New PDF release: Codes on Algebraic Curves

By Serguei A. Stepanov

ISBN-10: 1461371678

ISBN-13: 9781461371670

ISBN-10: 1461547857

ISBN-13: 9781461547853

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 easy techniques and evidence of the speculation of error-correcting codes (Part I). the second one is a whole presentation of the speculation 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 relief modulo a chief quantity (Part III). The fourth (and easy) is the development of geometric Goppa codes and the creation 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 idea of algebraic (specifically modular) curves over finite fields and the very concrete difficulties within the engineering of knowledge transmission. today there are primarily alternative ways to provide asymptotically solid codes coming from algebraic curves over a finite box with a really huge variety of rational issues. the 1st approach, built via M. A. Tsfasman, S. G. Vladut and Th. Zink [210], is very tough and assumes a major acquaintance with the speculation of modular curves and their aid modulo a primary quantity. the second one method, proposed lately by means of A.

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Check the following facts: (a) On the segment [0, (q -l)/q] the curveRGv(c5) is differentiable (of class COO) and convex; (b) RGv(O) = 1, RGv«q - l)/q) equality = 0 and for c5 -+ 0 there is the asymptotical RGv(c5) = 1 +c51o~ c5+o(c51ogq c5). In particular, the tangent at c5 = 0 is vertical; (c) For c5 -+ (q - 1)/q there is the asymptotical equality -1) = 2(q-1) logq W +o(w). q RGV ( -q--W q2 2 2 The tangent at (q - 1) I q is horizontal and the tangent order is two. 38 Chapter 2 (d) Tangents to RGv( a) are of the fonn Rt(a) = 1- (lo~(l +q -1) -t) -ta, Each Rt(a) is tangent toRGv(a) at the point q-l c50 = qt +q-l and RGv(c50)=I+ tqt t l-logq(l+q-l).

So, we are unable to solve even the following problems: Problem l Are the functions CYq(8) and cy~n (8) differentiable in the interval (0, (q -l)/q)? Problem Il Are these functions convex? Problem III Is it true that cy~n (8) = aq (8), or not? Therefore we are constrained to search some upper and lower bounds for the functions CYq ( 8) and cy~n(8) as close to each other as possible. For the set of all [n,k,d]q-codes overFq we define the functionAq(n,d) by Aq(n,d) = max {l/there exists an [n,k,d]q - code}.

Therefore the number of zeros of j in :P is at most (qm - 1) j (q - 1). The maximum cardinality of P is (qm+l - l)j(q - 1) (take for example all non-zero elements of F:;+l such that j(Yi) their first non-zero coordinate is 1). We obtain an [n,m + 1,n - q;_ll]q-code for n S; (qm + 1 - 1) j (q - 1), in particular a code C with parameters qm+l -1 ] [ q _I,m + l,qm q This is a very good code which lies on the Plotkin bound. 2) that the weight enumerator of the code C has the fonn 47 Examples and Constructions Hamming Codes The Hamming single-error-correcting codes form an important family of linear codes which are easy to encode and decode.

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Codes on Algebraic Curves by Serguei A. Stepanov

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