By Judy L. Walker

Whilst details is transmitted, mistakes are inclined to take place. This challenge has develop into more and more very important as super quantities of data are transferred electronically on a daily basis. Coding conception examines effective methods of packaging facts in order that those mistakes may be detected, or perhaps corrected.

The conventional instruments of coding conception have come from combinatorics and workforce conception. because the paintings of Goppa within the past due Nineteen Seventies, besides the fact that, coding theorists have extra ideas from algebraic geometry to their toolboxes. particularly, via re-interpreting the Reed-Solomon codes as coming from comparing capabilities linked to divisors at the projective line, you may 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 higher parameters than any formerly recognized codes.

This publication 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. the following, the reader is brought to the fascinating box of algebraic geometric coding concept. 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 recognition 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, Volume 7)**

**Example text**

If we homogenize, however, we see that Cf1 and Cf4 intersect at the point (0 : 1 : 0). By Bezout’s Theorem, the curves must intersect with multiplicity 2 there. In other words, the curves are tangent at the point (0 : 1 : 0). 12. Let f (x, y) = x3 + x2 y − 3xy 2 − 3y 3 + 2x2 − x + 5. Find all (complex) points at infinity on Cf , the projective closure of Cf . 13. Find C(F7 ) where C is the projective closure of the curve defined by the equation y 2 = x3 + x + 1. 1. Nonsingularity For coding theory, one only wants to work with “nice” curves.

F (Pn )) 6. Algebraic Geometry Codes 39 Since L(D) is a vector space over Fq and the evaluation map is a linear transformation, we see that C(X, P, D) is a linear code. Further, its length is obviously n = #P. What about the dimension? Clearly, it’s at most dim L(D), and it’s exactly dim L(D) if and only if is one-to-one. 23). So suppose (f ) = 0. Then f (P1 ) = · · · = f (Pn ) = 0, so the coefficient of each Pi in the divisor div(f ) is at least 1. Since no Pi is in suppD, we have that div(f ) + D − P1 − · · · − Pn ≥ 0, which means that f ∈ L(D − P1 − · · · − Pn ).

Q (δ) ≥ −δ + 1 − √ ( q − 1) By doing a little computation, it’s not difficult to see that the √ “Tsfasman-Vladut-Zink line” R = −δ + 1 − 1/( q − 1) and the “Gilbert-Varshamov curve” R = 1−Hq (δ) will intersect in exactly two points whenever q ≥ 49. Therefore, for all perfect squares q ≥ 49, the Tsfasman-Vladut-Zink Bound gives an improvement on the GilbertVarshamov bound for the possible asymptotic parameters of codes over the field Fq . 8. For each of the following values of q, draw a careful plot of the asymptotic Plotkin bound, the asymptotic GilbertVarshamov bound, and the Tsfasman-Vladut-Zink bound on a single set of axes: q = 25, q = 49, and q = 64.

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

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