1. Error Control Coding and Applications
Eric Chen Computer Science Group HKr
Allmän föreläsning för alla anställda vid Högskolan med intresse för EU-forskning. Föreläsare Professor Claes Magnusson Claes Magnusson arbetar f.n. med utveckling av maskinteknikprogrammet, Kristianstad Högskola. Han har sedan tidigare erfarenhet av EU:s forskningsprogram, både som projektledare och utvärderare och han arbetade i tre år som ”dörröppnare” i Bryssel åt svenska universitet och högskolor. Claes kommer i sitt anförande behandla frågor som: • Vilka forskningsområden finns? • Hur gör man en ansökan? • Hur är det att vara med? • Hur går det för Sverige? • Hur lång tid tar det innan pengarna kommer? • Vilken hjälp finns att få? • På vilka sätt kan jag delta? • Vilka fördelar kan deltagandet ge?

2. Internet Banking
Postgiro
Bankgior
OCR

3. OCR-Number (OCR-nummer ) ?
A reference number links your payment to the invoice
Typed OCR in-correctly ? Error detection?
E24.se reported (how it is possible?)
Företagaren Thomas Hultberg fyllde i fel OCR-nummer när han skulle betala in skatten. Nu tvingas han betala in de kronorna en gång till.

4. Outline of the presentation
About me
Errors and Error Effect
Error Control Coding
Personnummer, OCR
Hamming code
My Research and Results

5. About Me Education Work
1978.7— Electronics Engineering, Harbin C. U.
1982.7— Computer Science, SWJTU
1987.2— Electrical Engineering, SWJTU
Work
1986.1— Lecturer, Associate Prof. SWJTU
1993.4— Guest researcher, Linköping Uni.
1994.7– present HKr

6. Information Transmission System
Source encoding (remove redundancy)
Encoder ( add redundancy )
Decoder ( error detection/correction)
Source decoding
Information Sink
Receiver (Decoder)
Transmitter (Encoder)
Communication Channel
Information Source
Noise
n-digit
n-digit
k-digit
k-digit

7. Introduction to Coding Theory
Also called channel coding
The study of methods for efficient and accurate transfer of information
Detecting and correcting transmission errors

8. Errors in digital communications

9. Errors and Error Effect
Bits can be lost
Error effect
downloaded programs from Internet ?
CD music ?
Internet banking services ?
Errors must be detected/corrected !!!

10. Channel model -- BSC Binary symmetric channel
p: the bit error probability

11. Why Error Control Coding ?
Bit error rate p
p = 1/ = 10-5 for optical disks
p = for a fiber link
Some calculations
p = 10-6
download a file of length 107 bits  10 bit errors
Data rate at 10 Mbps  1 bit error in every 1 sec!!
p = 10-11, and data rate 10 Gigabits/sec
 1 bit error each 10 second !

12. Error Control Coding – Principle
add additional information, or redundancy to data
added by sender, checked by receiver
k data digits encoded to a codeword of n digits
Code rate r = k / n k n
Encoded as
codeword

13. Application Example– Swedish personal ID?
yy mm dd – nnnP
yy mm dd– year month day
nnn – serial number
odd– for male, even for female
P ? That is parity check digit
Used for error detection !
OCR number uses the same technique

14. Personal ID Encoding Method
position ?
2×odd
add
2-digits
sum = = 36
take the last digit of the sum: 6
parity check digit = 10 – 6 = 4 

15. Personal ID Error Detection
 ?
position ?
2×odd
add
2-digits
sum = = 43
take the last digit of the sum: 3
parity check digit = 10 – 3 = 7
 It is not equal to 4  Error in the number !

16. Parity Check Applications
The same coding methods have been used for
OCR reference number
Bankgironummer
Organisationsnummer
Reference

17. Error Correcting Code– Hamming Code
Binary Hamming [7, 4] code k = 4, n = 7
Encode 4 data bits by adding 3 parity bits
Can correct any single error
Encoding
a b c d  a b c d x y z
Where a, b, c, d are information bits
x, y, z are parity check bits

18. Given a, b, c, d. How to get x, y, z ?
Hamming Code
Given a, b, c, d. How to get x, y, z ?
Place a, b, c, d in the intersections
Label circles by x, y, z
Parity checking rule:
the sum of each circle is 0
x = a+b+c, y = a + c + d, z = b + c + d a b c d x y z

19. Given a, b, c, d. How to get x, y, z ?
Hamming Code Example
Given a, b, c, d. How to get x, y, z ?
0101  0101 xyz
so the codeword is a b c d x y z 1 x y z 1

20. Hamming Code for Error Correction
sent  received.
Encode 
Compare 101 with received 110  = 011, there is an error bit d must be in error, it affects y, z  correction 0101 1 1
reconstructed
received

21. Error Detecting Codes Only detect errors
Using protocol to correct errors:
ACK: positive acknowledgement ( I got it)
NAK: negative acknowledgement ( sorry )
Simple, reliable, high code rate
Used in data communications
sender
receiver
codeword
ACK/NAK

22. Error Correcting Codes
Detect and correct errors
No feedback channel required
Complicated, lower code rate (k/n)
Used in storage systems (computer storage, CD, DVD), and
space communications
sender
receiver
codeword

23. Generator Matrix and Encoding
Generator matrix G
Example Hamming [7, 4] code
Encoding: (a,b,c,d) G = c codeword
x = a+b+c, y = a + c + d, z = b + c + d a b c d x y z

24. Parity Check Matrix and Decoding
Parity check matrix H
HGtr = 0
Example Hamming [7, 4] code
Syndrome s: a column vector of length n-k
Decoding
Received codeword y: Hytr = s the syndrome of y
If s = 0,  no error detected
Otherwise, there must be errors a b c d x y z

25. Optimal codes Distance or d-optimal code Length or n-optimal code
A linear [n, k, d]q code is d-optimal if there does not exist an [n, k, d+1]q code.
Length or n-optimal code
A linear [n, k, d]q code is n-optimal if there does not exist an [n–1, k, d]q code.
k-optimal code
A linear [n, k, d]q code is k-optimal if there does not exist an [n, k+1, d]q code.

26. My Research Difference triangle sets Majority-logic decodable codes
A generalization of Golomb rulers
1989—1995
Majority-logic decodable codes
1988—1992
Quasi-Twisted codes (ex. a [112,13,48] code)
1989—present
Two-weight codes and graphs
2006—present

27. Difference Triangle Set
Golomb ruler
R = {0, 1, 4, 6}
Difference triangle
Difference triangle sets
A generalization of Golomb rulers
A set of t Golomb rulers
R1 = {0, 6, 11, 13}, R2 = {0, 8, 17, 18}, R3 = {0, 3, 15, 19}

28. Current Research Interests
Quasi-Twisted Codes
Many QT codes are good or optimal
Computer constructions
2-weight codes
Non-zero codewords of weights w1 or w2
Related to strongly regular graphs

29. Computer search for QT Codes
Given a cyclic weight matrix of order s
How to select p columns such that
Maximize the minimum row sums of p cols
Row sums of values w1 or w2

30. Computer search for QT Codes
Given a cyclic weight matrix of order s
Columns 0, 1, 3 produces a QT code with minimum distance of 6
row sums are of values 6 and 8  two-weight code found

31. Publications Co-authored books IEEE Trans. Information Theory
One text book ( VAX-11 Assembly lang. prog. )
One book ( combinatorial coding theory and appl.)
IEEE Trans. Information Theory
8 papers
IEE Electronics Letters
6 papers
Codes, Designs and Cryptography
1 paper (DDD disjoint distinct difference set)

32. Online Database on Codes
A web database of binary quasi-cyclic codes
see also: codetables
A Web database of two-weight codes

33. References
Personnummer