185 research outputs found
Transitive and self-dual codes attaining the Tsfasman-Vladut-Zink bound
A major problem in coding theory is the question of whether the class of cyclic codes is asymptotically good. In this correspondence-as a generalization of cyclic codes-the notion of transitive codes is introduced (see Definition 1.4 in Section I), and it is shown that the class of transitive codes is asymptotically good. Even more, transitive codes attain the Tsfasman-Vladut-Zink bound over F-q, for all squares q = l(2). It is also shown that self-orthogonal and self-dual codes attain the Tsfasman-Vladut-Zink bound, thus improving previous results about self-dual codes attaining the Gilbert-Varshamov bound. The main tool is a new asymptotically optimal tower E-0 subset of E-1 subset of E-2 subset of center dot center dot center dot of function fields over F-q (with q = l(2)), where all extensions E-n/E-0 are Galois
Asymptotic properties of Dedekind zeta functions in families of number fields
The main goal of this paper is to prove a formula that expresses the limit
behaviour of Dedekind zeta functions for in families of number
fields, assuming that the Generalized Riemann Hypothesis holds. This result can
be viewed as a generalization of the Brauer--Siegel theorem. As an application
we obtain a limit formula for Euler--Kronecker constants in families of number
fields
Generalised Mertens and Brauer-Siegel Theorems
In this article, we prove a generalisation of the Mertens theorem for prime
numbers to number fields and algebraic varieties over finite fields, paying
attention to the genus of the field (or the Betti numbers of the variety), in
order to make it tend to infinity and thus to point out the link between it and
the famous Brauer-Siegel theorem. Using this we deduce an explicit version of
the generalised Brauer-Siegel theorem under GRH, and a unified proof of this
theorem for asymptotically exact families of almost normal number fields
Schubert Varieties, Linear Codes and Enumerative Combinatorics
We consider linear error correcting codes associated to higher dimensional
projective varieties defined over a finite field. The problem of determining
the basic parameters of such codes often leads to some interesting and
difficult questions in combinatorics and algebraic geometry. This is
illustrated by codes associated to Schubert varieties in Grassmannians, called
Schubert codes, which have recently been studied. The basic parameters such as
the length, dimension and minimum distance of these codes are known only in
special cases. An upper bound for the minimum distance is known and it is
conjectured that this bound is achieved. We give explicit formulae for the
length and dimension of arbitrary Schubert codes and prove the minimum distance
conjecture in the affirmative for codes associated to Schubert divisors.Comment: 12 page
Asymptotic Bound on Binary Self-Orthogonal Codes
We present two constructions for binary self-orthogonal codes. It turns out
that our constructions yield a constructive bound on binary self-orthogonal
codes. In particular, when the information rate R=1/2, by our constructive
lower bound, the relative minimum distance \delta\approx 0.0595 (for GV bound,
\delta\approx 0.110). Moreover, we have proved that the binary self-orthogonal
codes asymptotically achieve the Gilbert-Varshamov bound.Comment: 4 pages 1 figur
Improved asymptotic bounds for codes using distinguished divisors of global function fields
For a prime power , let be the standard function in the
asymptotic theory of codes, that is, is the largest
asymptotic information rate that can be achieved for a given asymptotic
relative minimum distance of -ary codes. In recent years the
Tsfasman-Vl\u{a}du\c{t}-Zink lower bound on was improved by
Elkies, Xing, and Niederreiter and \"Ozbudak. In this paper we show further
improvements on these bounds by using distinguished divisors of global function
fields. We also show improved lower bounds on the corresponding function
for linear codes
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