By Serguei A. Stepanov
This is a self-contained creation to algebraic curves over finite fields and geometric Goppa codes. There are 4 major divisions within the ebook. the 1st is a quick exposition of uncomplicated innovations and evidence of the speculation of error-correcting codes (Part I). the second one is a whole presentation of the idea of algebraic curves, specially the curves outlined over finite fields (Part II). The 3rd is a close description of the speculation of classical modular curves and their aid modulo a primary quantity (Part III). The fourth (and easy) is the development of geometric Goppa codes and the construction of asymptotically stable linear codes coming from algebraic curves over finite fields (Part IV). the speculation of geometric Goppa codes is an engaging subject the place extremes meet: the hugely summary and deep idea of algebraic (specifically modular) curves over finite fields and the very concrete difficulties within the engineering of data transmission. today there are basically alternative ways to provide asymptotically reliable codes coming from algebraic curves over a finite box with an incredibly huge variety of rational issues. the 1st manner, built via M. A. Tsfasman, S. G. Vladut and Th. Zink [210], is quite tough and assumes a major acquaintance with the speculation of modular curves and their aid modulo a major quantity. the second one method, proposed lately through A.
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Extra info for Codes on Algebraic Curves
Example text
Yn) E F; we start with a calculation of syndromes: F; n Sj =Sj(y) = LYi Prove that Krawtchouk polynomials have the following properties: (a) Pi(U)=L]=O(-qy(q-l)i-J(~=j)0); (b) Pi(U) =L]=o(-I Yi/-J (n-;+j) (~=;); (c) Pi(U) is polynomial of degree i in u, with leading coefficient (-q)iji! and constant tenn (~) (q - l)i; Bounds on Codes 39 (d) Orthogonality relations: Ita G) (q - I)' Pi (l)Pj(l) = qn(q - l)i (~) aij; (e) (q-l)'G)Pi(l) = (q-l)iG)P,(i); (t) Il=oPi(l)P,(j) = qnaij; (g) Recurrence: (i + 1)Pi+ 1(u) = (( n - i) (q - 1) + i - qu )Pi (u) -(q-l)(n-i+l)Pi-l, Po = 1,Pl (u) ={q - l)n - qu; (h) Iff(u) is a polynomial of degree t and t feu) = I aiPi(U), i=O then n ai = q-n If(j)Pj(i). Consider a linear space Lm (r) of all polynomials of degree at most r in m variables over Fq • Fix a subset P = {YI, ... ,Yn} ~ F:; and consider the evaluation map: Ev : Lm(r) -+F;, fH- (f(yJ), ... ,J(yn)). Set C = Ev(Lm(r)). Finding out the parameters ofC is a rather difficult problem. For simplicity let us suppose that P = F:;, n = qm. The map Ev in general is not injective (in fact Ev(f) = Ev(fq) for everyf). Let L~ (r) be the space spanned by monomials of the form Ufl ... u~,m, 0 ::; Otj ::; q - I, L Otj = r.