10,895 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
List Decoding Algorithms based on Groebner Bases for General 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. By using the same
principle, we also generalize the unique decoding algorithm for one-point AG
codes over the Miura-Kamiya curves proposed by Lee, Bras-Amor\'os and
O'Sullivan to general one-point AG codes, without any assumption. Finally we
extend the latter 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 that has not been done in the original proposal, and
remove the unnecessary computational steps so that it can run faster.Comment: IEEEtran.cls, 5 pages, no figure. To appear in Proc. 2012 IEEE
International Symposium on Information Theory, July 1-6, 2012, Boston, MA,
USA. Version 4 corrected wrong description of the work by Lee, Bras-Amor\'os
and O'Sullivan, and added four reference
Feng-Rao decoding of primary codes
We show that the Feng-Rao bound for dual codes and a similar bound by
Andersen and Geil [H.E. Andersen and O. Geil, Evaluation codes from order
domain theory, Finite Fields Appl., 14 (2008), pp. 92-123] for primary codes
are consequences of each other. This implies that the Feng-Rao decoding
algorithm can be applied to decode primary codes up to half their designed
minimum distance. The technique applies to any linear code for which
information on well-behaving pairs is available. Consequently we are able to
decode efficiently a large class of codes for which no non-trivial decoding
algorithm was previously known. Among those are important families of
multivariate polynomial codes. Matsumoto and Miura in [R. Matsumoto and S.
Miura, On the Feng-Rao bound for the L-construction of algebraic geometry
codes, IEICE Trans. Fundamentals, E83-A (2000), pp. 926-930] (See also [P.
Beelen and T. H{\o}holdt, The decoding of algebraic geometry codes, in Advances
in algebraic geometry codes, pp. 49-98]) derived from the Feng-Rao bound a
bound for primary one-point algebraic geometric codes and showed how to decode
up to what is guaranteed by their bound. The exposition by Matsumoto and Miura
requires the use of differentials which was not needed in [Andersen and Geil
2008]. Nevertheless we demonstrate a very strong connection between Matsumoto
and Miura's bound and Andersen and Geil's bound when applied to primary
one-point algebraic geometric codes.Comment: elsarticle.cls, 23 pages, no figure. Version 3 added citations to the
works by I.M. Duursma and R. Pellikaa
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
Variations of the McEliece Cryptosystem
Two variations of the McEliece cryptosystem are presented. The first one is
based on a relaxation of the column permutation in the classical McEliece
scrambling process. This is done in such a way that the Hamming weight of the
error, added in the encryption process, can be controlled so that efficient
decryption remains possible. The second variation is based on the use of
spatially coupled moderate-density parity-check codes as secret codes. These
codes are known for their excellent error-correction performance and allow for
a relatively low key size in the cryptosystem. For both variants the security
with respect to known attacks is discussed
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