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 $C_{ab}$ 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

Sub-quadratic Decoding of One-point Hermitian Codes

We present the first two sub-quadratic complexity decoding algorithms for one-point Hermitian codes. The first is based on a fast realisation of the Guruswami-Sudan algorithm by using state-of-the-art algorithms from computer algebra for polynomial-ring matrix minimisation. The second is a Power decoding algorithm: an extension of classical key equation decoding which gives a probabilistic decoding algorithm up to the Sudan radius. We show how the resulting key equations can be solved by the same methods from computer algebra, yielding similar asymptotic complexities.Comment: New version includes simulation results, improves some complexity results, as well as a number of reviewer corrections. 20 page

Computational Aspects of Retrieving a Representation of an Algebraic Geometry Code

Producción CientíficaCode-based cryptography is an interesting alternative to classic number-theoretic public key cryptosystem since it is conjectured to be secure against quantum computer attacks. Many families of codes have been proposed for these cryptosystems such as algebraic geometry codes. In [Designs, Codes and Cryptography, pages 1-16, 2012] -for so called very strong algebraic geometry codes $\mathcal C=C_L(\mathcal X, \mathcal P, E)$, where $\mathcal X$ is an algebraic curve over $\mathbb F_q$, $\mathcal P$ is an $n$-tuple of mutually distinct $\mathbb F_q$-rational points of $\mathcal X$ and $E$ is a divisor of $\mathcal X$ with disjoint support from $\mathcal P$ --- it was shown that an equivalent representation $\mathcal C=C_L(\mathcal Y, \mathcal Q, F)$ can be found. The $n$-tuple of points is obtained directly from a generator matrix of $\mathcal C$, where the columns are viewed as homogeneous coordinates of these points. The curve $\mathcal Y$ is given by $I_2(\mathcal Y)$, the homogeneous elements of degree $2$ of the vanishing ideal $I(\mathcal Y)$. Furthermore, it was shown that $I_2(\mathcal Y)$ can be computed efficiently as the kernel of certain linear map. What was not shown was how to get the divisor $F$ and how to obtain efficiently an adequate decoding algorithm for the new representation. The main result of this paper is an efficient computational approach to the first problem, that is getting $F$. The security status of the McEliece public key cryptosystem using algebraic geometry codes is still not completely settled and is left as an open problemThis research was partly supported by the Danish National Research Foundation and the National Science Foundation of China (Grant No.\ 11061130539) for the Danish-Chinese Center for Applications of Algebraic Geometry in Coding Theory and Cryptography and by Spanish grants MTM2007-64704, MTM2010-21580-C02-02 and MTM2012-36917-C03-03. Part of the research of the second author is also funded by the Vernon Wilson Endowed Chair at Eastern Kentucky University during his sabbatical leave

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum