Error-Correcting Codes and Curves over Finite Fields
It is often asserted that when the conditions of society permit the development of mathematics, it is autonomous, and the simultaneous but independent discovery of new mathematical notions is cited as evidence for this point of view. But against this view
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1. Introduction It is often asserted that when the conditions of society permit the development of mathematics, it is autonomous, and the simultaneous but independent discovery of new mathematical notions is cited as evidence for this point of view. But against this view one could point to white spots which remained for a long time on the charts of mathematics and which disappeared only after events outside (pure) mathematics spurred an exploration of the uncharted waters and ,stimulated a development which eventually coloured the white spot. An example of such a white spot is the question, how many points a curve over a finite field can have. It seems that this question in pure mathematics was not only neglected for a long time, but that at least twice interest disappeared once it had been there and it was eventually brought back only by a stimulus from applied mathematics. We could start our story, as is customary for such stories, with Gauss. He studied in the early years of the 19th century congruences of the form f(x , y)
== 0 (mod p)
for a prime p and a polynomial f and found the number of solutions. We now interpret these solutions as points on a curve defined over the finite field F p with p elements. This interpretation involves two concepts: that of an algebraic curve and that of a finite field . The notion of a finite field is due to Galois (1830), but although this notion is much simpler than such a complicated notion as the real numbers, it took a long time before finite fields were accepted, let alone treated on an equal footing with the real or complex numbers. The notion of a curve dates back much further and curves played an important role in the mathematics of the 19th century. But although a purely algebraic approach to algebraic curves was advocated by Kronecker, Dedekind and others, it took many years before there was interest in curves over finite fields and more than a century elapsed after Gauss before mathematicians returned to the problem of counting the number of points on curves defined over finite fields. After Artin advanced in 1924 his analogue of the Riemann hypothesis for curves defined over finite fields it drew quite some attention and Artin's conjecture was proved within 16 years by Wei!. In the years thereafter curves over finite fields seemed an almost dead and forgotten topic, until Goppa discovered in 1977 an unexpected link with error correcting codes and thus brought it back to life. B. Engquist et al. (eds.), Mathematics Unlimited — 2001 and Beyond © Springer-Verlag Berlin Heidelberg 2001
1116
G. VAN DER GEER
This relation between two different fields has been fruitful for both of them, resulting on the one hand in the construction of codes with the help of algebraic curves that beat earlier records, and on the other hand in bringing back interest in curves over finite fields and leading among other things to the construction of towers of curves over finite fields with many points. Applications of curves over finite fields to cryptography have now added to the renewe
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