University of California San Diego
ECE 259A:
Problem Set #2
1
. Recall that the syndrome
s
is a linear function of the errorpattern
e
. A
linear decoder
finds its
estimate
e
of the errorpattern as a linear function of
s
. That is
e
L
s
, where
L
satisfies
L
s
s
L
s
L
s
. Show that a linear decoder for a binary linear code can correct at most
n
k
of the
n
single erorrs (and, hence, it is not a particularly useful decoder).
2
. Another proof of the GilbertVarshamov bound (due to Gilbert):
(a) Given an
n
, 2
k
,
d
binary code
– not necessarily linear – show that for any integer
r
, the
average number of codewords in a Hamming sphere of radius
r
is given by
r
def
∑
x
n
2
x
,
r
2
n
V
n
,
r
2
n
k
where
V
n
,
r
∑
r
i
0
n
i
is the volume of the Hamming sphere of radius
r
.
(b) Use the above relation with
r
d
to show that there exist binary codes of rate
R
and mini
mum distance
d
, which satisfy
R
1
1
n
log
2
V
n
,
d
.
3
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Prof.AlexanderVardy
 Coding theory, Hamming Code, Linear code, binary linear code, following operation

Click to edit the document details