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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.

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The paper has two purposes. First, we start to develop a theory of infinite global fields, i.e., of infinite algebraic extensions either of Q or of F r (t). We produce a series of invariants of such fields, and we introduce and study a kind of zeta-function for them. Second, for sequences of number fields with growing discriminant we prove generalizations of the Odlyzko-Serre bounds and of the Brauer-Siegel theorem, taking into account non-archimedean places. This leads to asymptotic bounds on the ratio log hR/ log |D| valid without the standard assumption n/ log |D| → 0, thus including, in particular, the case of unramified towers. Then we produce examples of class field towers, showing that this assumption is indeed necessary for the Brauer-Siegel theorem to hold. As an easy consequence we ameliorate on existing bounds for regulators.

Using algebraic geometry codes we give a polynomial construction of quantum codes with asymptotically non-zero rate and relative distance.