111 research outputs found
List Decoding Algorithm based on Voting in Groebner Bases for General One-Point AG Codes
We generalize the unique decoding algorithm for one-point AG codes over the
Miura-Kamiya Cab curves proposed by Lee, Bras-Amor\'os and O'Sullivan (2012) to
general one-point AG codes, without any assumption. We also extend their unique
decoding algorithm to list decoding, modify it so that it can be used with the
Feng-Rao improved code construction, prove equality between its error
correcting capability and half the minimum distance lower bound by Andersen and
Geil (2008) that has not been done in the original proposal except for
one-point Hermitian codes, remove the unnecessary computational steps so that
it can run faster, and analyze its computational complexity in terms of
multiplications and divisions in the finite field. As a unique decoding
algorithm, the proposed one is empirically and theoretically as fast as the BMS
algorithm for one-point Hermitian codes. As a list decoding algorithm,
extensive experiments suggest that it can be much faster for many moderate
size/usual inputs than the algorithm by Beelen and Brander (2010). It should be
noted that as a list decoding algorithm the proposed method seems to have
exponential worst-case computational complexity while the previous proposals
(Beelen and Brander, 2010; Guruswami and Sudan, 1999) have polynomial ones, and
that the proposed method is expected to be slower than the previous proposals
for very large/special inputs.Comment: Accepted for publication in J. Symbolic Computation. LaTeX2e
article.cls, 42 pages, 4 tables, no figures. Ver. 6 added an illustrative
example of the algorithm executio
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
On maximal curves
We study arithmetical and geometrical properties of maximal curves, that is,
curves defined over the finite field F_{q^2} whose number of F_{q^2}-rational
points reaches the Hasse-Weil upper bound. Under a hypothesis on non-gaps at a
rational point, we prove that maximal curves are F_{q^2}-isomorphic to y^q + y
= x^m, for some . As a consequence we show that a maximal curve of
genus g=(q-1)^2/4 is F_{q^2}-isomorphic to the curve y^q + y = x^{(q+1)/2}.Comment: LaTex2e, 17 pages; this article is an improved version of the paper
alg-geom/9603013 (by Fuhrmann and Torres
Hermitian codes from higher degree places
Matthews and Michel investigated the minimum distances in certain
algebraic-geometry codes arising from a higher degree place . In terms of
the Weierstrass gap sequence at , they proved a bound that gives an
improvement on the designed minimum distance. In this paper, we consider those
of such codes which are constructed from the Hermitian function field. We
determine the Weierstrass gap sequence where is a degree 3 place,
and compute the Matthews and Michel bound with the corresponding improvement.
We show more improvements using a different approach based on geometry. We also
compare our results with the true values of the minimum distances of Hermitian
1-point codes, as well as with estimates due Xing and Chen
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