Some recent results in mathematics related to data transmission: Michel Waldschmidt Université P....

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

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

India October- November 2007

Some recent results in mathematics related to data transmission

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.

Some recent results in mathematics related to data transmission

India November 2007

Mathematical aspects of Coding Theory in France:

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''.

The most important are:

INRIA RocquencourtUniversité de Bordeaux ENST Télécom BretagneUniversité de Limoges Université de MarseilleUniversité de Toulon

Université de Toulouse

Bordeaux

MarseilleToulouse

Limoges

Toulon

Institut National de Recherche en Informatique et en Automatique

http://www-rocq.inria.fr/codes/

National Research

Institute in Computer

Science and Automatic

Institut de Mathématiques de Bordeaux

http://www.math.u-bordeaux1.fr/maths/

Lattices and combinatorics

École Nationale Supérieure des Télécommunications de Bretagne

http://departements.enst-bretagne.fr/sc/recherche/turbo/

Turbocodes

Research Laboratory of LIMOGES

http://www.xlim.fr/

Marseille: Institut de Mathématiques de Luminy

Arithmetic and Information TheoryAlgebraic geometry over finite fields

Université du Sud Toulon-Var

http://grim.univ-tln.fr/

Boolean functions

Université de ToulouseLe Mirail

http://www.univ-tlse2.fr/grimm/algo

Algebraic geometry over finite fields

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/

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/

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

Bangalore

Kanpur

Bombay

Chandigarh

IIT BombayIndian Institute of Technology

• Department of Mathematics • Department of Electrical Engineering

http://www.iitb.ac.in/

• 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

IIT KanpurIndian Institute of Technology

http://www.iitk.ac.in/

http://www.puchd.ac.in/

Department of Mathematics

http://www.du.ac.in/

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/

Playing cards

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.

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.

2 cards: one question suffices

• Question: is-it this one?

4 cards

First question: is-it one of these two?

Second question: is-it one of these two ?

4 cards: 2 questions suffice

Y Y Y N

N Y N N

8 Cards

First question: is-it one of these?

Second question: is-it one of these?

Third question: is-it one of these?

8 Cards: 3 questions

YYY YYN YNY YNN

NYY NYN NNY NNN

Yes / No

• 0 / 1

• Yin — / Yang - -

• True / False

• White / Black

• + / -

• Heads / Tails (tossing a coin)

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

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.

Label the 16 cards

0 1 32

4 5 76

8 9 1110

12 13 1514

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

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

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!

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?

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.

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!).

4 cards

First question: is-it one of these two?

Second question: is-it one of these two?

Third question: is-it one of these two?

4 cards: 3 questions

Y Y Y Y N N

N Y N N N Y

4 cards: 3 questions

0 0 0 0 1 1

1 0 1 1 1 0

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

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

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.

8 Cards

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

First question: is-it one of these?

Second question: is-it one of these?

Third question: is-it one of these?

Fourth question: is-it one of these?

16 cards, at most one wrong answer:

5 questions to detect the mistake

Ask the 5 questions so that the answers are:

YYYYY YYYNN YYNNYYYNYN

YNYYN YNYNY YNNNNYNNYY

NYYYN NYYNY NYNNNNYNYY

NNYYY NNYNN NNNNYNNNYN

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.

With 2 cards

• I repeat the same question three times.

• The most frequent answer is the right one: vote with the majority.

With 4 cards

I repeat my two questions three times each, which makes 6 questions

With 8 Cards

I repeat 3 timesmy 3 questions,which makes 9 questions

With 16 cards, this process requires

34=12 questions

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

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!

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.

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.

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! "

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.

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.

Error Correcting Codes Data Transmission

• Telephone

• CD or DVD

• Image transmission

• Sending information through the Internet

• Radio control of satellites

Applications of error correcting codes

• Transmitions by satellites

• Compact discs

• Cellular phones

Olympus Mons on Mars Planet

Image from Mariner 2 in 1971.

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.

• 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.

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.

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

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.

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.

sont requis pour visionner cette image.

Sojourner rover and Mars Pathfinder lander

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.

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.

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.

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

Codes and Mathematics

• Algebra (discrete mathematics

finite fields, linear algebra,…)

• Geometry

• Probability and statistics

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)

Coding Theory

transmission

Source Coded Text

Coded Text

Receiver

Error correcting codes

Transmission

with noise

Source Coded Text

Coded Text

Receiver

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.

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)

Rate: 1/2

Principle of codes detecting one error:

Two distinct code words have at least two distinct letters

Detecting one error with the parity bit

Code words (three letters):

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

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.

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.

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)

Rate: 1/3

• Correct 0 0 1 as 0 0 0

• 0 1 0 as 0 0 0

• 1 0 0 as 0 0 0

• 1 1 0 as 1 1 1

• 1 0 1 as 1 1 1

• 0 1 1 as 1 1 1

Principle of codes correcting one error: Two distinct code words have at least

three distinct letters

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

Richard W. Hamming (1915-1998)

Hamming distance 1

Hamming’s unit sphere

• The unit sphere around a word includes the words at distance at most 1

At most one error

Words at distance at least 3

Decoding

• 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)

(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

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!

Hamming’s code

e=a+b+d

g=a+b+c

f=a+c+d

How to compute e , f , g from a , b , c , d.

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

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

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).

Playing cards

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

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

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.

• 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?

• 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?

• Hint:

Improve the odds by using the available information: everybody sees the color of the hat on everyone’s head but himself.

• 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.

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!

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!

They lose

Everybody sees two white hats, and therefore writes black on the paper.

They lose

Everybody sees two black hats, and therefore writes white on the paper.

Winning:

two whiteor

two black

Losing:

three whiteor

three black

• 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.

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.

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.

• 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.

• 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.

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

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

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?

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.

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

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.

Sphere packing

The kissing number is 12

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.

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.

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

India, October-November 2007

Some recent results in mathematics related to data transmission: Michel Waldschmidt Université P....

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