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 Some recent results in mathematics related to data transmission 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. http://www.math.jussieu.fr/~miw/
French Science Today India 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. http://www.math.jussieu.fr/~miw/
French Science Today 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''. http://www.math.jussieu.fr/~miw/
French Science Today The most important are: INRIA Rocquencourt Université de Bordeaux ENST Télécom Bretagne Université de Limoges Université de Marseille Université de Toulon Université de Toulouse http://www.math.jussieu.fr/~miw/
INRIA Brest Limoges Bordeaux Marseille Toulon Toulouse
http://www-rocq.inria.fr/codes/ Institut National de Recherche en Informatique et en Automatique National Research Institute in Computer Science and Automatic
http://www.math.u-bordeaux1.fr/maths/ Institut de Mathématiques de Bordeaux Lattices and combinatorics
http://departements.enst-bretagne.fr/sc/recherche/turbo/ École Nationale Supérieure des Télécommunications de Bretagne Turbocodes
http://www.xlim.fr/ Research Laboratory of LIMOGES
Marseille: Institut de Mathématiques de Luminy Arithmetic and Information Theory Algebraic geometry over finite fields
http://grim.univ-tln.fr/ Université du Sud Toulon-Var Boolean functions
Université de Toulouse Le Mirail Algebraic geometry over finite fields http://www.univ-tlse2.fr/grimm/algo
http://www.gdr-im.fr/ 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)
French Science Today 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
Chandigarh Delhi Kanpur Bombay Bangalore
http://www.iitb.ac.in/ IIT BombayIndian Institute of Technology • Department of Mathematics • Department of Electrical Engineering
http://www.iisc.ernet.in/ • Finite fields and Coding Theory classification of permutation polynomials, study of PAPR of families of codes, construction of codes with low PAPR. • Department of Mathematics peak-to-average power
http://www.iitk.ac.in/ IIT KanpurIndian Institute of Technology
http://www.puchd.ac.in/ Department of Mathematics
http://www.du.ac.in/ Department of Mathematics
http://www.ias.ac.in/resonance/ Error Correcting Codesby Priti Shankar • How Numbers Protect Themselves • The Hamming Codes Volume 2 Number 1 • Reed Solomon Codes Volume 2 Number 3
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: 2 questions suffice Y Y Y N N Y N N
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 or flipping a coin)
0 0 0 0 0 0 1 1 0 1 0 2 0 1 1 3 1 0 0 4 1 0 1 5 1 1 0 6 1 1 1 7 3 questions, 8 solutions
Exponential law Add one question = multiply the number of cards by 2
16 Cards 4 questions • 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.
12 0 4 8 13 1 5 9 10 14 6 2 11 15 7 3 Label the 16 cards
0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 0 0 1 0 0 1 0 1 1 1 0 0 1 1 0 1 0 1 0 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 1 Binary representation:
Y Y Y Y N N Y Y Y N Y Y N Y Y Y Y Y Y N N N Y N Y N Y N N Y Y N Y Y N Y N N N Y Y N N Y N Y N Y N N N N N Y N N Y N N N Y Y N N Ask the questions so that the answers are:
0000 1100 0100 1000 1101 0001 0101 1001 0110 1010 1110 0010 0111 1111 0011 1011 The 4 questions: • Is the first digit 0 ? • Is the second digit 0? • Is the third digit 0? • Is the fourth digit 0?
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!).