17 research outputs found
Fast Erasure-and-Error Decoding and Systematic Encoding of a Class of Affine Variety Codes
In this paper, a lemma in algebraic coding theory is established, which is
frequently appeared in the encoding and decoding for algebraic codes such as
Reed-Solomon codes and algebraic geometry codes. This lemma states that two
vector spaces, one corresponds to information symbols and the other is indexed
by the support of Grobner basis, are canonically isomorphic, and moreover, the
isomorphism is given by the extension through linear feedback shift registers
from Grobner basis and discrete Fourier transforms. Next, the lemma is applied
to fast unified system of encoding and decoding erasures and errors in a
certain class of affine variety codes.Comment: 6 pages, 2 columns, presented at The 34th Symposium on Information
Theory and Its Applications (SITA2011
Lemma for Linear Feedback Shift Registers and DFTs Applied to Affine Variety Codes
In this paper, we establish a lemma in algebraic coding theory that
frequently appears in the encoding and decoding of, e.g., Reed-Solomon codes,
algebraic geometry codes, and affine variety codes. Our lemma corresponds to
the non-systematic encoding of affine variety codes, and can be stated by
giving a canonical linear map as the composition of an extension through linear
feedback shift registers from a Grobner basis and a generalized inverse
discrete Fourier transform. We clarify that our lemma yields the error-value
estimation in the fast erasure-and-error decoding of a class of dual affine
variety codes. Moreover, we show that systematic encoding corresponds to a
special case of erasure-only decoding. The lemma enables us to reduce the
computational complexity of error-evaluation from O(n^3) using Gaussian
elimination to O(qn^2) with some mild conditions on n and q, where n is the
code length and q is the finite-field size.Comment: 37 pages, 1 column, 10 figures, 2 tables, resubmitted to IEEE
Transactions on Information Theory on Jan. 8, 201
Quantum codes from a new construction of self-orthogonal algebraic geometry codes
[EN] We present new quantum codes with good parameters which are constructed from self-orthogonal algebraic geometry codes. Our method permits a wide class of curves to be used in the formation of these codes. These results demonstrate that there is a lot more scope for constructing self-orthogonal AG codes than was previously known.G. McGuire was partially supported by Science Foundation Ireland Grant 13/IA/1914. The remainder authors were partially supported by the Spanish Government and the EU funding program FEDER, Grants MTM2015-65764-C3-2-P and PGC2018-096446-B-C22. F. Hernando and J. J. Moyano-Fernandez are also partially supported by Universitat Jaume I, Grant UJI-B2018-10.Hernando, F.; Mcguire, G.; Monserrat Delpalillo, FJ.; Moyano-Fernández, JJ. (2020). Quantum codes from a new construction of self-orthogonal algebraic geometry codes. 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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.Fil: Cabaña, Gustavo Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe; Argentina. Universidad Nacional del Litoral. Facultad de Ciencias Economicas; Argentina. Universidad Nacional del Litoral. Facultad de Ingeniería Química. Departamento de Matemáticas; ArgentinaFil: Chara, María de Los Ángeles. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe; Argentina. Universidad Nacional del Litoral. Facultad de Ingeniería Química. Departamento de Matemáticas; ArgentinaFil: Podestá, Ricardo César. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; ArgentinaFil: Toledano, Ricardo. Universidad Nacional del Litoral. Facultad de Ingeniería Química. Departamento de Matemáticas; Argentin
Lifting iso-dual algebraic geometry codes
In this work we investigate the problem of producing iso-dual algebraic
geometry (AG) codes over a finite field with elements. Given
a finite separable extension of function fields and
an iso-dual AG-code defined over , we provide a
general method to lift the code to another iso-dual AG-code
defined over under some assumptions on the
parity of the involved different exponents. We apply this method to lift
iso-dual AG-codes over the rational function field to elementary abelian
-extensions, like the maximal function fields defined by the Hermitian,
Suzuki, and one covered by the function field. We also obtain long binary
and ternary iso-dual AG-codes defined over cyclotomic extensions.Comment: 26 pages, 3 figure
Algebraic Geometry Codes from Castle curves
The quality of an algebraic geometry code depends on the curve from which the code has been defined. In this paper we consider codes obtained from Castle curves, namely those whose number of rational points attains Lewittes' bound for some rational point Q and the Weierstrass semigroup at Q is symmetric
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