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A bit of magic A.J. Han Vinck A.J. Han Vinck, China 2012 1

A bit of magic A.J. Han Vinck A.J. Han Vinck, China 20121

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Page 1: A bit of magic A.J. Han Vinck A.J. Han Vinck, China 20121

A.J. Han Vinck, China 2012 1

A bit of magic

A.J. Han Vinck

Page 2: A bit of magic A.J. Han Vinck A.J. Han Vinck, China 20121

Some German history in China (1897-1914)

Tsingtao (Germania) brewery

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A.J. Han Vinck, Jinan 2011 3

My cooperation with Chinese scientistsBackground Application

• Fang-Wei Fu Information theory Flash Memory

• Yuan Luo Error Control Coding Networking

• Yanling Chen Wiretap channel Biometrics

• Feng-Wen Sun Mathematics Convolutional Codes

Common link? Fang-Wei Fu from Nankai University

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A.J. Han Vinck, China 2012 4

Error correction is needed in high tech devices

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A.J. Han Vinck, China 2012 5

fundamental limits set by Shannon (1948)

Bell Labs, 1955.

Fundamental problem: reproducing at the receiver a message selected at the transmitter

Shannon‘s contribution: - bound on the efficiency (capacity) - how to achieve the bound

The Bell System Technical Journal, Vol. 27, pp. 379–423, 623–656, October, 1948.

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A.J. Han Vinck, Jinan 2011 6

message: Alpha

codeword: (0 1 0 1 0 1 0 1)

transmission error (0 0 0 1 0 1 0 1)

decode (0 1 0 1 0 1 0 1)

translate: Alpha

How can we do that?

Introduction to the basic concepts

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A.J. Han Vinck, Jinan 2011 7

It works as follows

• Use the difference in transmitted vectors to recover the transmitted original

Example:

4 messages encoded as A 00 000B 01 011C 10 101D 11 110the words differ in at least 3 positions

Hence, one difference (error) can be detected and corrected:

receive: 10110

the transmitted message is? 11110

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A.J. Han Vinck, China 2012 8

1-Error correction started by Hamming (1950)

(31,26)(7,4)

(3,1)

1 2 3

123

000111

3567 124

0000 0001000 1100100 101•••1111 111 minimum distance dmin = 3

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A.J. Han Vinck, China 2012 9

What are the classical research problems

Codebook: maximize - # of codewords of length n - minimum distance dmin

Channel characterization:- Types of errors; memory in the noise behavior ? - Channel Capacity?

Decoder:- Design of a decoding/decision algorithm- Minimize complexity

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A.J. Han Vinck, China 2012 10

Lecture content

Goal: give examples, different from just error correction

- improve storage efficiency in flash memory

- give biometrics privacy

- code design for variable efficiency in networking

- permutation codes for non-traditional error models

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A.J. Han Vinck, China 2012

Ibm punch card „a flash memory“

On the Capacity of Write-Unidirectional Memories with Nonperiodic Codes”, IEEE Transactions on Information theory, April 2004, pp. 649-656, (Fang-Wei Fu, A.J. Han Vinck, Victor Wei, Raymond Yeung)

Write, read and throw away, but •••

Write, read and refresh

Write, read and erase, but •••

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A.J. Han Vinck, China 2012

A modern ROM: the Flash Memory

Cell array in a flash memory A flash memory cell

(Floating gate)

- A flash memory cell stores charge / data- We can only increase the cell charge- If charge top level is reached, erase memory block

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A.J. Han Vinck, China 2012

Writing in a q-ary flash memory cell

cell charge level value

1, 0, 1, 0, 0, 0, 1, 1, 0, 1 1, 2, 3, 4, 4, 4, 5, 5, 6, 7

A.J. Han Vinck, 2011

Cell Level 0

Input

Strategy for writing: increase charge content or stay at present content

Start at 0

7 16 05 14 03 12 01 10 0

bit

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Use the memory T=4 times: how many sequences we can write into the

memory?

q = 3: 5 +T(T-1)/2 = 11 words

write 0000 0011 0111 1111 0001 0012 0112 1112

0122 1122 1222

q = 2: T+1 = 5 words

write 0000 0011 0111 1111 0001

Capacity: log2|#sequences | ≈ (q-1) log2 (T+1)

Note: writer and reader must know the content!

A.J. Han Vinck, China 2012

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A.J. Han Vinck, China 2012

cell level

average number of writes to reach top level

Performance of the 1 step up strategy

probabilityP(1) = 1 – P(0) = p

p

1-p

Probability (level change) = 2p(1-p)

p)2p(11q

T

capacity! the of 50% is which

1)(Tlog2

1qpT

Tlog :is stored bits of# the 22

10101010

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Example of Codes: 2 bits in 3 cellsbefore erasing

A.J. Han Vinck, China 2012

2 bits

2 bits

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A.J. Han Vinck, China 2012 17

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Coding for Noisy Biometrics used as a key/password

- verify claimed identity with stored information

problems:

• Errors at authentication: False Rejection (FRR)/ False Acceptance (FAR)

• Leakage: attacker finds information about bio from Data Base (DB)

I am Han, can I come in?

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the Juels-Wattenberg scheme (binary)

b(io)

c bRandomCodeword c

b‘(io)

DataBase

c b b‘ = c e

Hash(c)

Enrollment Authentication c b

Hash(c)

Hash(c)

c b

= ?Y/N

Our contributions: use knowledge of bio at enrollment to encodeuse c b from database to improve decision on c

Yanling Chen and Han Vinck, A fresh look into the biometric authentication: Perspective from Shannon’s secrecy system and a special wiretap channel, Secrypt, 2011V. B. Balakirsky, A. R. G. and A. J. Han Vinck, Permutation block coding from biometrical authentication, Proc. 7th Int. Conf. on Computer Science and Info. Technologies, 2009

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A.J. Han Vinck, China 2012 20

Juels-Wattenberg for RS codes

Secret s { 1,2,•••,2k } → RS codeword c = c1,c2,c3,••• ,cn

c* b(io) changes n - t positions b => --- --- -----

Enrollment: store c*

Authentication: b‘ points at n- t „incorrect“ positions in c*

these positions are considered as erasures: for t k : dmin n– k + 1 – (n-t) = t – k + 1

kt for nt

k

nk

t

Rate Acceptance FalseFARk

b' in errors of number the is e where ne

Rate Rejection False FRR2k-t

Increasing k reduces FAR, but increases FRR

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A.J. Han Vinck, China 2012 21

• Translation from Bio to Template– Iris, fingerprint, etc

• Investigation of error behavior– Burst, impulsive, non-linear

• Performance estimation for real dataPractical performance FAR, FRR, etc.

• Using other techniques, such as data compression– Fault tolerant hashing

Research topics we work on

011001…

000001…

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A.J. Han Vinck, China 2012 22

Mondriaan

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A.J. Han Vinck, China 2012 23

Optimum distance profile (intro)

Encoder for binary linear (n,k) codes:

info generator matrix codeword

k bits n n bits

x •k G = cExample: equivalent Hamming code generators with dmin = 3 ( minimum # of differences)

1 0 0 0 1 1 1 1 1 1 1 1 1 10 1 0 0 1 0 1 1 1 1 0 1 0 0

0 0 1 0 1 1 0 1 1 0 1 0 0 10 0 0 1 0 1 1 1 0 1 1 0 1 0

G = ≡

The matrices generate the same code words

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24

Motivation (2)• UMTS: starting code (32,6) Reed Muller code, dmin = 16

6

32

• UMTS: extend to (32,10) code in best possible way

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 9, SEPTEMBER 2010 4309

A Lower Bound on the Optimum Distance Profiles of the Second-Order Reed–Muller CodesYanling Chen and A. J. Han Vinck

A.J. Han Vinck, China 2012

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 3, MARCH 2010 1007

On the Optimum Distance Profiles About Linear Block CodesYuan Luo, A. J. Han Vinck, and Yanling Chen

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Motivation (2)

• From (32,6), dmin = 16 to bound

– (32,7) dmin 14

– (32,8) dmin 13

– (32,9) dmin 12

– (32,10) dmin 12

– (32,11) dmin 12 is also possible !

QUESTION: is this possible?

REFERENCE: Markus Grassl ([email protected]).

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Optimum distance profile (for the Hamming code)

G0 = one row of G: dmin = 7

1 1 1 1 1 1 1

1 0 1 1 0 1 0

1 1 0 1 0 0 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 0 1 1 0 1 0

1 1 1 1 1 1 1

1 0 1 1 0 1 0

1 1 0 1 0 0 1

0 1 1 1 1 0 0

G1 = add one row of G: dmin = 3

G2 = add one row of G: dmin = 3

G3 = add one row of G: dmin = 3

G for the Hamming CODE

PROFILE :(7, 3, 3, 3)

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Optimum distance profile (idea 2)

G0 = one row of G: dmin = 4

0 1 1 1 1 0 0

1 0 1 1 0 1 0

1 1 0 1 0 0 1

0 1 1 1 1 0 0

0 1 1 1 1 0 0

1 0 1 1 0 1 0

1 1 1 1 1 1 1

1 0 1 1 0 1 0

1 1 0 1 0 0 1

0 1 1 1 1 0 0

G1 = add one row of G: dmin = 4

G2 = add one row of G: dmin = 4

G3 = add one row of G: dmin = 3

G for the Hamming CODE

PROFILE :(4, 4, 4, 3)

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Research problem: optimize distance profile

Inverse Optimum Distance profile:

Д:= (0, 1, 2, 3 ) Ex: (7,3,3,3) better than (4,4,4,3)

General problem: find ODP for classes of codes

like: Reed-Muller

BCH

Golay code

•••

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29A.J. Han Vinck, China 2012

Binary Extended Golay code (24,12,8)» Use Grassl (http://www.codetables.de/)

# rows: 12 11 10 9 8 7 6 5 4 3 2 1

– Dinvupper-bound := (8,8,8,8,8, 10,10, 12, 12,13,16,24)

– ODPinv-BGolay =(8,8,8,8,8, 8, 8, 8, 12,12,12,24)

Why? 1. All ones should be the first row (weight 24)hence 10 = 16 cannot occur Codeword weights: 0, 8, 12, 16, 24

2. there is a unique (24,5,12) code ( code word weights: 12 and 16)

D.B. Jaffe, optimal binary linear codes of length 30, Discr. Math., vol 223, pp. 135-155, 2000

3. a code with n = 24 and k = 4, d = 12 exists

123

Codewords: (1111111-1111111-11111111), (10100110-10100110-10100110) (01001110-01001110-01001110), (00111010-00111010-00111010)

Binary Extended Golay code (24,12,8)

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The Optimum distance profile in practice

(32,10) sub-code ofthe second orderReed-Muller code

T F C I co d e w o rdb 0 ...b 3 1

T F C I (1 0 b its )a 0 ...a 9

Figure 6: Channel coding of TFCI bits

China Wireless Telecommunication Standard (CWTS); Working Group 1 (WG1);

Multiplexing and channel coding

CWTS STD-TDD-103

Currently, a [32,10,12] sub-code of the second order Reed-Muller Code is used in the TFCI coding. An easy improvement can be obtained by extending it to a [32,11,12] sub-code .

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A.J. Han Vinck, China 2012 31

Escher

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A.J. Han Vinck, China 2012 32

Permutation codes

• Idea: present codewords as balls in an array

• One ball per column and one per row

word 1

word 2

= 4

number

1

2

6

time1 2 6

3

4

5

3 4 5

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A.J. Han Vinck, China 2012 33

effect of noise on permutation codes (simplified)

Transmitted Background-insertion Background-deletion

Impulsive-broadband narrowband-jammer frequency selective fading

f

t

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A.J. Han Vinck, China 2012 34

A practical example: Spread-FSK

Send: f0 or f1 :

Detector: uses FSK or On-Off keying mode

Advantages: robust against narrowband noise

alternative

Send: (f0,f1 ) or (f1,f0 ) i.e. R = 1/2

Detector: select max. # of agreements

Advantages: robust

against: narrowband + impulsive noise + random

f

f0 f1

2-FSK mode

f

f1

On-off mode

Permutation code, dmin = 2

f0

f1

t

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A.J. Han Vinck, China 2012 35

Code parameters

• Upperbound on cardinality of the code

Q1: when do we achieve equality?

Q2: if not, what is the upperbound

• References: - Ian Blake, Permutation codes for discrete channels (1975, IT)

- P. Frankl and M. Deza, On the max. # of Permutations with given Max. or Min. Distance (1977, Jrnl of Comb. Th.)

- T. Klöve: |C| = 18 for M = 6, D = 5 instead of 30

1)!(DM!

|C|p

# of permutations

reduction for

distance

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A.J. Han Vinck, China 2012 36

• all cases M < 7 solved• Interesting cases left ?

Dp = 2 3 4 5 6 7 8 9 10

6 x x x 18 x |C| = 18 is the Klöve (2000) result

7 x x ? ? x x proof that the max code size is 18

M 8 x x ? ? x x x for Dp = 5, M = 6.

9 x x ? ? ? x x x M! = 720 possible code words

10 x x ? ? ? ? x ? x

Some answers to the questions

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A.J. Han Vinck, China 2012 37

Existing code constructions

• Dp = M; Dp = M-1 , where M is prime (power)

• Dp = 2; Dp = 3

• The Mathieu groups– M11 Dp = 8 |C| = 7920

– M12 Dp = 8 |C| = 95040

• Codes with distance Dp = M-2 exist for– M = 1 + pn |C| = pn(1+pn)(pn-1)

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A.J. Han Vinck, China 2012 38

Performance: conform

• Correct decoding if # error events ( impulse + narrow + background + fading ) < Dp

an error event hits a codeword only at one place

• An efficient trellis based decoding scheme has been designed for low complexity decoding

Hendrik C. Ferreira, A. J. Han Vinck, Theo G. Swart, Ian de Beer. Permutation trellis codes. IEEE Transactions on Communications, 2005: 1782~1789

Code performance

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A.J. Han Vinck, China 2012 39

• extremely suited for dirty channels– Impulse-, narrow band -, background noise, fading

• Problems to work on:– Cardinality of the codes: upperbound can be obtained, but not always– Encoding complexity no linear encoding– Decoding complexity no easy decoding algorithm due to non-linearity

• Performance analysis on real channels

• Applications in Powerline Communications and Flash memories - Coded Modulation for Powerline Communications, In AEÜ Journal, 2000, pp. 45-49, Jan 2000, A.J. Han Vinck

- Rank Modulation for Flash Memories, A. Jiang, Member, R. Mateescu, M. Schwartz, J.Bruck, IEEE Tr. IT, JUNE 2009

Conclusions

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A.J. Han Vinck, China 2012 40

Our (Olaf hooijen and me) modem design based on permutation codes,

1997, Cebit Hannover

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• Coding has a mathematical and a systems aspect

• Applications are widespread: from networking to chip level

• Implementations lead to low complexity algorithms– Viterbi (Convolutional Codes)– Berlekamp Massey (RS-codes)

• Background knowledge in Information and Communication theory and combinatorics necessary

• Cooperation with different disciplines important

A.J. Han Vinck, China 2012 41

Some lessons I learned

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A.J. Han Vinck, Jinan 2011 42

My scientific genealogy

• Friedrich Leibniz (1597, Leipzig)• Carl Friedrich Gauß (Göttingen)• Ernst Adolph Guillemin (München)• Robert Mario Fano (MIT)• John McReynolds Wozencraft (MIT)• Thomas Kailath (Stanford)• Johan Pieter M. Schalkwijk (Stanford-TUE)• Me (University Duisburg-Essen, Germany)

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my scientific Genealogy (coincidence?)http://genealogy.math.ndsu.nodak.edu/

• Friedrich Leibnitz (1597, Leipzig)

• Carl Friedrich Gauß (Göttingen)

• John M. Wozencraft (MIT) (1925-2009)

• Me (University Duisburg-Essen, Germany)– Digital Communications and - Coding techniques

A.J. Han Vinck, China 2012 43

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EARLIEST DEPICTION OF JUGGLING known shows skillful Egyptian women on the 15th Beni Hassan

tomb of an unknown prince from the Middle Kingdom period of about 1994 to 1781 B.C

Shannon juggling machine and juggling theorem

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A.J. Han Vinck, China 2012 46

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Traditional Jeju (Korean) Channel Code

is there a Chinese equivalent?

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A Related problem: code generator partitioning

Code generator

make d2 as large as possible for any generator for a specific code

d1

d2

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Optimum distance profile (idea 1)

G0 = 4 rows of G: dmin = 3

G1 = delete one row of G: dmin = 4

G2 = delete one row of G: dmin = 4

G3 = 1 row of G: dmin = 4

1 0 1 1 0 1 0

1 1 0 1 0 0 1

0 1 1 1 1 0 0

1 1 0 1 0 0 1

0 1 1 1 1 0 0

1 1 0 1 0 0 1

Start with G for the Hamming CODE

1 1 1 1 1 1 1

1 0 1 1 0 1 0

1 1 0 1 0 0 1

0 1 1 1 1 0 0 PROFILE (3, 4, 4, 4)

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A.J. Han Vinck, China 2012 50

Optimum distance profile (idea 1)

G0 = 4 rows of G: dmin = 3

G1 = delete one row of G: dmin = 3

G2 = delete one row of G: dmin = 3

G3 = 1 row of G: dmin = 7

1 1 1 1 1 1 1

1 1 0 1 0 0 1

0 1 1 1 1 0 0

1 1 1 1 1 1 1

0 1 1 1 1 0 0

1 1 1 1 1 1 1

Start with G for the Hamming CODE

1 1 1 1 1 1 1

1 0 1 1 0 1 0

1 1 0 1 0 0 1

0 1 1 1 1 0 0 PROFILE (3, 3, 3, 7)

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Research problem: optimize distance profile

Optimum Distance profile for the Hamming code:

D:= ( d0, d1, d2, d3 ) Ex: (3,4,4,4) better than (3,3,3,7)

General problem: find ODP for classes of codes

like: Reed-Muller

BCH

Golay code

•••

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1 2

A.J. Han Vinck, China 2012 52

Binary Extended Golay code (24,12,8)

• Use Grassl (http://www.codetables.de/)

– # rows: 12 11 10 9 8 7 6 5 4 3 2 1

– Ddicupper-bound := (8,8,8,8,8, 10,10, 12,12,13,16,24)

– ODPdic-BGolay =(8,8,8,8,8, 8, 8, 12,12,12,16,16)

Why? 1. Codeword weights: 0, 8, 12, 16, 24 2. there is a unique (24,5,12) code ( code word

weights: 12 and 16)D.B. Jaffe, optimal binary linear codes of length 30, Discr. Math., vol 223, pp.

135-155, 2000

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Simple code construction with Dp = M-1

• M prime (power)

Same as Reed Solomon codes except for I 0

CI(m) = ( m , I ) 1 1 1 Dp = M – 1 0 1 M-1

result: M-1 sets of M codewords; (M-1) M/2 pairs at distance M !

----- ----- -----M ----- ----- -----

••• ••• ••• ••• ---------- -----

M-1 sets

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A.J. Han Vinck, China 2012 54

Example: M = 3

M = 3; M(M-1) = 6; minimum distance = M-1 = 2

(m,I) 1 1 1 C0 1 2

I = 1 I = 2+ 000 0 1 2 0 2 1 + 111 1 2 3 1 0 2+ 222 2 0 1 2 1 0

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• Design of optimal codes

• Complexity of decoding (exhaustive search)

• Low efficiency

Problems

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My scientific genealogy

• Friedrich Leibniz (1597, Leipzig)• Carl Friedrich Gauß (Göttingen)• Ernst Adolph Guillemin (München)• Robert Mario Fano (MIT)• John McReynolds Wozencraft (MIT)• Thomas Kailath (Stanford)• Johan Pieter M. Schalkwijk (Stanford-TUE)• Me (University Duisburg-Essen, Germany)

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A.J. Han Vinck, China 2012 57

The first topic is impulsive noise• Mobile radio with electromagnetic interference

– D. Middleton ""Statistical-physical models of electromagnetic interference,"", IEEE Trans. Electromagn. Compat., vol. EMC-19, pp. 106-127, Aug. 1977.

• Measurements and Models of Radio Frequency Impulsive Noise for Indoor Wireless Communications

K. Blackard, T. S. Rappaport, and C. W. Bostian, IEEE JSAC, Sept. 1993

• Impulsive Noise Environment of High Voltage Electricity Transmission Substations and its Impact of the Performance of ZigBee

• Impulse noise, is characterized by long quiet intervals of time followed by high-amplitude bursts. This noise results from natural causes— such as lightning— as well as man-made causes.

• Impulse noise is responsible for most errors in digital communication systems and generally provokes errors to occur dependently in groups.

http://www.wireless-center.net/WLANs-WPANs/2425.html

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Some German history in China (1897-1914)

Tsingtao (Germania) brewery

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A.J. Han Vinck, China 2012 59

My cooperation with Chinese scientists

Background Application• Fang-Wei Fu Information theory Flash Memory

• Yuan Luo Error Control Coding Networking

• Yanling Chen Wiretap channel Biometrics

• Feng-Wen Sun Mathematics Convolutional Codes

Common link? Fang-Wei Fu from Nankai University

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Error correction started by Hamming (1950)

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The classical starting point

http://commons.wikimedia.org/wiki/File:Hamming(7,4)_as_bits.svg

Hamming Code (7,4)

- Message bits 3,5,6,7

- check bits 1,2,4

Conclusion:1 error can be found

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Can we do a general representation?Example: 5 check digits

(31,26)

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What is error correction?

Shannon does the math at Bell Labs, 1955.

The Bell System Technical Journal, Vol. 27, pp. 379–423, 623–656, October, 1948.

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message: Alpha

codeword: (0 1 0 1 0 1 0 1)

transmission error (0 0 0 1 0 1 0 1)

decode (0 1 0 1 0 1 0 1)

translate: Alpha

How can we do that?

Introduction to the basic concepts

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It works as follows

• Use the difference in transmitted vectors to recover the transmitted original

Example:

4 messages encoded as A 00 000B 01 011C 10 101D 11 110the words differ in at least 3 positions

Hence, one difference (error) can be detected and corrected:

receive: 10110

the transmitted message is? 11110