GEORGIA INSTITUTE OF TECHNOLOGY
School of Electrical and Computer Engineering
ECE 6606
Coding Theory
HWK #2, Due: Friday June 5
Problem 1-4: Solve the questions 2,11,12,21 from Chapter 4 of the textbook.
Problem 5: Prove that the Hamming distance dH(x, y) satisfies the triangle inequality,
i.e., for any three vectors x,y, and z, we have:
dH(x, y)≤dH(x, z) + dH(z , y)
Problem 6: Hamming bound states that for a binary code {c1,...,cM}of length n, which
is capable of correcting terrors, we have
M≤2n
V2(n, t)
where V2(n, t) denotes the discrete “Volume” of an n-dimensional Hamming sphere of
radius t.
(a) Compute the Hamming bound on Mfor the following pairs of (n, t):(3,1), (5,2),
(7,3), (9,4), ....
(b) Do you see a pattern? If so, state the generalization and prove that it is true.
Problem 7: Let PS,PE, and PFbe the “probability of decoder success”, “probability
of decoder error”, and “probability of decoder failure”, respectively. In this problem
you are asked to actually compute these three probabilities, as a function of p, for the
following two codes on a binary symmetric channel with crossover probability p.
(a) The (7,4) Hamming code, whose decoder is designed to correct all error patterns
of weight 0 and 1, and no others.
(b) The expurgated (7,3) Hamming code, whose decoder is designed to correct all
error patterns of weight 0 and 1, and no others. Note that the parity-check
matrix for this code is:
H=
0111100
1011010
1101001
1111111
1
