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 booklet. the 1st is a quick exposition of easy ideas and evidence of the speculation of error-correcting codes (Part I). the second one is a whole presentation of the idea of algebraic curves, particularly 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 main quantity (Part III). The fourth (and uncomplicated) is the development of geometric Goppa codes and the construction of asymptotically reliable 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 primarily other ways to provide asymptotically stable codes coming from algebraic curves over a finite box with a really huge variety of rational issues. the 1st approach, constructed by way of M. A. Tsfasman, S. G. Vladut and Th. Zink [210], is quite tricky and assumes a major acquaintance with the idea of modular curves and their relief modulo a chief quantity. the second one approach, proposed lately via A.

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The image of the evaluation map is an [n,m + 1,n - qm-l]q-code with n S; qm, which is called the Reed-Muller code oj the first order. This construction can be easily generalized as follows. Consider all homogeneous linear fonns in (m + 1) variables. Together with zero they fonn a linear space L~ of dimension (m + 1) over Fq. Let:P = {Yl, . ,Yn} C F:;+ 1 be such that Yi i= (0, ... ,0),1 S; i S; n, andYi E P implies aYi I$:P for every a E F; \ {I}. Consider again the evaluation map ° ° ° A non-zero fonnj E L~ has at most qm zeros in F:;+l (recall that 1$ :P and if = for Yi E P then alsoj(aYi) = for all a i= 0, 1).

D-n,0- < M' < - q we obtain the required result. 7. The volume o/a ball Bt(x) o/radius t in F; equals Proof: The ball Bt (x) is the union of disjoint spheres Si(X) We have ISi(x) I = = {Y E F; Illy -x 11= i}, 0:::; i :::; t. (~) (q -' I)i and hence This gives us the desired result. • The following result is an easy consequence of the fact that the balls Bt(x) of radius t :::; (d - 1)/2 centered at code-vectors of an [n,k,d]q-code C provide a sphere packing of the space F; . 8 (the Hamming bound).

Q -' I)i and hence This gives us the desired result. • The following result is an easy consequence of the fact that the balls Bt(x) of radius t :::; (d - 1)/2 centered at code-vectors of an [n,k,d]q-code C provide a sphere packing of the space F; . 8 (the Hamming bound). 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.

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