9,534 research outputs found
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
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
(2,1)-separating systems beyond the probabilistic bound
Building on previous results of Xing, we give new lower bounds on the rate of
intersecting codes over large alphabets. The proof is constructive, and uses
algebraic geometry, although nothing beyond the basic theory of linear systems
on curves. Then, using these new bounds within a concatenation argument, we
construct binary (2,1)-separating systems of asymptotic rate exceeding the one
given by the probabilistic method, which was the best lower bound available up
to now. This answers (negatively) the question of whether this probabilistic
bound was exact, which has remained open for more than 30 years. (By the way,
we also give a formulation of the separation property in terms of metric
convexity, which may be an inspirational source for new research problems.)Comment: Version 7 is a shortened version, so that numbering should match with
the journal version (to appear soon). Material on convexity and separation in
discrete and continuous spaces has been removed. Readers interested in this
material should consult version 6 instea
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