179 research outputs found
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
On the complexity of computing Gr\"obner bases for weighted homogeneous systems
Solving polynomial systems arising from applications is frequently made
easier by the structure of the systems. Weighted homogeneity (or
quasi-homogeneity) is one example of such a structure: given a system of
weights , -homogeneous polynomials are polynomials
which are homogeneous w.r.t the weighted degree
. Gr\"obner bases for weighted homogeneous systems can be
computed by adapting existing algorithms for homogeneous systems to the
weighted homogeneous case. We show that in this case, the complexity estimate
for Algorithm~\F5 \left(\binom{n+\dmax-1}{\dmax}^{\omega}\right) can be
divided by a factor . For zero-dimensional
systems, the complexity of Algorithm~\FGLM (where is the
number of solutions of the system) can be divided by the same factor
. Under genericity assumptions, for
zero-dimensional weighted homogeneous systems of -degree
, these complexity estimates are polynomial in the
weighted B\'ezout bound .
Furthermore, the maximum degree reached in a run of Algorithm \F5 is bounded by
the weighted Macaulay bound , and this bound is
sharp if we can order the weights so that . For overdetermined
semi-regular systems, estimates from the homogeneous case can be adapted to the
weighted case. We provide some experimental results based on systems arising
from a cryptography problem and from polynomial inversion problems. They show
that taking advantage of the weighted homogeneous structure yields substantial
speed-ups, and allows us to solve systems which were otherwise out of reach
A polyhedral approach to computing border bases
Border bases can be considered to be the natural extension of Gr\"obner bases
that have several advantages. Unfortunately, to date the classical border basis
algorithm relies on (degree-compatible) term orderings and implicitly on
reduced Gr\"obner bases. We adapt the classical border basis algorithm to allow
for calculating border bases for arbitrary degree-compatible order ideals,
which is \emph{independent} from term orderings. Moreover, the algorithm also
supports calculating degree-compatible order ideals with \emph{preference} on
contained elements, even though finding a preferred order ideal is NP-hard.
Effectively we retain degree-compatibility only to successively extend our
computation degree-by-degree. The adaptation is based on our polyhedral
characterization: order ideals that support a border basis correspond
one-to-one to integral points of the order ideal polytope. This establishes a
crucial connection between the ideal and the combinatorial structure of the
associated factor spaces
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
Generic interpolation polynomial for list decoding
AbstractWe extend results of K. Lee and M.E. OʼSullivan by showing how to use Gröbner bases to find the interpolation polynomial for list decoding a one-point AG code C=CL(rP,D) on any curve X, where P is an Fq-rational point on X and D=P1+P2+⋯+Pn is the sum of other Fq-rational points on X. We then define the generic interpolation polynomial for list decoding such a code. The generic interpolation polynomial should specialize to the interpolation polynomial for most received strings. We give an example of a family of Reed–Solomon 1-error correcting codes for which a single error can be decoded by a very simple process involving substituting into the generic interpolation polynomial
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