Computer Networking
Error Control Coding
Dr Sandra I. Woolley
An Introduction to Error Control Coding
Data transmission and channel
errors
Introduction to error control
Parity
Hamming codes
–
example of a simple linear block
code
Interleaving and product codes
–
(+ demonstration).
IP checksums (Polynomial/Cyclic
Redundancy Codes)
We will work through examples in
class.
http://www.thinkgeek.com/interests/exclusives/7a5c/images/1293/
3
Introduction
Error control coding involves the addition of redundancy to
enable error detection and/or correction.
There are two basic approaches to error control
–
Automatic retransmission request (ARQ)
–
Forward error correction (FEC)
More generally, when errors are detected data can be resent,
concealed or corrected.
4
Data Transmission
Compress
Data
Encrypt
Error control
Line code
Channel
errors
Remove redundancy
for efficient
communication
Encrypt message
for added security
Add systematic
redundancy to protect
data against channel
errors
Encode signal to suit
communication
channel
characteristics
..011
101
10..
..01101010..
5
Designing Error Protection Systems
“Know your enemy”
–
Understand the channel burst length distribution and gap
length distribution, i.e., how large and how frequent are the
errors.
Trade

off the correction of data with the addition of redundancy
–
System cost/delay
Should errors be detected and corrected?
–
Detect and request retransmission? Detect and conceal?
How important is the data.
–
Is all data equally important? Should some data elements be
protected more than others?
Is accepting bad data as good worse than rejecting good data as
bad?
6
Channel Errors
Errors can occur singly or in bursts.
Most channel errors can be approximated by simple state models.
The BER (Byte error rate) is a simple measure of channel quality.
In the Good state there are no errors.
Bad Type 1 and Bad Type 2 represent two types of error events;
non

burst
and
burst
.
p1 and p2 are the probabilities of starting the errored states.
q1 and q2 are the probabilities of ending the errored states.
Good
Bad
Type 2
1

q2
p2
Bad
Type 1
1

q1
p1
1

p1

p2
The modified
Gilbert model
q1
q2
7
Channel Errors
Burst length distributions provide important information about
channel error activity.
Good
Bad
Type 2
1

q2
p2
Bad
Type 1
1

q1
p1
1

p1

p2
increasing p1
decreasing q1
increasing p2
decreasing q2
error length (log scale)
Probability (log scale)
q1
q2
The Simplest Error Detection
–
Parity Bit
Odd or even parity requires
that the sum total of
codewords be odd or even,
respectively.
So for odd parity there will be
an odd number of ones and
for even parity there will be an
even number of ones.
For example, if we have data
bits 0011010 and we want
even parity. We need to add
a parity bit of 1 to have an
even number on ones. So, if
we append our parity bit to the
end, we have a codeword of
0011010
1
(where
1
is our
even parity bit).
7 bits of data
(number of 1s)
8 bits including parity
even
odd
0000000 (0)
0
0000000
1
0000000
1010001 (3)
1
1010001
0
1010001
1101001 (4)
0
1101001
1
1101001
1111111 (7)
1
1111111
0
1111111
Examples from Wikipedia
The parity bit is added to the front
http://en.wikipedia.org/wiki/Parity_bit
9
The Hamming (7,4) Code
An example of an (
n,k
) linear block code. Each codeword contains n bits,
k information bits and (n

k) check bits.
The Hamming (7,4) is a nice easy code but it is not very efficient (it has a
75% overhead!). It generates
codewords
with a ‘Hamming distance’ of 3,
i.e., all
codewords
differ in 3 locations. It can correct one bit in error and
detect two.
In the codeword
[C]= k1 k2 k3 k4 c1 c2 c3
,
k1

k4
are information bits and
c1

c3
are check bits (the ‘systematic
redundancy’). “+” here denotes binary addition (it is the same as XOR).
c1=k1+k2+k4
c2=k1+k3+k4
c3=k2+k3+k4
So the data
[X] = [0 1 1 0]
becomes the codeword
[0 1 1 0 (0+1+0) (0+1+0) (1+1+0)] = [0 1 1 0 1 1 0]
10
The Hamming (7,4) Code
If we insert an error at bit 2
[ x ]
(syndrome)
[0 1 1 0 1 1 0]
becomes
[0 0 1 0 1 1 0]
Note
c1
and
c3
are
wrong

these intersect
at
k2
hence
k2
is in error
c1=k1+k2+k4
c3=k2+k3+k4
c2=k1+k3+k4
k1
k2
k4
k3
0
0
1
1
0
1
0
11
The Hamming (7,4) Code
If we insert an error at bit 4
[ x ]
(
syndrome
)
[0 1 1 0 1 1 0]
becomes
[0 1 1 1 1 1 0]
Note
c1
,
c2
and
c3
are all wrong
hence
k4
is in error
c1=k1+k2+k4
c3=k2+k3+k4
c2=k1+k3+k4
k1
k2
k4
k3
0
0
1
1
1
1
1
12
Bi

Directional/Product Codes
Imagine we have 4 x 4 data bits
k
1
1

k
1
4, k
2
1

k
2
4, k
3
1

k
3
4, k
4
1

k
4
4
Arranging them horizontally we can add error protection in the
vertical direction as well.
k
1
1 k
1
2 k
1
3 k
1
4
c
1
1 c
1
2 c
1
3
k
2
1 k
2
2 k
2
3 k
2
4
c
2
1 c
2
2 c
2
3
k
3
1 k
3
2 k
3
3 k
3
4
c
3
1 c
3
2 c
3
3
k
4
1 k
4
2 k
4
3 k
4
4
c
4
1 c
4
2 c
4
3
d
1
1 d
1
2 d
1
3 d
1
4 f
1
1 f
1
2 f
1
3
d
2
1 d
2
2 d
2
3 d
2
4 f
2
1 f
2
2 f
2
3
d
3
1 d
3
2 d
3
3 d
3
4 f
3
1 f
3
2 f
3
3
13
Bi

Directional/Product Codes
In the class we observe animated demonstrations of these codes
at work using the product code demonstrator.
–
First we generate errors using the Modified Gilbert Error
Model. The model requires just 4 probability values to
describe the errors it generates.
–
Next the data was interleaved.
–
Then the data was iteratively corrected vertically and
horizontally.
The demonstration shows how, with more sophisticated error
control coding, we can significantly increase correction capacity
making even severely corrupted data correctable.
The complex and time

consuming nature of this method makes it
inappropriate for Internet protocols*. However, robust correction
is desirable in storage systems and these methods can be found
in the more sophisticated server room RAID

type** storage
systems.
*but important payload data could be protected in this way
**RAID
–
Redundant Array of Inexpensive Disks
14
Interleaving
Interleaving (systematically reordering) the protected data stream means that
errors are distributed, i.e., more correctable.
For example, consider the three (7,4) Hamming
codewords
below in transmission
on a channel with a small burst of 3
errored
bits (syndrome
x
x
x
).
k1 k2 k3 k4 c1 c2 c3
k1 k2 k3 k4 c1 c2 c3
k1 k2 k3 k4 c1 c2 c3
x
x
x
These 3 bits all fall in one codeword, so would be
uncorrectable
.
But with a three

way interleave:

k1
k1
k1
k2
k2
k2
k3
k3
k3
k4
k4
k4
c1
c1
c1
c2
c2
c2
c3
c3
c3
x
x
x
Assuming the same syndrome of 3 bits in error, if we
unscamble
the bits (shown
below) we can see now there is now just one bit in error in each codeword, and
so our data is correctable.
k1 k2 k3 k4 c1 c2 c3
k1 k2 k3 k4 c1 c2 c3
k1 k2 k3 k4 c1 c2 c3
x
x
x
15
More Sophisticated Codes
Sophisticated interleaving strategies are used in most advanced digital
recording systems.
CDs use product codes but with Reed

Solomon codes (not Hamming.)
They also use interleaving. DVDs and Blu

ray discs also use Reed
Solomon block codes.
Reed

Solomon (RS) codes work on groups (e.g., bytes) of inputs. For
bytes n<2
8
(codewords are a maximum of 255 bytes). There is only a
very small probability of ‘crypto

errors’, i.e., correcting good bytes by
mistake.
RS block codes can correct (n

k)/2 bytes and detect (n

k) bytes. For
example, RS(122,106) can correct (122

106)/2 = 8 bytes in error.
Other systems use layers of error correction. If a lower (simpler) layer
detects errors, the next (more powerful) layer is inspected. If errors are
still detected the final layer is interrogated. This method increases the
speed of decoding by only computing check bytes when errors are
suspected. DAT tape drives uses 3 layers of error control.
16
IP Checksum
The IP checksum is an IP header field. It is calculated using the
contents of the header.
It was designed for ease of implementation in software (rather
than its error

detection ability) because it has needed to be
recalculated (IPv4) at every router.
Consider L 16

bit words making up the information bits (IP
header).
Checksum,
b
L
, is given by :
b
L
=

x
So that adding the 16

bit words and the checksum (as below)
gives 0.
(Assessment does not require worked examples.
For interest only, this is a tutorial example of how the checksum is calculated with 1’s complement arithmetic is
http://www.netfor2.com/checksum.html
)
17
Polynomial/Cyclic Redundancy Check (CRC) Codes
k information bits
(
i
k

1
, i
k

2
, .... i
1
,i
0
) create an information polynomial
i
(x)
, of degree
k

1
.
i
(x)=i
k

1
x
k

1
+ i
k

2
x
k

2
+ ... + i
1
x + i
0
i
(x) and a generator polynomial, g(x), are used to calculate a
codeword polynomial, b(x).
Checksum calculation :
Divide
x
n

k
i
(x)
by
g(x)
to obtain the remainder
r(x)
.
x
n

k
i
(x)=g(x).q(x) + r(x)
(
q(x)
= quotient and
r(x)
=remainder)
b(x) = x
n

k
i(x) + r(x)
n
bits
k
bits
n

k
bits
18
Binary Polynomial Division
Division with Decimal Numbers
32
35 ) 1222
3
105
17
2
4
140
divisor
quotient
remainder
dividend
1222 = 34 x 35 + 32
dividend = quotient x divisor +remainder
Polynomial Division
x
3
+
x
+ 1 )
x
6
+
x
5
x
6
+
x
4
+
x
3
x
5
+
x
4
+
x
3
x
5
+
x
3
+
x
2
x
4
+
x
2
x
4
+
x
2
+
x
x
=
q(x)
quotient
=
r(x)
remainder
divisor
dividend
+
x
+
x
2
x
3
Note: Degree of r(x) is less than
degree of divisor
Transmitted codeword:
b(x) = x
6
+ x
5
+ x
b
= (1,1,0,0,0,1,0)
1011 ) 1100000
1110
1011
1110
1011
1010
1011
010
x
3
+ x
+
1
) x
6
+ x
5
x
3
+ x
2
+ x
x
6
+ x
4
+ x
3
x
5
+ x
4
+ x
3
x
5
+ x
3
+ x
2
x
4
+ x
2
x
4
+ x
2
+ x
x
Polynomial Example:
k
= 4,
n
–
k
= 3
Generator polynomial:
g(x)= x
3
+ x +
1
Information: (1,1,0,0)
i(x) = x
3
+ x
2
Encoding:
x
3
i(x) = x
6
+ x
5
Examples of Standard Generator Polynomials
CRC

8:
CRC

16:
CCITT

16:
CCITT

32:
CRC = cyclic redundancy check
ISO HDLC, XMODEM, V.41, Bluetooth
IEEE 802, DoD, V.42, MPEG

2
IBM Bisync
ATM
= x
8
+ x
2
+ x +
1
= x
16
+ x
15
+ x
2
+
1
= (
x
+ 1)(
x
15
+ x +
1)
= x
16
+ x
12
+ x
5
+
1
= x
32
+ x
26
+ x
23
+
x
22
+
x
16
+ x
12
+ x
11
+ x
10
+ x
8
+
x
7
+ x
5
+ x
4
+ x
2
+ x
+
1
Thank You
Recommended private study exercise
: read the error control
coding section from Chapter 3 of the recommended text.
Use the content of the slides to guide your revision. Methods
not referred to in the slides will not be assessed.
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