By Judy L. Walker

Whilst details is transmitted, blunders are inclined to happen. This challenge has turn into more and more very important as large quantities of knowledge are transferred electronically each day. Coding concept examines effective methods of packaging facts in order that those error might be detected, or perhaps corrected.

The conventional instruments of coding concept have come from combinatorics and staff conception. because the paintings of Goppa within the past due Nineteen Seventies, besides the fact that, coding theorists have extra suggestions from algebraic geometry to their toolboxes. particularly, through re-interpreting the Reed-Solomon codes as coming from comparing features linked to divisors at the projective line, you can still see how to find new codes in response to 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 series of codes with asymptotically higher parameters than any formerly identified codes.

This e-book 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. providing the cloth 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 awareness given to projective curves, rational services 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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**Example text**

18. 19. Let k be any field, n a positive integer, and let a0 , . . , an−1 ∈ k. Compute x(a0 + a1 x + · · · + an−1 xn−1 ) in Rn . 3. 20. Let k be a field. A vector space V over k is an abelian group which admits a scalar multiplication by elements of k. If we let + denote the group operation and · (or concatenation) denote the scalar multiplication, then the following properties must be satisfied for any v, w ∈ V and any α, β ∈ k: • α(v + w) = αv + αw • (αβ)v = α(βv) • (α + β)v = αv + βv • 1k · v = v, where 1k is the multiplicative identity of k Elements of V are called vectors.

Let h(x) = x3 + x2 + 1 ∈ F2 [x]. a) Show that h(x) is irreducible in F2 [x]. Conclude that F := F2 [x]/ h(x) is a field. b) How many elements does F have? List them. Make an addition table and a multiplication table. 6 above. 8. Let m be a positive integer. Then there is a field with exactly m elements if and only if m = pn for some prime p and some positive integer n. Further, up to isomorphism, there is only one field with exactly pn elements, and it is of the form Fp [x]/ f (x) for some irreducible polynomial f (x) ∈ Fp [x] of degree n.

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 , . . , pk and positive integers n1 , . . , nk such that G∼ = C n1 ⊕ · · · ⊕ C nk p1 pk • There are integers r1 , . . 2. 3. A ring is a set R equipped with two operations, usually denoted by + and · (or concatenation). 2. Rings, Fields, Ideals, and Factor Rings 47 ring, but we tend to call + addition and · multiplication in general. Every ring must satisfy all of the following properties: • Existence of Additive Identity: There is an element 0 ∈ R such that 0 + a = a = a + 0 for all a ∈ R.