Algebraic Geometric Codes: Basic Notions

Abstract

PREFACE xv (for many interesting discussions of elliptic modules and modular curves); Gregory Katsman (who taught us coding theory); Leonid Bassalygo (who explained to us many coding subtleties); Gilles Lachaud (for many years of fruitful cooperation and hospitality, which helped us to write many chapters of the book); Alexander Barg (for many fruitful remarks and for the first version of tables of asymptotic bounds); Sergei Gelfand (for inducing us to publish our first paper improving the Gilbert-Varshamov bound); Gregory Kabatiansky (who made lots of valuable remarks reading the text of this book); Simon Litsyn (who attracted our attention to sphere packings in R n); Michael Rosenbloom (who worked with us on the problem of analogues); Alexei Skorobogatov (for many discussions); Andries Brouwer, Gerard van der Geer, and Marcel van der Vlught (who wrote appendices to the Russian edition of this book); members of the coding theory seminar and our colleagues from the Institute for Information Transmission Problems; and many other mathematicians for their friendship and help. Dmitry Nogin thanks not only the above-named mathematicians but also his co-authors, who attracted him to undertake this work. We are deeply grateful to our parents for their care and to our wives for their tender love.

Lower bounds on the class number of algebraic function fields defined over any finite field

“…This result was improved in certain cases by the following result of Perret in [10]: Moreover, Tsfasman [12] and Tsfasman-Vladut [13] [14] obtain asymptotics for the Jacobian; these best known results can also be found in [15]. We recall the following three important theorems contained in this book.…”

Minimum distance of orthogonal line-Grassmann codes in even characteristic

“…as Π ranges among all hyperplanes of the space PG( Ω ); we refer to [13] for further details. The codes associated with polar k-Grassmannians of either orthogonal, symplectic or Hermitian type have been introduced respectively in [1], [4] and [5].…”

Algebraic Geometric Codes over Rings

“…Definition 1.1 (Goppa, [13]; see also [40], [42]). Let X be a smooth, absolutely irreducible, projective curve over the finite field F q .…”

“…, f (P n )) | f ∈ L(D)} and C Ω (X, P, D) = {(res P1 The following theorem gives the basic properties of algebraic geometric codes over finite fields. Theorem 1.2 (Goppa, [13]; see also [40], [42]). Let X, P = {P 1 , .…”