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On nested code pairs from the Hermitian curve
Nested code pairs play a crucial role in the construction of ramp secret
sharing schemes [Kurihara et al. 2012] and in the CSS construction of quantum
codes [Ketkar et al. 2006]. The important parameters are (1) the codimension,
(2) the relative minimum distance of the codes, and (3) the relative minimum
distance of the dual set of codes. Given values for two of them, one aims at
finding a set of nested codes having parameters with these values and with the
remaining parameter being as large as possible. In this work we study nested
codes from the Hermitian curve. For not too small codimension, we present
improved constructions and provide closed formula estimates on their
performance. For small codimension we show how to choose pairs of one-point
algebraic geometric codes in such a way that one of the relative minimum
distances is larger than the corresponding non-relative minimum distance.Comment: 28 page
Sobre la distancia mínima de códigos AG unipuntuales castillo
We present a characterization of the lower bound d* for minimum distance of algebraic geometry one-point codes coming from castle curves. This article shows explicit calculations of this bound in the case of a Weierstrass semigroup generated by two consecutive elements. In particular, we obtain a simple characterization of the exact value of the minimum distance Hermitian codes.Presentamos una caracterización del límite inferior d * para la distancia mínima de los códigos de un punto de geometría algebraica provenientes de las curvas del castillo. Este artículo muestra cálculos explícitos de este límite en el caso de un semigrupo de Weierstrass generado por dos elementos consecutivos. En particular, obtenemos una caracterización simple del valor exacto de los códigos hermitianos de distancia mínima
Minimum-weight codewords of the Hermitian codes are supported on complete intersections
Let be the Hermitian curve defined over a finite field
. In this paper we complete the geometrical characterization
of the supports of the minimum-weight codewords of the algebraic-geometry codes
over , started in [1]: if is the distance of the code, the
supports are all the sets of distinct -points on
complete intersection of two curves defined by polynomials with
prescribed initial monomials w.r.t. \texttt{DegRevLex}.
For most Hermitian codes, and especially for all those with distance studied in [1], one of the two curves is always the Hermitian curve
itself, while if the supports are complete intersection of
two curves none of which can be .
Finally, for some special codes among those with intermediate distance
between and , both possibilities occur.
We provide simple and explicit numerical criteria that allow to decide for
each code what kind of supports its minimum-weight codewords have and to obtain
a parametric description of the family (or the two families) of the supports.
[1] C. Marcolla and M. Roggero, Hermitian codes and complete intersections,
arXiv preprint arXiv:1510.03670 (2015)
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