University of Illinois Fall 2005
ECE 556/CS 577/MATH 579: Problem Set 10
Due: Tuesday November 15, 8:30 a.m.
Reading: Blahut, Algebraic Codes for Data Transmission, Chapters 9, 10, and 11
Reminder: The Second MidSemester Exam is Thursday November 17, 8:30 a.m. – 9:50 a.m. in 169 EL.
This Problem Set contains three problems
1. Consider all (2,1) systematic convolutional codes with polynomial generator matrices [1, g(x)]
where deg g(x)<3. What is the largest free distance achieved by such codes? How does
it compare to the free distance of the nonsystematic ad nasueam (2,1) code with generator
matrix [1 + x+x2,1 + x2]?
2. Blahut Problem 9.3 on page 310. Note that there is a typographical error in the problem:
the rate of the punctured code is 2/3, not 3/4.
3. Consider a (3, 2) convolutional code with polynomial generator matrix
1 + x1 1 + x
x1 + x0
(a) Draw a neat sketch of an encoder for this code.
(b) Draw the state transition diagram for the encoder and use it to find the path enumerator
A(x) for this code. What is the free distance of this code?
4. Let E(m) denote the number of binary polynomials of degree exactly m. Let L(m) de-
note the number of linear binary convolutional codes with polynomial generator matrix
[g(0)(x), g(1) (x), . . . , g(n−1)(x)] where all the polynomials are nonzero and of degree at most
m, with at least one of them having degree exactly m, that is,
max{deg g(0)(x),deg g(1) (x), . . . , deg g(n−1)(x)}=m.
Let N(m) denote the number of noncatastrophic encodings which satisfy the above conditions
and for which gcd(g(0)(x), g(1) (x), . . . , g(n−1)(x)) = 1. (Note that noncatastrophic encodings
for which gcd(g(0)(x), g(1) (x), . . . , g(n−1)(x)) = xiare not included in N(m)).
(a) Find an expression for E(m) as a function of m.
(b) Show that
L(m) =
m
X
i=0
N(i)E(m−i) = (2m+1 −1)n−(2m−1)n.
(c) For m > 0, show that
m
X
i=0
N(i) = (2m+1 −1)n−2·(2m−1)n.
Thus, almost all (n, 1) codes are noncatastrophic.
