179 research outputs found

    Generalization of the Lee-O'Sullivan List Decoding for One-Point AG Codes

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    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

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    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 W=(w_1,,w_n)W=(w\_{1},\dots,w\_{n}), WW-homogeneous polynomials are polynomials which are homogeneous w.r.t the weighted degree deg_W(X_1α_1,,X_nα_n)=w_iα_i\deg\_{W}(X\_{1}^{\alpha\_{1}},\dots,X\_{n}^{\alpha\_{n}}) = \sum w\_{i}\alpha\_{i}. 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 (w_i)ω\left(\prod w\_{i} \right)^{\omega}. For zero-dimensional systems, the complexity of Algorithm~\FGLM nDωnD^{\omega} (where DD is the number of solutions of the system) can be divided by the same factor (w_i)ω\left(\prod w\_{i} \right)^{\omega}. Under genericity assumptions, for zero-dimensional weighted homogeneous systems of WW-degree (d_1,,d_n)(d\_{1},\dots,d\_{n}), these complexity estimates are polynomial in the weighted B\'ezout bound _i=1nd_i/_i=1nw_i\prod\_{i=1}^{n}d\_{i} / \prod\_{i=1}^{n}w\_{i}. Furthermore, the maximum degree reached in a run of Algorithm \F5 is bounded by the weighted Macaulay bound (d_iw_i)+w_n\sum (d\_{i}-w\_{i}) + w\_{n}, and this bound is sharp if we can order the weights so that w_n=1w\_{n}=1. 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

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    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

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    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

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    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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