By Judy L. Walker

Whilst info is transmitted, error are inclined to take place. This challenge has develop into more and more vital as super quantities of knowledge are transferred electronically each day. Coding concept examines effective methods of packaging info in order that those mistakes should be detected, or perhaps corrected. the normal instruments of coding conception have come from combinatorics and staff concept. because the paintings of Goppa within the overdue Nineteen Seventies, notwithstanding, coding theorists have extra thoughts 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, you can still 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 you can actually outline a series of codes with asymptotically higher parameters than any formerly recognized codes. This publication is predicated 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 interesting box of algebraic geometric coding conception. proposing 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 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 info for Codes and Curves (Student Mathematical Library, Vol. 7) (Student Mathematical Library, V. 7.)**

**Sample text**

B) What happens if k has characteristic 2? 2. Genus Topologically, every nonsingular curve over C can be realized as a surface in R3 . For example, an elliptic curve has an equation of the form y 2 = f (x), where f (x) is a cubic polynomial in x with no repeated roots, and can be thought of as a torus (a donut) in R3 . In general, every nonsingular curve can be realized as a torus with some number of holes, and that number of holes is called the topological genus of the curve. In particular, an elliptic curve has genus 1.

An ideal I of a (commutative) ring R is called principal if there is some a ∈ I such that I = {ar | r ∈ R}. In this case we write I = a or I = aR. Two examples of principal ideals are: the even integers (as an ideal of the integers) and the set of all polynomials f (x) ∈ Q[x] satisfying f (1) = 0 (this is (x − 1)Q[x]). An example of an ideal which is not principal is x, y := {xf (x, y)+yg(x, y) | f, g ∈ Q[x, y]} ⊆ Q[x, y]. 11. Let I be an ideal of the ring R. Show that I = R if and only if some unit of R is in I.

Letting α be the element of F9 corresponding to t, we have F9 = {a + bα | a, b ∈ F3 }, where α2 = −1 = 2. Some computations yield C0 (F9 ) = {(0 : α : 1), (0 : 2α : 1), (1 : α : 1), (1 : 2α : 1), (2 : α : 1), (2 : 2α : 1), P∞ }. The Frobenius σ3,2 : F9 → F9 satisfies σ3,2 (α) = α3 = α·α2 = 2α, so we see that C0 (F9 ) = Q1 ∪ Q2 ∪ Q3 ∪ {P∞ }, where Q1 = {(0 : α : 1), (0 : 2α : 1)}, Q2 = {(1 : α : 1), (1 : 2α : 1)}, and Q3 = {(2 : α : 1), (2 : 2α : 1)} are the only three points of degree two on C0 .