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

By Judy L. Walker

Whilst details is transmitted, error tend to take place. This challenge has turn into more and more very important as super quantities of data are transferred electronically on a daily basis. Coding thought examines effective methods of packaging facts in order that those blunders may be detected, or maybe corrected.
The conventional instruments of coding idea have come from combinatorics and team conception. because the paintings of Goppa within the past due Seventies, notwithstanding, coding theorists have additional strategies from algebraic geometry to their toolboxes. specifically, by means of re-interpreting the Reed-Solomon codes as coming from comparing features linked to divisors at the projective line, you could see how to find new codes in keeping with different divisors or on different algebraic curves. for example, utilizing modular curves over finite fields, Tsfasman, Vladut, and Zink confirmed that you may outline a chain of codes with asymptotically larger parameters than any formerly recognized codes.
This booklet relies on a chain 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 thought. offering the cloth within the related 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 consciousness 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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Codes and Curves (Student Mathematical Library, Volume 7)

While details is transmitted, blunders are inclined to ensue. This challenge has turn into more and more very important as large quantities of knowledge are transferred electronically on a daily basis. Coding idea examines effective methods of packaging facts in order that those error might be detected, or maybe corrected.
The conventional instruments of coding conception have come from combinatorics and staff concept. because the paintings of Goppa within the past due Seventies, despite the fact that, coding theorists have extra suggestions from algebraic geometry to their toolboxes. specifically, via re-interpreting the Reed-Solomon codes as coming from comparing capabilities linked to divisors at the projective line, one could see how to find new codes in keeping with different divisors or on different algebraic curves. for example, utilizing modular curves over finite fields, Tsfasman, Vladut, and Zink confirmed that you can actually outline a series of codes with asymptotically higher parameters than any formerly identified codes.
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Extra resources for Codes and Curves (Student Mathematical Library, Volume 7)

Example text

6. Let i = −1 and set Q[i] = {a + bi | a, b ∈ Q}. Show that Q[i] is a ring under normal addition and multiplication of complex numbers. What is the “0”? What is the “1”? Is this ring commutative? What are the units of this ring? 7. A field is a ring which satisfies two additional properties: • Commutativity of Multiplication: For all a, b ∈ R, ab = ba. • Existence of Multiplicative Inverses: For all a ∈ R \ {0} there is a b ∈ R \ {0} such that ab = 1 = ba. Some familiar examples of fields are: Q, R (the reals), C (the complex numbers), Z/pZ (the integers modulo p, where p is prime), Q(x) (quotients of polynomials with rational coefficients).

4. Let X be a nonsingular, projective plane curve of genus g, defined over the field Fq . Let P ⊂ X(Fq ) be a set of n distinct Fq -rational points on X, and let D be a divisor on X satisfying 2g − 2 < deg D < n. Then the algebraic geometric code C := C(X, P, D) is linear of length n, dimension k := deg D + 1 − g, and minimum distance d, where d ≥ n − deg D. Proof. We’ve already shown that C is linear of length n and dimension dim L(D), since deg D < n. That dim L(D) = deg D + 1 − g is exactly the statement of the Riemann-Roch Theorem, since deg D > 2g − 2.

A singular point of Cf is a point (x0 , y0 ) ∈ k¯ × k¯ such that f (x0 , y0 ) = 0 and fx (x0 , y0 ) = 0 and fy (x0 , y0 ) = 0. The curve Cf is nonsingular if it has no singular points. , if F (X0 , Y0 , Z0 ) = FX (X0 , Y0 , Z0 ) = FY (X0 , Y0 , Z0 ) = FZ (X0 , Y0 , Z0 ) = 0. The curve Cf is nonsingular if it has no singular points. 2. Let f (x, y) ∈ R[x, y] and suppose (0, 0) is a nonsingular point on Cf . If fy (0, 0) = 0, show that the line y = mx, where m = fx (0, 0)/fy (0, 0), is the tangent line to Cf at (0, 0).

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