By Serguei A. Stepanov

This is a self-contained advent to algebraic curves over finite fields and geometric Goppa codes. There are 4 major divisions within the booklet. the 1st is a quick exposition of simple options and evidence of the speculation of error-correcting codes (Part I). the second one is an entire presentation of the speculation of algebraic curves, specifically the curves outlined over finite fields (Part II). The 3rd is a close description of the speculation of classical modular curves and their relief modulo a chief quantity (Part III). The fourth (and uncomplicated) is the development of geometric Goppa codes and the construction of asymptotically solid linear codes coming from algebraic curves over finite fields (Part IV). the idea of geometric Goppa codes is an interesting subject the place extremes meet: the hugely summary and deep conception of algebraic (specifically modular) curves over finite fields and the very concrete difficulties within the engineering of data transmission. this present day there are primarily other ways to supply asymptotically solid codes coming from algebraic curves over a finite box with a very huge variety of rational issues. the 1st approach, built via M. A. Tsfasman, S. G. Vladut and Th. Zink [210], is quite tricky and assumes a significant acquaintance with the idea of modular curves and their aid modulo a primary quantity. the second one manner, proposed lately by means of A.

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**Example text**

Ynun-I. Examples and Constructions Call e(u) 51 = y(u) -x(u) = el + e2u + ... + enu n- 1 the error-vector. The set 1= is the set of positions where an error has occurred and we assume that the number of errors I = III does not exceed t. Define {i II :s; i :s; n, ei =1= O} a(z) = TI(I -(h), iEI the so called error-locator polynomial, and w(z) = LeiaiTI(I - ciz), iEI JEI Hi the error-evaluator polynomial. Clearly a(z) is a polynomial of degree I :s; t and w(z) has degree less than I. If we know these polynomials, then we know I (by factoring a(z)) and from w(z) we can then find the values ofthe ei by substituting z = a-i.

For an [n,k,d]q-code C we have F; Proof: Consider balls in of radius t = These balls do not intersect and hence l dZ-i J centered at the code-vectors x. 30 Chapter 2 • This proves the theorem. A code C that attains the Hamming bound is called perfect. 9. 10 (the asymptotic Hamming bound). llq(8) :::; RH(8) = I-Hq(8/2). 11. Let Aq(n,d, w) be the maximum possible number of vectors of weight w in F; such that distance between any two of them is at least d. Then for each w, 0 :::; w :::; n, we have qn Aq(n,d, w) ( d) Aq n, :::; ( ) .

9. The rational Goppa code C has parameters l]q. 2). Since the code is linear, we can consider the weight II x II of x = (XI, ... ,xn). If this is w then the degree of the numerator a(z) of the sum ±~ i=1 Z-Cii is w - 1 (in fact less if LXi more. Define = 0). So w - 1 is at least t. If q j(z) = = 2 we can say a lot n IT(Z-Cii)Xi, ;=1 so that n j'(z) Xi L - =j(z) -' i=IZ-Ci; Since all exponents inj'(z) are even, this is a perfect square. If we assume that g(z) has no multiple zeros, then the fact that g(z) dividesj' (z) implies that g2(z) divides j' (z ) .