GEORGIA INSTITUTE OF TECHNOLOGY
School of Electrical and Computer Engineering
ECE 6606
Coding Theory
Assigned: Monday, June 15, 2009
Due: Friday, July 3, 2009
Problem Set #4
Problem 1: (a) Does there exist a binary cyclic code of length 14? Justify your answer.
(b) For what even lengths does there exist a cyclic code? Justify your answer.
Problem 2: (a) Give two different construction of GF(8) (using two different primitive
polynomials).
(b) For both construction in part (a), find the minimal polynomials associated with
the conjugacy. Verify that the minimal polynomials of each construction are the
same.
(c) Suppose a cyclic code C1is generated using the primitive polynomial p1(x) used
in the first construction of GF (8). Let another cyclic code C2be generated using
the second primitive polynomial p2(x). Are the codewords in C1identical to the
codewords in C2? Explain your answer.
Problem 3: Let Cbe a binary cyclic code of length nwith generator polynomial g(x) =
x8+x2+x+ 1.
(a) Show that a code with this g(x) can detect all odd weight error patterns.
(b) Show that g(x) can be factored into the product of exactly two binary polynomials.
Give the two polynomials and explain why there are only two.
(c) What is the code length n?
Problem 4: Consider the general (2m−1,2m−1−m) binary cyclic Hamming code. The
code is defined as: (c0, . . . , cn−1) is a codeword if and only if c0+c1α+· · ·+cn−1αn−1=
0, where αis a primitive element in GF (2m), and n= 2m−1. Find a general
formula for the number of codewords of weight 3. [Hint: Codewords of weight 3 are
1
