Chapter 7: Error Correcting Codes
Part I
ECE 204
Information Theory and Coding
Jamshid Abouei, ECE 204 Information Theory and Coding, Yazd University, Iran, Spring 2011
Yazd University
2
Outline
Introduction
Parity Check Codes
Hamming Distance
Yazd
University
Jamshid Abouei, ECE 204 Information Theory and Coding, Yazd University, Iran, Spring 2011
Syndrome and Error Detection/Error Correction
Application of Group Theory in Parity Check Codes
Standard Array and Syndrome Decoding
3
Introduction
Yazd
University
Jamshid Abouei, ECE 204 Information Theory and Coding, Yazd University, Iran, Spring 2011
So far, we have restricted our attention to the
source coding
and the
capacity
.
We will focus on the theory and practice of
error correcting codes
(ECCs) in
communication systems corrupted with noise.
We limit our attention to binary
forward error correcting
(FEC) block codes.
Symbol alphabet consists of
0
and
1
,
Receiver can correct a transmission error without asking the sender for
more
information
or for a
retransmission
.
Transmissions consist of a sequence of
fixed length blocks
, called
codewords.
4
Introduction
Yazd
University
Jamshid Abouei, ECE 204 Information Theory and Coding, Yazd University, Iran, Spring 2011
In this chapter, we present two categories of the channel codes:
Error detecting codes,
According to the Shannon’s Theorem, “
there exist
”
codes that allow transmission at
any rate below channel capacity with an arbitrarily small probability of error.
Problem 1:
The proofs do not provide clue on “
how to construct”
the codes
;
Error correcting codes.
Problem 2:
In the Shannon’s Theorem, it is assumed that the codeword block
lengths are infinity in order to achieve
Problem 3:
In real communication systems, the codeword block lengths are
finite and we have an error in transmitting symbols.
5
Introduction
Yazd
University
Jamshid Abouei, ECE 204 Information Theory and Coding, Yazd University, Iran, Spring 2011
The main goal in this chapter:
Find a good encoding scheme to
detect
(in the first step) the errors and then
correct
them.
Assumptions:
The channel is assumed as
BSC
with parameter p,
Precise bounds on the error correcting ability of codes.
The
inputs
to the channel is chosen from a set of binary codewords of
length
The codewords occur with
equal probability
,
The
Ideal-Observer
(I.O.) decision scheme is used
6
Decision Rule
Yazd
University
Jamshid Abouei, ECE 204 Information Theory and Coding, Yazd University, Iran, Spring 2011
Since an error may be made in any digit, the
set of possible encoder output sequences
is the set of all
binary sequences of length
n
. Our first problem is to find the form
of the I.O. decision scheme for such a code.
Decision Rule:
The
channel output
sequences are
where we assume that the
channel output could be any
sequence due to the error in the channel.
