University of Illinois Fall 2005
ECE 556/CS 577/MATH 579: Problem Set 6
Due: October 4, 8:30 a.m.
Reading: Blahut, Algebraic Codes for Data Transmission,
Chapters 4 and 5.
Reminder: Class will not meet on Thursday September 29. Instead, you are urged to
attend the 43rd Annual Allerton Conference on Communication, Control
and Computing, Allerton House, Monticello IL, September 28 - 30, 2005.
http://www.csl.uiuc.edu/allerton/
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. (a) Let C(1) and C(2) denote linear cyclic codes of length nwith generator polynomials
g(1)(x) and g(2) (x) respectively. Then the codes
C(3) =C(1) ∩ C(2) and C(4) ={a+b:a∈ C(1),b∈ C(2)}
are also linear cyclic codes. Find their generator polynomials.
(b) Let Aidenote the number of codewords of Hamming weight iin a linear binary
cyclic code. Show that i·Aiis a multiple of n. (Note: In Problem 1(c) of Problem
Set 3, you proved that, for any code, Pi·Aius a multiple of n. This result is
stronger . . . )
(c) Show that if the generator polynomial g(x) of a linear binary cyclic code is not
divisible by (x−1), then Ai=An−ifor 0 ≤i≤n.
3. (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.
4. 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).