Codes and Curves (Student Mathematical Library, Volume 7) by Judy L. Walker

By Judy L. Walker

While info is transmitted, error are inclined to ensue. This challenge has turn into more and more very important as super quantities of knowledge are transferred electronically each day. Coding thought examines effective methods of packaging info in order that those error should be detected, or perhaps corrected.
The conventional instruments of coding conception have come from combinatorics and crew concept. because the paintings of Goppa within the past due Nineteen Seventies, although, coding theorists have extra concepts from algebraic geometry to their toolboxes. particularly, by way of re-interpreting the Reed-Solomon codes as coming from comparing features linked to divisors at the projective line, it is easy to see how to find new codes according to different divisors or on different algebraic curves. for example, utilizing modular curves over finite fields, Tsfasman, Vladut, and Zink confirmed that you can outline a chain of codes with asymptotically higher parameters than any formerly identified codes.
This e-book relies on a sequence of lectures the writer gave as a part of the IAS/Park urban arithmetic Institute (Utah) application on mathematics algebraic geometry. right here, the reader is brought to the fascinating box of algebraic geometric coding concept. providing the fabric within the similar conversational tone of the lectures, the writer covers linear codes, together with cyclic codes, and either bounds and asymptotic bounds at the parameters of codes. Algebraic geometry is brought, with specific recognition given to projective curves, rational features and divisors. the development of algebraic geometric codes is given, and the Tsfasman-Vladut-Zink end result pointed out above is mentioned.

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Extra resources for Codes and Curves (Student Mathematical Library, Volume 7)

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If fy (0, 0) = 0, show that the line x = 0 is the tangent line to Cf at (0, 0). In general, if P is a nonsingular point on Cf , then the line through P with slope fx (P )/fy (P ) is the tangent line to Cf at P . If fy (P ) = 0, the tangent line is the vertical line through P . 2 proves this (after a change of coordinates). 3. 1 is to make sense, one would expect that if Cf is nonsingular then the only possible singular points of Cf are at infinity. This is true, and follows from the definition of the homogenization of f and the chain rule for partial derivatives.

One way of thinking about the fact that we want both R and δ large while acknowledging that there is a trade-off between these values is to say that we want R + δ large. In our situation, we have R+δ ≥ n − deg D n+1−g deg D + 1 − g + = = 1 + 1/n − g/n. n n n 41 42 7. Good Codes from Algebraic Geometry For long codes, we consider the limit as n gets large. This means we consider a sequence of algebraic geometric codes of increasing length. To construct these codes, we need a sequence of curves Xi of genus gi , a set of ni rational points on Xi , and a chosen divisor Di on Xi .

Riemann-Roch Theorem) Let C be a nonsingular projective plane curve of genus g defined over the field Fq and let D be a divisor on X. Then dim L(D) ≥ deg D + 1 − g. Further, if 36 5. Points, Functions, and Divisors on Curves deg D > 2g − 2, then dim L(D) = deg D + 1 − g. Let us return one final time to our ongoing example. We have the curve C0 defined over F3 by the equation Y 2 Z − X 3 − 2XZ 2 − 2Z 3 . Recall that Q1 is the point {(0 : α : 1), (0 : 2α : 1)} of degree 2 on C0 , where α2 + 1 = 0.

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