3,773 research outputs found
Evaluation codes defined by finite families of plane valuations at infinity
We construct evaluation codes given by weight functions defined over polynomial rings in m a parts per thousand yen 2 indeterminates. These weight functions are determined by sets of m-1 weight functions over polynomial rings in two indeterminates defined by plane valuations at infinity. Well-suited families in totally ordered commutative groups are an important tool in our procedureSupported by Spain Ministry of Education MTM2007-64704 and Bancaixa P1-1B2009-03. The authors thank to the referees for their valuable suggestions.Galindo Pastor, C.; Monserrat Delpalillo, FJ. (2014). Evaluation codes defined by finite families of plane valuations at infinity. Designs, Codes and Cryptography. 70(1-2):189-213. https://doi.org/10.1007/s10623-012-9738-7S189213701-2Abhyankar S.S.: Local uniformization on algebraic surfaces over ground field of characteristic p ≠ 0. Ann. Math. 63, 491–526 (1956)Abhyankar S.S.: On the valuations centered in a local domain. Am. J. 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Soc. 98, 714–740 (2009)Galindo C., Monserrat F.: The Abhyankar-Moh theorem for plane valuations at infinity. Preprint 2010. ArXiv:0910.2613v2.Galindo C., Sanchis M.: Evaluation codes and plane valuations. Des. Codes Cryptogr. 41(2), 199–219 (2006)Geil O.: Codes based on an -algebra. PhD Thesis, Aalborg University, June (2000).Geil O., Matsumoto R.: Generalized Sudan’s list decoding for order domain codes. Lecture Notes in Computer Science, vol. 4851, pp. 50–59 (2007)Geil O., Pellikaan R.: On the structure of order domains. Finite Fields Appl. 8, 369–396 (2002)Goppa V.D.: Codes associated with divisors. Probl. Inf. Transm. 13, 22–26 (1997)Goppa V.D.: Geometry and Codes. Mathematics and Its Applications, vol. 24. Kluwer, Dordrecht (1991).Greco S., Kiyek K.: General elements in complete ideals and valuations centered at a two-dimensional regular local ring. In: Algebra, Arithmetic, and Geometry, with Applications, pp. 381–455. 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Subquadratic time encodable codes beating the Gilbert-Varshamov bound
We construct explicit algebraic geometry codes built from the
Garcia-Stichtenoth function field tower beating the Gilbert-Varshamov bound for
alphabet sizes at least 192. Messages are identied with functions in certain
Riemann-Roch spaces associated with divisors supported on multiple places.
Encoding amounts to evaluating these functions at degree one places. By
exploiting algebraic structures particular to the Garcia-Stichtenoth tower, we
devise an intricate deterministic \omega/2 < 1.19 runtime exponent encoding and
1+\omega/2 < 2.19 expected runtime exponent randomized (unique and list)
decoding algorithms. Here \omega < 2.373 is the matrix multiplication exponent.
If \omega = 2, as widely believed, the encoding and decoding runtimes are
respectively nearly linear and nearly quadratic. Prior to this work, encoding
(resp. decoding) time of code families beating the Gilbert-Varshamov bound were
quadratic (resp. cubic) or worse
Generalization of the Lee-O'Sullivan List Decoding for One-Point AG Codes
We generalize the list decoding algorithm for Hermitian codes proposed by Lee
and O'Sullivan based on Gr\"obner bases to general one-point AG codes, under an
assumption weaker than one used by Beelen and Brander. Our generalization
enables us to apply the fast algorithm to compute a Gr\"obner basis of a module
proposed by Lee and O'Sullivan, which was not possible in another
generalization by Lax.Comment: article.cls, 14 pages, no figure. The order of authors was changed.
To appear in Journal of Symbolic Computation. This is an extended journal
paper version of our earlier conference paper arXiv:1201.624
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
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
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