985 research outputs found
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
Distance bounds for algebraic geometric codes
Various methods have been used to obtain improvements of the Goppa lower
bound for the minimum distance of an algebraic geometric code. The main methods
divide into two categories and all but a few of the known bounds are special
cases of either the Lundell-McCullough floor bound or the Beelen order bound.
The exceptions are recent improvements of the floor bound by
Guneri-Stichtenoth-Taskin, and Duursma-Park, and of the order bound by
Duursma-Park and Duursma-Kirov. In this paper we provide short proofs for all
floor bounds and most order bounds in the setting of the van Lint and Wilson AB
method. Moreover, we formulate unifying theorems for order bounds and formulate
the DP and DK order bounds as natural but different generalizations of the
Feng-Rao bound for one-point codes.Comment: 29 page
Steane-Enlargement of Quantum Codes from the Hermitian Curve
In this paper, we study the construction of quantum codes by applying
Steane-enlargement to codes from the Hermitian curve. We cover
Steane-enlargement of both usual one-point Hermitian codes and of order bound
improved Hermitian codes. In particular, the paper contains two constructions
of quantum codes whose parameters are described by explicit formulae, and we
show that these codes compare favourably to existing, comparable constructions
in the literature.Comment: 11 page
Good Codes From Generalised Algebraic Geometry Codes
Algebraic geometry codes or Goppa codes are defined with places of degree
one. In constructing generalised algebraic geometry codes places of higher
degree are used. In this paper we present 41 new codes over GF(16) which
improve on the best known codes of the same length and rate. The construction
method uses places of small degree with a technique originally published over
10 years ago for the construction of generalised algebraic geometry codes.Comment: 3 pages, to be presented at the IEEE Symposium on Information Theory
(ISIT 2010) in Austin, Texas, June 201
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