Get Codes and Curves (Student Mathematical Library, Vol. 7) PDF

By Judy L. Walker

ISBN-10: 082182628X

ISBN-13: 9780821826287

While details is transmitted, blunders tend to ensue. This challenge has develop into more and more very important as super quantities of knowledge are transferred electronically on a daily basis. Coding conception examines effective methods of packaging facts in order that those error could be detected, or perhaps corrected. the normal instruments of coding concept have come from combinatorics and staff concept. because the paintings of Goppa within the past due Nineteen Seventies, even though, coding theorists have further recommendations 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, you'll see how to find new codes in accordance with different divisors or on different algebraic curves. for example, utilizing modular curves over finite fields, Tsfasman, Vladut, and Zink confirmed that it is easy to outline a chain of codes with asymptotically larger parameters than any formerly identified codes. This ebook relies on a chain of lectures the writer gave as a part of the IAS/Park urban arithmetic Institute (Utah) software on mathematics algebraic geometry. right here, the reader is brought to the intriguing box of algebraic geometric coding concept. providing the fabric within the comparable 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 realization given to projective curves, rational features and divisors. the development of algebraic geometric codes is given, and the Tsfasman-Vladut-Zink outcome pointed out above is mentioned.

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Extra resources for Codes and Curves (Student Mathematical Library, Vol. 7) (Student Mathematical Library, V. 7.)

Example text

Recall that the projective plane was defined as P2 (Fq ) = (F3q \ {(0, 0, 0)})/ ∼, where (X0 , Y0 , Z0 ) ∼ (X1 , Y1 , Z1 ) if and only if there is some α ∈ k × with X1 = αX0 , Y1 = αY0 , and Z1 = αZ0 . In the same spirit, we have: 37 38 6. 1. The projective line P1 (Fq ) is defined to be (F2q \ {(0, 0)})/ ∼, where (X0 , Y0 ) ∼ (X1 , Y1 ) if and only if there is some α ∈ F× q with X1 = αX0 and Y1 = αY0 . Writing (X0 : Y0 ) for the equivalence class of the point (X0 , Y0 ), we have that P1 (Fq ) = {(α : 1) | α ∈ Fq } ∪ {(1 : 0)} We may think of P1 as the line defined by the equation Z = 0 in P2 .

A cyclic group is a group C which has an element a such that C = {ak | k ∈ Z}. In this case we write C = a . The order of a group is the number of elements it has. 21 below) there is only one cyclic group of order n for each positive integer n. We will use Cn to denote this group. 2. (Fundamental Theorem of Finite Abelian Groups) Let G be a finite abelian group. Then G can be written as a direct sum of cyclic groups. In fact, there are two canonical ways of doing this: • There are primes p1 , .

11) applies. Then it is ¯ q ) guaranteed not hard to see that the de points of intersection in P2 (F by Bezout’s theorem cluster into points of varying degrees over Fq , with the sum of those degrees being de. As an example of a curve with points of higher degree, let C0 be the projective plane curve over F3 corresponding to the affine equation y 2 = x3 + 2x + 2. 5. Check that C0 is nonsingular and show that it has genus 1. By plugging in the values 0, 1, 2 for x, we see that there are no F3 rational affine points on C.

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


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