University of Illinois Fall 2004
ECE 556/CS 577/MATH 579: Problem Set 6
Due: Thursday October 7, 8:30 a.m.
Reading: Blahut, Algebraic Codes for Data Transmission, Chapters 5 and 8
This Problem Set contains four problems
1. Consider a generator matrix G= [P|I] for a systematic (n, k) cyclic code with parity-
check polynomial h(x) = Pk
i=0 hixi. Show that the rightmost column of Pis
(−hk−1,−hk−2,...,−h1,−h0)T.
2. (Blahut, Problem 5.8 with embellishments) Suppose that a binary cyclic code has the
property that whenever c(x) is a codeword, so is xn−1c(x−1) (which is just the codeword
with the code bits in reverse order.) Such a code is called a reversible code.
(a) Show that g(x) = xn−kg(x−1).
(b) What is the corresponding result for nonbinary codes?
(c) Prove that g0=±1.
(d) Show that if there is an integer msuch that qm≡ −1 mod n, then every cyclic
code of length nover GF(q) is reversible.
3. Consider a linear binary cyclic code of length n= 2m−1.
(a) Show that if c(x) is a codeword, then so is [c(x)]2mod (xn−1).
(b) Let gcd(n, L) = 1. A decimation by Loperation on a sequence ζof period n
results in another sequence ˆ
ζof period nwhere ˆ
ζi=ζLi =ζLi mod n. Regarding
c(x) as one period of a periodic sequence, show that the codeword corresponding
to decimation by 2m−1is [c(x)]2mod (xn−1), the codeword you found in part
(a).
4. Blahut Problem 5.17 on page 130.
