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Some recent results in mathematics related to data transmission:. Michel Waldschmidt Université P. et M. Curie - Paris VI Centre International de Mathématiques Pures et Appliquées - CIMPA. India, October-November 2007. http://www.math.jussieu.fr/~miw/. French Science Today. - PowerPoint PPT Presentation
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Some recent results in mathematics related to data transmission:
Michel Waldschmidt
Université P. et M. Curie - Paris VI
Centre International de Mathématiques Pures et Appliquées - CIMPA
http://www.math.jussieu.fr/~miw/
India, October-November 2007
2
Starting with card tricks, we show how mathematical tools are used to detect and to correct errors occuring in the transmission of data.
These so-called "error-detecting codes" and "error-correcting codes" enable identification and correction of the errors caused by noise or other impairments during transmission from the transmitter to the receiver. They are used in compact disks to correct errors caused by scratches, in satellite broadcasting, in digital money transfers, in telephone connexions, they are useful for improving the reliability of data storage media as well as to correct errors cause when a hard drive fails. The National Aeronautics and Space Administration (NASA) has used many different error-correcting codes for deep-space telecommunications and orbital missions.
French Science Today
http://www.math.jussieu.fr/~miw/
India October- November 2007
Some recent results in mathematics related to data transmission
3
Most of the theory arises from earlier developments of mathematics which were far removed from any concrete application. One of the main tools is the theory of finite fields, which was invented by Galois in the XIXth century, for solving polynomial equations by means of radicals. The first error-correcting code happened to occur in a sport newspaper in Finland in 1930. The mathematical theory of information was created half a century ago by Claude Shannon. The mathematics behind these technical devices are being developped in a number of scientific centers all around the world, including in India and in France.
French Science Today
http://www.math.jussieu.fr/~miw/
Some recent results in mathematics related to data transmission
India November 2007
4
Mathematical aspects of Coding Theory in France:
French Science Today
http://www.math.jussieu.fr/~miw/
The main teams in the domain are gathered in the group
C2 ''Coding Theory and Cryptography'' ,which belongs to a more general group
(GDR) ''Mathematical Informatics''.
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The most important are:
INRIA RocquencourtUniversité de Bordeaux ENST Télécom BretagneUniversité de Limoges Université de MarseilleUniversité de Toulon
Université de Toulouse
French Science Today
http://www.math.jussieu.fr/~miw/
6
Brest
Bordeaux
MarseilleToulouse
Limoges
INRIA
Toulon
7
Institut National de Recherche en Informatique et en Automatique
http://www-rocq.inria.fr/codes/
National Research
Institute in Computer
Science and Automatic
8
Institut de Mathématiques de Bordeaux
http://www.math.u-bordeaux1.fr/maths/
Lattices and combinatorics
9
École Nationale Supérieure des Télécommunications de Bretagne
http://departements.enst-bretagne.fr/sc/recherche/turbo/
Turbocodes
Research Laboratory of LIMOGES
http://www.xlim.fr/
11
Marseille: Institut de Mathématiques de Luminy
Arithmetic and Information TheoryAlgebraic geometry over finite fields
12
Université du Sud Toulon-Var
http://grim.univ-tln.fr/
Boolean functions
13
Université de ToulouseLe Mirail
http://www.univ-tlse2.fr/grimm/algo
Algebraic geometry over finite fields
14
GDR IMGroupe de Recherche
Informatique Mathématique
• The GDR ''Mathematical Informatics'' gathers all the french teams which work on computer science problems with mathematical methods.
http://www.gdr-im.fr/
15
Some instances of scientific domains of the GDR IM:
• Calcul Formel (Symbolic computation)
• ARITH: Arithmétique (Arithmetics)
• COMBALG : Combinatoire algébrique (Algebraic Combinatorics)
http://www.gdr-im.fr/
16
Mathematical Aspects of Coding Theory in India:
Indian Institute of Technology Bombay
Indian Institute of Science Bangalore
Indian Institute of Technology Kanpur
Panjab University Chandigarh
University of Delhi Delhi
French Science Today
17
Delhi
Bangalore
Kanpur
Bombay
Chandigarh
18
IIT BombayIndian Institute of Technology
• Department of Mathematics • Department of Electrical Engineering
http://www.iitb.ac.in/
19
• Finite fields and Coding Theory classification of permutation polynomials, study of PAPR of families of codes, construction of codes with low PAPR.
http://www.iisc.ernet.in/
• Department of Mathematics
peak-to-average power
20
IIT KanpurIndian Institute of Technology
http://www.iitk.ac.in/
21
http://www.puchd.ac.in/
Department of Mathematics
22
Department of Mathematics
http://www.du.ac.in/
23
Error Correcting Codesby Priti Shankar
• How Numbers Protect Themselves • The Hamming Codes Volume 2 Number 1• Reed Solomon Codes Volume 2 Number 3
http://www.ias.ac.in/resonance/
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Playing cards
25
I know which card you selected
• Among a collection of playing cards, you select one without telling me which one it is.
• I ask you some questions where you answer yes or no.
• Then I am able to tell you which card you selected.
26
2 cards
• You select one of these two cards
• I ask you one question and you answer yes or no.
• I am able to tell you which card you selected.
27
2 cards: one question suffices
• Question: is-it this one?
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4 cards
29
First question: is-it one of these two?
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Second question: is-it one of these two ?
31
4 cards: 2 questions suffice
Y Y Y N
N Y N N
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8 Cards
33
First question: is-it one of these?
34
Second question: is-it one of these?
35
Third question: is-it one of these?
36
8 Cards: 3 questions
YYY YYN YNY YNN
NYY NYN NNY NNN
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Yes / No
• 0 / 1
• Yin — / Yang - -
• True / False
• White / Black
• + / -
• Heads / Tails (tossing a coin)
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3 questions, 8 solutions
0 0 0 0 0 1 0 1 0 0 1 1
1 0 0 1 0 1 1 1 0 1 1 1
39
16 Cards
• If you select one card among a set of 16, I shall know which one it is, once you answer my 4 questions by yes or no.
40
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Label the 16 cards
0 1 32
4 5 76
8 9 1110
12 13 1514
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In binary expansion:
0 0 0 0 0 0 0 1 0 0 1 10 0 1 0
0 1 0 0 0 1 0 1 0 1 1 10 1 1 0
1 0 0 0 1 0 0 1 1 0 1 11 0 1 0
1 1 0 0 1 1 0 1 1 1 1 11 1 1 0
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Ask the questions so that the answers are:
Y Y Y Y Y Y Y N Y Y N NY Y N Y
Y N Y Y Y N Y N Y N N NY N N Y
N Y Y Y N Y Y N N Y N NN Y N Y
N N Y Y N N Y N N N N NN N N Y
44
The 4 questions:
• Is the first digit 0 ?
• Is the second digit 0 ?
• Is the third digit 0 ?
• Is the fourth digit 0 ?
0000 0001 00110010
0100 0101 01110110
1000 1001 10111010
1100 1101 11111110
More difficult:
One answer may be wrong!
46
One answer may be wrong
• Consider the same problem, but you are allowed to give (at most) one wrong answer.
• How many questions are required so that I am able to know whether your answers are right or not? And if they are right, to know the card you selected?
47
Detecting one mistake
• If I ask one more question, I shall be able to detect if there is one of your answers which is not compatible with the others.
• And if you made no mistake, I shall tell you which is the card you selected.
48
Detecting one mistake with 2 cards
• With two cards I just repeat twice the same question.
• If both your answers are the same, you did not lie and I know which card you selected
• If your answers are not the same, I know that one is right and one is wrong (but I don’t know which one is correct!).
49
4 cards
50
First question: is-it one of these two?
51
Second question: is-it one of these two?
52
Third question: is-it one of these two?
53
4 cards: 3 questions
Y Y Y Y N N
N Y N N N Y
54
4 cards: 3 questions
0 0 0 0 1 1
1 0 1 1 1 0
55
Correct triple of answers:
0 0 0 0 1 1 1 0 1 1 1 0
Wrong triple of answers
0 0 1 0 1 0 1 0 0 1 1 1
One change in a correct triple of answers yields a wrong triple of answers
56
Boolean addition
• even + even = even
• even + odd = odd
• odd + even = odd
• odd + odd = even
• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 0
57
Parity bit• Use one more bit which is the Boolean
sum of the previous ones.
• Now for a correct answer the sum of the bits should be 0.
• If there is exactly one error, the parity bit will detect it: the sum of the bits will be 1 instead of 0.
58
8 Cards
59
4 questions for 8 cards
0000 0011 0101 0110
1001 1010 1100 1111
YYYY YYNN YNYN YNNY
NYYN NYNY NNYY NNNN
Use the 3 previous questions plus the parity bit question
60
First question: is-it one of these?
61
Second question: is-it one of these?
62
Third question: is-it one of these?
63
Fourth question: is-it one of these?
64
16 cards, at most one wrong answer:
5 questions to detect the mistake
65
Ask the 5 questions so that the answers are:
YYYYY YYYNN YYNNYYYNYN
YNYYN YNYNY YNNNNYNNYY
NYYYN NYYNY NYNNNNYNYY
NNYYY NNYNN NNNNYNNNYN
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Correcting one mistake
• Again I ask you questions where your answer is yes or no, again you are allowed to give at most one wrong answer, but now I want to be able to know which card you selected - and also to tell you whether and when you lied.
67
With 2 cards
• I repeat the same question three times.
• The most frequent answer is the right one: vote with the majority.
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With 4 cards
I repeat my two questions three times each, which makes 6 questions
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With 8 Cards
I repeat 3 timesmy 3 questions,which makes 9 questions
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With 16 cards, this process requires
34=12 questions
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16 cards, 7 questions
• We shall see that 7 questions allow to recover the exact result if there is at most one wrong answer.
Error correcting codes
73
Coding Theory
• Coding theory is the branch of mathematics concerned with transmitting data across noisy channels and recovering the message. Coding theory is about making messages easy to read: don't confuse it with cryptography which is the art of making messages hard to read!
74
Claude Shannon
• In 1948, Claude Shannon, working at Bell Laboratories in the USA, inaugurated the whole subject of coding theory by showing that it was possible to encode messages in such a way that the number of extra bits transmitted was as small as possible. Unfortunately his proof did not give any explicit recipes for these optimal codes.
75
Richard Hamming
Around the same time, Richard Hamming, also at Bell Labs, was using machines with lamps and relays having an error detecting code. The digits from 1 to 9 were send on ramps of 5 lamps with two on and three out. There were very frequent errors which were easy to detect and then one had to restart the process.
76
The first correcting codes
• For his researches, Hamming was allowed to have the machine working during the week-end only, and they were on the automatic mode. At each error the machine stopped until the next monday morning.
• "If it can detect the error," complained Hamming, "why can't it correct it! "
77
The origin of Hamming’s code• He decided to find a device so that the machine would
not only detect the errors but also correct them. • In 1950, he published details of his work on explicit
error-correcting codes with information transmission rates more efficient than simple repetition.
• His first attempt produced a code in which four data bits were followed by three check bits which allowed not only the detection but the correction of a single error.
78
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Codes and Geometry
• 1949: Marcel Golay (specialist of radars): produced two remarkably efficient codes.
• Eruptions on Io (Jupiter’s volcanic moon)• 1963 John Leech: uses Golay’s ideas for
sphere packing in dimension 24 - classification of finite simple groups
• 1971: no other perfect code than the two found by Golay.
80
Error Correcting Codes Data Transmission
• Telephone
• CD or DVD
• Image transmission
• Sending information through the Internet
• Radio control of satellites
81
Applications of error correcting codes
• Transmitions by satellites
• Compact discs
• Cellular phones
82
Olympus Mons on Mars Planet
Image from Mariner 2 in 1971.
83
Information was sent to the earth using an error correcting code which corrected 7 bits among 32.
In each group of 32 bits, 26 are control bits and the 6 others contain the information.
84
• Between 1969 and 1973 the NASA Mariner probes used a powerful Reed--Muller code capable of correcting 7 errors out of 32 bits transmitted, consisting now of 6 data bits and 26 check bits! Over 16,000 bits per second were relayed back to Earth.
QuickTime™ et undécompresseur TIFF (non compressé)
sont requis pour visionner cette image.The North polar cap of Mars, taken by Mariner 9 in 1972.
85
Voyager 1 and 2 (1977)
• Journey: Cape Canaveral, Jupiter, Saturn, Uranus, Neptune.
• Sent information by means of a binary code which corrected 3 errors on words of length 24.
86
Mariner spacecraft 9 (1979)
• Sent black and white photographs of Mars
• Grid of 600 by 600, each pixel being assigned one of 64 brightness levels
• Reed-Muller code with 64 words of 32 letters, minimal distance 16, correcting 7 errors, rate 3/16
87
Voyager (1979-81)
• Color photos of Jupiter and Saturn• Golay code with 4096=212 words of 24
letters, minimal distance 8, corrects 3 errors, rate 1/2.
• 1998: lost of control of Soho satellite recovered thanks to double correction by turbo code.
88
NASA's Pathfinder mission on Mars
• The power of the radio transmitters on these craft is only a few watts, yet this information is reliably transmitted across hundreds of millions of miles without being completely swamped by noise.
QuickTime™ et undécompresseur TIFF (non compressé)
sont requis pour visionner cette image.
QuickTime™ et undécompresseur TIFF (non compressé)
sont requis pour visionner cette image.
Sojourner rover and Mars Pathfinder lander
89
Listening to a CD• On a CD as well as on a computer, each
sound is coded by a sequence of 0’s and 1’s, grouped in octets
• Further octets are added which detect and correct small mistakes.
• In a CD, two codes join forces and manage to handle situations with vast number of errors.
90
Coding the sound on a CD
Using a finite field with 256 elements, it is possible to correct 2 errors in each word of 32 octets with 4 control octets for 28 information octets.
On CDs the signal in encoded digitally. To guard against scratches, cracks and similar damage, two "interleaved" codes which can correct up to 4,000 consecutive errors (about 2.5 mm of track) are used.
91
A CD of high quality may have more than 500 000 errors!
• After processing of the signal in the CD player, these errors do not lead to any disturbing noise.
• Without error-correcting codes, there would be no CD.
92
1 second of radio signal = 1 411 200 bits
• The mathematical theory of error correcting codes provides more reliability and at the same time decreases the cost. It is used also for data transmission via the internet or satellites
93
Codes and Mathematics
• Algebra (discrete mathematics
finite fields, linear algebra,…)
• Geometry
• Probability and statistics
94
Finite fields and coding theory
• Solving algebraic equations with radicals: Finite fields theory Evariste Galois (1811-1832)
• Construction of regular polygons with rule and compass
• Group theorySrinivasa Ramanujan (1887-1920)
95
Coding Theory
transmission
Source Coded Text
Coded Text
Receiver
96
Error correcting codes
Transmission
with noise
Source Coded Text
Coded Text
Receiver
97
Principle of coding theory
Only certain words are allowed (code = dictionary of valid words).
The « useful » letters carry the information, the other ones (control bits) allow detecting or correcting errors.
98
Detecting one error by sending twice the message
• Send twice each bit
2 code words among 4=22 possible words
(1 useful letter among 2)
Code words
(two letters)
0 0
and
1 1
Rate: 1/2
99
Principle of codes detecting one error:
Two distinct code words have at least two distinct letters
100
Detecting one error with the parity bit
Code words (three letters):
0 0 0
0 1 1
1 0 1
1 1 0
Parity bit : (x y z) with z=x+y.
4 code words (among 8 words with 3 letters),
2 useful letters (among 3).
Rate: 2/3
2
101
Code Words Non Code Words
0 0 0 0 0 10 1 1 0 1 01 0 1 1 0 0
1 1 0 1 1 1
Two distinct code words have at least two distinct letters.
2
102
Check bit
• In the International Standard Book Number (ISBN) system used to identify books, the last of the ten-digit number is a check bit.
• The Chemical Abstracts Service (CAS) method of identifying chemical compounds, the United States Postal Service (USPS) use check digits.
• Modems, computer memory chips compute checksums.
• One or more check digits are commonly embedded in credit card numbers.
103
Correcting one errorby repeating three times
• Send each bit three times
2 code words
among 8 possible ones
(1 useful letter among 3)
Code words
(three letters)
0 0 0
1 1 1
Rate: 1/3
104
• Correct 0 0 1 as 0 0 0
• 0 1 0 as 0 0 0
• 1 0 0 as 0 0 0
and
• 1 1 0 as 1 1 1
• 1 0 1 as 1 1 1
• 0 1 1 as 1 1 1
105
Principle of codes correcting one error: Two distinct code words have at least
three distinct letters
106
Hamming Distance between two words:
= number of places in which the two words
differ
Examples:
(0,0,1) and (0,0,0) have distance 1
(1,0,1) and (1,1,0) have distance 2
(0,0,1) and (1,1,0) have distance 3
Richard W. Hamming (1915-1998)
107
Hamming distance 1
QuickTime™ et undécompresseur TIFF (non compressé)
sont requis pour visionner cette image.
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Hamming’s unit sphere
• The unit sphere around a word includes the words at distance at most 1
109
At most one error
QuickTime™ et undécompresseur TIFF (non compressé)
sont requis pour visionner cette image.
110
Words at distance at least 3
111
Decoding
112
• The set of words with three letters (eight elements) splits into two balls
• The centers are (0,0,0) and (1,1,1)
• Each of the two balls consists of elements at distance at most 1 from the center
The code (0 0 0) (1 1 1)
113
(0,0,1)
(0,0,0) (0,1,0)
(1,0,0)
(1,1,0)
(1,1,1) (1,0,1)
(0,1,1)
Two or three 0 Two or three 1
114
Generalization of the check bitfor correcting one error
The check bit allows to detect one error by adding one bit.
Is-it possible to extend this idea in order to correct one error?
Answer by Hamming in 1950: YES!
115
Hamming’s code
116
ab
d
c
e=a+b+d
g=a+b+c
f=a+c+d
How to compute e , f , g from a , b , c , d.
117
Hamming code
Words of 7 letters Code words: (16=24 among 128=27)
(a b c d e f g)with
e=a+b+d f=a+c+d g=a+b+c
Rate: 4/7
118
16 code words of 7 letters
0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0 0
1 0 0 0 1 1 1 1 0 0 1 0 0 1 1 0 1 0 1 0 0 1 0 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1
Two distinct code words have at least three distinct letters
119
The binary code of Hamming (1950)
It is a linear code (the sum of two code words is a code word) and the 16 balls of radius 1 with centers in the code words cover all the space of the 128 binary words of length 7
(each word has 7 neighbors (7+1)16= 256).
120
Playing cards
121
7 questions to find the selected card among 16, with one possible wrong answer
Replace the cards by labels from 0 to 15 and write the binary expansions of these:
0000, 0001, 0010, 0011
0100, 0101, 0110, 0111
1000, 1001, 1010, 1011
1100, 1101, 1110, 1111
Using the Hamming code, get 7 digits.
Select the questions so that Yes=0 and No=1
122
7 questions to find the selected number in {0,1,2,…,15} with one possible wrong answer
• Is the first binary digit 1?
• Is the second binary digit 1?
• Is the third binary digit 1?
• Is the fourth binary digit 1?
• Is the number in {1,2,4,7,9,10,12,15}?
• Is the number in {1,2,5,6,8,11,12,15}?
• Is the number in {1,3,4,6,8,10,13,15}?
The Hat Problem
124
The Hat Problem• Three people are in a room, each has a hat on
his head, the color of which is black or white. Hat colors are chosen randomly. Everybody sees the color of the hat on everyone’s head, but not on their own. People do not communicate with each other.
• Everyone gets to guess (by writing on a piece of paper) the color of their hat. They may write: Black/White/Abstain.
125
• The people in the room win together or lose together.
• The team wins if at least one of the three people did not abstain, and everyone who did not abstain guessed the color of their hat correctly.
• How will this team decide a good strategy with a high probability of winning?
126
• Simple strategy: they agree that two of them abstain and the other guesses randomly.
• Probability of winning: 1/2.
• Is it possible to do better?
127
• Hint:
Improve the odds by using the available information: everybody sees the color of the hat on everyone’s head but himself.
128
• Better strategy: if a member sees two different colors, he abstains. If he sees the same color twice, he guesses that his hat has the other color.
129
The two people with white hats see one white hat and one black hat, so they abstain.
The one with a black hat sees two white hats, so he writes black.
They win!
130
The two people with black hats see one white hat and one black hat, so they abstain.
The one with a white hat sees two black hats, so he writes white.
They win!
131
They lose
Everybody sees two white hats, and therefore writes black on the paper.
132
They lose
Everybody sees two black hats, and therefore writes white on the paper.
133
Winning:
two whiteor
two black
134
Losing:
three whiteor
three black
135
• The team wins exactly when the three hats do not have all the same color, that is in 6 cases out of a total of 8
• Probability of winning: 3/4.
136
Connection with error detecting codes
• Replace white by 0 and black by 1 hence the distribution of colors becomes a
word of three letters on the alphabet {0 , 1}• Consider the centers of the balls (0,0,0) and
(1,1,1).• The team bets that the distribution of colors
is not one of the two centers.
137
Assume the distribution of hats does not correspond to one of the centers (0, 0, 0) and (1, 1, 1). Then:
• One color occurs exactly twice (the word has both digits 0 and 1).
• Exactly one member of the team sees twice the same color: this corresponds to 0 0 in case he sees two white hats, 1 1 in case he sees two black hats.
• Hence he knows the center of the ball: (0 , 0 , 0) in the first case, (1, 1, 1) in the second case.
• He bets the missing digit does not yield the center.
138
• The two others see two different colors, hence they do not know the center of the ball. They abstain.
• Therefore the team wins when the distribution of colors does not correspond to one of the centers of the balls.
• This is why the team wins in 6 cases.
139
• Now if the word corresponding to the distribution of the hats is one of the centers, all members of the team bet the wrong answer!
• They lose in 2 cases.
140
Hat problem with 7 people
For 7 people in the room in place of 3,which is the best strategy
and its probability of winning?
Answer: the best strategy gives a
probability of winning of 7/8
141
The Hat Problem with 7 people
• The team bets that the distribution of the hats does not correspond to the 16 elements of the Hamming code
• Loses in 16 cases (they all fail)
• Wins in 128-16=112 cases (one bets correctly, the 6 others abstain)
• Probability of winning: 112/128=7/8
142
SPORT TOTO: the oldest error correcting code
• A match between two players (or teams) may give three possible results: either player 1 wins, or player 2 wins, or else there is a draw.
• There is a lottery, and a winning ticket needs to have at least three correct bets. How many tickets should one buy to be sure to win?
143
4 matches, 3 correct forecasts
• For 4 matches, there are 34 = 81 possibilities. • A bet on 4 matches is a sequence of 4 symbols
{1, 2, x}. Each such ticket has exactly 3 correct answers 8 times.
• Hence each ticket is winning in 9 cases. • Therefore a minimum of 9 tickets is required to be
sure to win.
144
9 tickets (in columns)
x x x 1 1 1 2 2 2
x 1 2 x 1 2 x 1 2
x 1 2 1 2 x 2 x 1
x 1 2 2 x 1 1 2 x
This is an error correcting code on the alphabet{1,2,x} with rate 1/2
145
Sphere Packing
• While Shannon and Hamming were working on information transmission in the States, John Leech invented similar codes while working on Group Theory at Cambridge. This research included work on the sphere packing problem and culminated in the remarkable, 24-dimensional Leech lattice, the study of which was a key element in the programme to understand and classify finite symmetry groups.
146
Sphere packing
The kissing number is 12
147
Sphere Packing
• Kepler Problem: maximal density of a packing of identical sphères :
/ 18= 0.740 480 49…
Conjectured in 1611.
Proved in 1999 by Thomas Hales.
• Connections with crystallography.
148
Current trends In the past two years the goal of finding explicit codes
which reach the limits predicted by Shannon's original work has been achieved. The constructions require techniques from a surprisingly wide range of pure mathematics: linear algebra, the theory of fields and algebraic geometry all play a vital role. Not only has coding theory helped to solve problems of vital importance in the world outside mathematics, it has enriched other branches of mathematics, with new problems as well as new solutions.
149
Directions of research
• Theoretical questions of existence of specific codes
• connection with cryptography
• lattices and combinatoric designs
• algebraic geometry over finite fields
• equations over finite fields
Some recent results in mathematics related to data transmission:
Michel Waldschmidt
Université P. et M. Curie - Paris VI
Centre International de Mathématiques Pures et Appliquées - CIMPA
http://www.math.jussieu.fr/~miw/
India, October-November 2007