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 publication. the 1st is a short exposition of uncomplicated recommendations and proof 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 idea of classical modular curves and their relief modulo a main quantity (Part III). The fourth (and simple) 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 interesting subject the place extremes meet: the hugely summary and deep concept of algebraic (specifically modular) curves over finite fields and the very concrete difficulties within the engineering of data transmission. today there are basically other ways to supply asymptotically solid codes coming from algebraic curves over a finite box with an exceptionally huge variety of rational issues. the 1st means, constructed by way of M. A. Tsfasman, S. G. Vladut and Th. Zink , is quite tricky and assumes a significant acquaintance with the speculation of modular curves and their relief modulo a chief quantity. the second one method, proposed lately by way of A.
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Additional resources for Codes on Algebraic Curves
Xl, ... ,Xn) E C yields (X2,'" ,Xn ,Xl) E C. Note that cyclicity is not an invariant of the equivalence class of codes. From now on we make the convention (n,q) = 1. To describe cyclic codes algebraically we observe that F; as vector space is isomorphic to the ring Rn = Fq[u]/(u n -1), if only we ignore the multiplication in this ring. We now identify the vector (Xl, ... ,Xn) E F; with the corresponding polynomial Xl + X2U + ... + xnu n- l . Observe that multiplication by u now is nothing but a cyclic shift of the vector.
3. IfC is [n,k,d]q-code ofgenus zero then and for i =1= 0 Ai= (~X~(-IYG)(qi-d-j+I-I) = (~)(q-I)~~(-IYCj l)qi-d-j . Finally we describe a decoding procedure of Reed-Solomon codes. Consider [n, n - m - 1, m + 2]q-code C dual to a Reed-Solomon code of degree m. Recall that decoding up to t = l d 21 J means an algorithm that makes it possible, starting with some y E which is at most at distance t from some code-vector x E C, to find this x. Let e = (el, ... , en) = y - x, II e II::; t, be the error-vector.
Ii) Let us "correct" each non-linear [n,k,dlq-code, by crossing out at most n-:zl qk code vectors. Then almost every code can be corrected so that the parameters of the resulting code lie on the curve Rov (8). , such correction does not change asymptotic parameters. The asymptotic Gilbert-Varshamov bound was not improved until recently. The algebraic-geometric bound that we shall find later on is Bounds on Codes 37 where q ~ 49 and q is square (see Fig. 7). 1. lolL n-+oo n ""I (f (~)(q_1)i) i=O I =Hq(c5).