Computer Networking
Error Control Coding
Dr Sandra I. Woolley
An Introduction to Error Control Coding
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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)
http://www.thinkgeek.com/interests/exclusives/7a5c/images/1293/
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We will work through examples in
class.
Introduction
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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.
3
Data Transmission
Channel
errors
Data
..01101010..
Compress
Encrypt
Error control
Encrypt message
for added security
Remove redundancy
for efficient
communication
Add systematic
redundancy to protect
data against channel
errors
Line code
..01110110..
Encode signal to suit
communication
channel
characteristics
4
Designing Error Protection Systems
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“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?
5
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.
p1
1-q1
Bad
Type 1
1-p1-p2
p2
Bad
Type 2
Good
q1
1-q2
The modified
Gilbert model
q2
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In the Good state there are no errors.
 Bad Type 1 and Bad Type 2 represent two types of error events; nonburst and burst.
 p1 and p2 are the probabilities of starting the errored states.
 q1 and q2 are the probabilities of ending the errored states.
6
Channel Errors
Burst length distributions provide important information about
channel error activity.
p1
1-q1
Bad
Type 1
1-p1-p2
p2
Bad
Type 2
Good
q1
Probability (log scale)

1-q2
q2
increasing p1
decreasing q1
increasing p2
decreasing q2
error length (log scale)
7
The Simplest Error Detection – Parity Bit
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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
00110101 (where 1 is our
even parity bit).
7 bits of data 8 bits including parity
(number of 1s) even
odd
0000000 (0)
00000000
10000000
1010001 (3)
11010001
01010001
1101001 (4)
01101001
11101001
1111111 (7)
11111111
01111111
Examples from Wikipedia
The parity bit is added to the front
http://en.wikipedia.org/wiki/Parity_bit
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]
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The Hamming (7,4) Code
If we insert an error at bit 2
[ x
](syndrome)
[0 1 1 0 1 1 0]
c1=k1+k2+k4
becomes
[0 0 1 0 1 1 0]
1
k1
Note c1 and c3 are
k2
0
wrong - these intersect
at k2 hence
k2 is in error
0
0
k4
1
c2=k1+k3+k4
1
k3
0
c3=k2+k3+k4
10
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]
c1=k1+k2+k4
1
k1
Note c1,c2 and c3
k2
0
are all wrong
hence
k4 is in error
1
1
k4
1
c2=k1+k3+k4
1
k3
0
c3=k2+k3+k4
11
Bi-Directional/Product Codes
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Imagine we have 4 x 4 data bits k11-k14, k21-k24, k31k34, k41-k44

Arranging them horizontally we can add error protection in the
vertical direction as well.
k11
k21
k31
k41
d11
d21
d31
k12
k22
k32
k42
d12
d22
d32
k13
k23
k33
k43
d13
d23
d33
k14
k24
k34
k44
d14
d24
d34
c11
c21
c31
c41
f11
f21
f31
c12
c22
c32
c42
f12
f22
f32
c13
c23
c33
c43
f13
f23
f33
12
Bi-Directional/Product Codes
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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
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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
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More Sophisticated Codes
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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<28 (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.
15
IP Checksum
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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).
x  b0  b1  .....  bL 1 (mod 216  1)
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Checksum, bL, is given by : bL=-x
So that adding the 16-bit words and the checksum (as below)
gives 0.
16
0  b0  b1  .....  bL 1  bL (mod 2  1)
(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 )
16
Polynomial/Cyclic Redundancy Check (CRC) Codes
k information bits (ik-1, ik-2, .... i1,i0) create an information polynomial
i(x), of degree k-1.
i(x)=ik-1xk-1 + ik-2xk-2 + ... + i1x + i0
i(x) and a generator polynomial, g(x), are used to calculate a
codeword polynomial, b(x).
Checksum calculation :
Divide xn-ki(x) by g(x) to obtain the remainder r(x).
xn-ki(x)=g(x).q(x) + r(x)
(q(x)= quotient and r(x)=remainder)
b(x) = xn-ki(x) + r(x)
n bits
k bits
n-k bits
17
Binary Polynomial Division
Division with Decimal Numbers
34
35 ) 1222
105
divisor
17 2
140
32
quotient
dividend
dividend = quotient x divisor +remainder
1222 = 34 x 35 + 32
remainder
x3 + x2 + x
Polynomial Division
= q(x) quotient
x3 + x + 1 ) x6 + x5
x6 +
x4 + x3
dividend
divisor
x5 + x4 + x3
x5 +
x3 + x2
Note: Degree of r(x) is less than
degree of divisor
x4 +
x4 +
x2
x2 + x
x = r(x) remainder
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Polynomial Example: k = 4, n–k = 3
g(x)= x3 + x + 1
i(x) = x3 + x2
x3i(x) = x6 + x5
Generator polynomial:
Information: (1,1,0,0)
Encoding:
x3 + x2 + x
x3 + x + 1 ) x6 + x5
x6 +
1110
1011 ) 1100000
1011
x 4 + x3
x5 + x4 + x3
x5 +
x3 + x 2
x4 +
x4 +
1110
1011
x2
x2 + x
x
Transmitted codeword:
b(x) = x6 + x5 + x
b = (1,1,0,0,0,1,0)
1010
1011
010
Examples of Standard Generator Polynomials
CRC = cyclic redundancy check
ATM
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CRC-8:
= x8 + x2 + x + 1
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CRC-16:
= x16 + x15 + x2 + 1
= (x + 1)(x15 + x + 1)
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CCITT-16: = x16 + x12 + x5 + 1
ISO HDLC, XMODEM, V.41, Bluetooth
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CCITT-32: = x32 + x26 + x23 + x22 +
IEEE 802, DoD, V.42, MPEG-2
x16 + x12 + x11 + x10 + x8 +
x7 + x5 + x4 + x2 + x + 1
IBM Bisync
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.