64 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
Codes and Curves
When information is transmitted, errors are likely to occur. Coding theory examines effi cient ways of packaging data so that these errors can be detected, or even corrected. The traditional tools of coding theory have come from combinatorics and group theory. Lately, however, coding theorists have added techniques from algebraic geometry to their toolboxes. In particular, by re-interpreting the Reed- Solomon codes, one can see how to defi ne new codes based on divisors on algebraic curves. For instance, using modular curves over fi nite fi elds, Tsfasman, Vladut, and Zink showed that one can defi ne a sequence of codes with asymptotically better parameters than any previously known codes. This monograph is based on a series of lectures the author gave as part of the IAS/PCMI program on arithmetic algebraic geometry. Here, the reader is introduced to the exciting fi eld of algebraic geometric coding theory. Presenting the material in the same conversational tone of the lectures, the author covers linear codes, including cyclic codes, and both bounds and asymptotic bounds on the parameters of codes. Algebraic geometry is introduced, with particular attention given to projective curves, rational functions and divisors. The construction of algebraic geometric codes is given, and the Tsfasman-Vladut-Zink result mentioned above is discussed
Weierstrass Semigroup, Pure Gaps and Codes on Kummer Extensions
We determine the Weierstrass semigroup at one and two totally ramified places
in a Kummer extension defined by the affine equation over , the algebraic closure of ,
where are pairwise distinct elements, and
. For an arbitrary function field, from the
knowledge of the minimal generating set of the Weierstrass semigroup at two
rational places, the set of pure gaps is characterized. We apply these results
to construct algebraic geometry codes over certain function fields with many
rational places.Comment: 24 page
On Linear Complementary Pairs of Algebraic Geometry Codes over Finite Fields
Linear complementary dual (LCD) codes and linear complementary pairs (LCP) of
codes have been proposed for new applications as countermeasures against
side-channel attacks (SCA) and fault injection attacks (FIA) in the context of
direct sum masking (DSM). The countermeasure against FIA may lead to a
vulnerability for SCA when the whole algorithm needs to be masked (in
environments like smart cards). This led to a variant of the LCD and LCP
problems, where several results have been obtained intensively for LCD codes,
but only partial results have been derived for LCP codes. Given the gap between
the thin results and their particular importance, this paper aims to reduce
this by further studying the LCP of codes in special code families and,
precisely, the characterisation and construction mechanism of LCP codes of
algebraic geometry codes over finite fields. Notably, we propose constructing
explicit LCP of codes from elliptic curves. Besides, we also study the security
parameters of the derived LCP of codes (notably
for cyclic codes), which are given by the minimum distances
and . Further, we show that for LCP algebraic geometry
codes , the dual code is
equivalent to under some specific conditions we exhibit. Finally,
we investigate whether MDS LCP of algebraic geometry codes exist (MDS codes are
among the most important in coding theory due to their theoretical significance
and practical interests). Construction schemes for obtaining LCD codes from any
algebraic curve were given in 2018 by Mesnager, Tang and Qi in [``Complementary
dual algebraic geometry codes", IEEE Trans. Inform Theory, vol. 64(4),
2390--3297, 2018]. To our knowledge, it is the first time LCP of algebraic
geometry codes has been studied
On cyclic algebraic-geometry codes
In this paper we initiate the study of cyclic algebraic geometry codes. We
give conditions to construct cyclic algebraic geometry codes in the context of
algebraic function fields over a finite field by using their group of
automorphisms. We prove that cyclic algebraic geometry codes constructed in
this way are closely related to cyclic extensions. We also give a detailed
study of the monomial equivalence of cyclic algebraic geometry codes
constructed with our method in the case of a rational function field.Comment: 25 pages, 1 figur
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