: Solve questions 2 (parts a, b), 5 (parts a,b) from Chapter 9 of the textbook.
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