By Previato E. (ed.)

Our wisdom of items of algebraic geometry comparable to moduli of curves, (real) Schubert periods, basic teams of enhances of hyperplane preparations, toric kinds, and version of Hodge constructions, has been greater lately through rules and structures of quantum box thought, resembling replicate symmetry, Gromov-Witten invariants, quantum cohomology, and gravitational descendants.

These are many of the topics of this refereed selection of papers, which grew out of the distinctive consultation, "Enumerative Geometry in Physics," held on the AMS assembly in Lowell, MA, April 2000. This consultation introduced jointly mathematicians and physicists who pronounced at the most up-to-date effects and open questions; the entire abstracts are incorporated as an Appendix, and in addition integrated are papers via a few who couldn't attend.

The assortment offers an outline of state of the art instruments, hyperlinks that attach classical and sleek difficulties, and the most recent wisdom available.

Readership: Graduate scholars and examine mathematicians drawn to algebraic geometry and comparable disciplines.

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S v < n - d is a code vector of would Ho (that is so already The parameters of the corresponding Ho). code of which is not a linear combination of v We get a matrix Ho. 8, + 1 dl. Hence k+d=n + n - n which row k < then we we have is not of kl. < n ). Consider a [6,4,2] 2 -code C generated by the matrix [ ~ i ~ ~ ~ ~l 000 110 000 011 Calculate the spectra of (Ansver: B 4 = 6 , B B' 0 = 0 3 15, B' 1 C B 1 = B'2 = and cl. 42 6 in terms of B B' 3 2 2 ) . 15, recall that an additive character of homomorphism X ~q from the additive group of 3 B.

We are going to study spectra of linear codes. Let e be a linear [n,k,d]q-code, define as the number of code vectors of weight r for course, Ar 2! y r=O r it is easy to see that '\' L. y IIvll Sometimes non-homogeneous coordinates convenient, then we consider polynomials n n r '\' L. y are more r r=O r A code e has only one vector of weight 0 and has no other vectors of weight less that d. 1 15 Since in many cases we know not the precise value of d but only some lower estimate for it, the following form is rather convenient.

State and prove the spoiling lemma for non-linear codes. (Hint: In this case (k - 1) is changed by logq(lqk-l Exercise r) ). In many cases it is difficult to calculate precise values of parameters but it is possible to bound them. The spoiling lemma makes it possible to pass from an [:s n, d]q-code C to an [n,k,d]q-code. In such a situation we say that up to a spoiling the code C is an [n,k,d]q-code. 2: k, 2: So we can always spoil parameters, but of course we cannot always make them better. Here are some restrictions.