4,238 research outputs found
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
Algebraic Codes For Error Correction In Digital Communication Systems
Access to the full-text thesis is no longer available at the author's request, due to 3rd party copyright restrictions. Access removed on 29.11.2016 by CS (TIS).Metadata merged with duplicate record (http://hdl.handle.net/10026.1/899) on 20.12.2016 by CS (TIS).C. Shannon presented theoretical conditions under which communication was possible
error-free in the presence of noise. Subsequently the notion of using error
correcting codes to mitigate the effects of noise in digital transmission was introduced
by R. Hamming. Algebraic codes, codes described using powerful tools from
algebra took to the fore early on in the search for good error correcting codes. Many
classes of algebraic codes now exist and are known to have the best properties of
any known classes of codes. An error correcting code can be described by three of its
most important properties length, dimension and minimum distance. Given codes
with the same length and dimension, one with the largest minimum distance will
provide better error correction. As a result the research focuses on finding improved
codes with better minimum distances than any known codes.
Algebraic geometry codes are obtained from curves. They are a culmination of years
of research into algebraic codes and generalise most known algebraic codes. Additionally
they have exceptional distance properties as their lengths become arbitrarily
large. Algebraic geometry codes are studied in great detail with special attention
given to their construction and decoding. The practical performance of these codes
is evaluated and compared with previously known codes in different communication
channels. Furthermore many new codes that have better minimum distance
to the best known codes with the same length and dimension are presented from
a generalised construction of algebraic geometry codes. Goppa codes are also an
important class of algebraic codes. A construction of binary extended Goppa codes
is generalised to codes with nonbinary alphabets and as a result many new codes
are found. This construction is shown as an efficient way to extend another well
known class of algebraic codes, BCH codes. A generic method of shortening codes
whilst increasing the minimum distance is generalised. An analysis of this method
reveals a close relationship with methods of extending codes. Some new codes from
Goppa codes are found by exploiting this relationship. Finally an extension method
for BCH codes is presented and this method is shown be as good as a well known
method of extension in certain cases
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 ā Ā 0. Ann. Math. 63, 491ā526 (1956)Abhyankar S.S.: On the valuations centered in a local domain. Am. J. Math. 78, 321ā348 (1956)Abhyankar S.S.: Lectures on expansion techniques in algebraic geometry. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 57. Tata Institute of Fundamental Research, Bombay (1977).Abhyankar S.S.: On the semigroup of a meromorphic curve (part I). In: Proceedings of the International Symposium on Algebraic Geometry (Kyoto) Kinokunio Tokio, pp. 249ā414 (1977).Abhyankar S.S., Moh T.T.: Newton-Puiseux expansion and generalized Tschirnhausen transformation (I). J. Reine Angew. Math. 260, 47ā83 (1973)Abhyankar S.S., Moh T.T.: Newton-Puiseux expansion and generalized Tschirnhausen transformation (II). J. Reine Angew. Math. 261, 29ā54 (1973)Berlekamp E.R.: Algebraic Coding Theory. McGraw-Hill, New York (1968)Campillo A., FarrĆ”n J.I.: Computing Weierstrass semigroups and the Feng-Rao distance from singular plane models. Finite Fields Appl. 6, 71ā92 (2000)Carvalho C., Munuera C., Silva E., Torres F.: Near orders and codes. IEEE Trans. Inf. Theory 53, 1919ā1924 (2007)Decker W., Greuel G.M., Pfister G., Schƶenemann H.: Singular 3.1.3, a computer algebra system for polynomial computations (2011) http://www.singular.uni-kl.de .Feng G.L., Rao T.R.N.: Decoding of algebraic geometric codes up to the designed minimum distance. IEEE Trans. Inf. Theory 39, 37ā45 (1993)Feng G.L., Rao T.R.N.: A simple approach for construction of algebraic-geometric codes from affine plane curves. IEEE Trans. Inf. Theory 40, 1003ā1012 (1994)Feng G.L., Rao T.R.N.: Improved geometric Goppa codes, part I: basic theory. IEEE Trans. Inf. Theory 41, 1678ā1693 (1995)Fujimoto M., Suzuki M.: Construction of affine plane curves with one place at infinity. Osaka J. Math. 39(4), 1005ā1027 (2002)Galindo C.: Plane valuations and their completions. Commun. Algebra 23(6), 2107ā2123 (1995)Galindo C., Monserrat F.: Ī“-sequences and evaluation codes defined by plane valuations at infinity. Proc. Lond. Math. Soc. 98, 714ā740 (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), 199ā219 (2006)Geil O.: Codes based on an -algebra. PhD Thesis, Aalborg University, June (2000).Geil O., Matsumoto R.: Generalized Sudanās list decoding for order domain codes. Lecture Notes in Computer Science, vol. 4851, pp. 50ā59 (2007)Geil O., Pellikaan R.: On the structure of order domains. Finite Fields Appl. 8, 369ā396 (2002)Goppa V.D.: Codes associated with divisors. Probl. Inf. Transm. 13, 22ā26 (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. 381ā455. Springer, Berlin (2003).HĆøholdt T., van Lint J.H., Pellikaan R.: Algebraic geometry codes. In: Handbook of Coding Theory, vol. 1, pp. 871ā961. Elsevier, Amsterdam (1998).Jensen C.D.: Fast decoding of codes from algebraic geometry. IEEE Trans. Inf. Theory 40, 223ā230 (1994)Justesen J., Larsen K.J., Jensen H.E., Havemose A., HĆøholdt T.: Construction and decoding of a class of algebraic geometric codes. IEEE Trans. Inf. Theory 35, 811ā821 (1989)Justesen J., Larsen K.J., Jensen H.E., HĆøholdt T.: Fast decoding of codes from algebraic plane curves. IEEE Trans. Inf. Theory 38, 111ā119 (1992)Massey J.L.: Shift-register synthesis and BCH decoding. IEEE Trans. Inf. Theory 15, 122ā127 (1969)Matsumoto R.: Miuraās generalization of one point AG codes is equivalent to HĆøholdt, van Lint and Pellikaanās generalization. IEICE Trans. Fundam. E82-A(10), 2007ā2010 (1999)Moghaddam M.: Realization of a certain class of semigroups as value semigroups of valuations. Bull. Iran. Math. Soc. 35, 61ā95 (2009)OāSullivan M.E.: Decoding of codes defined by a single point on a curve. IEEE Trans. Inf. Theory 41, 1709ā1719 (1995)OāSullivan M.E.: New codes for the Belekamp-Massey-Sakata algorithm. Finite Fields Appl. 7, 293ā317 (2001)Pinkham H.: SĆ©minaire sur les singularitĆ©s des surfaces (Demazure-Pinkham-Teissier), Course donnĆ© au Centre de Math. de lāEcole Polytechnique (1977ā1978).Sakata S.: Extension of the Berlekamp-Massey algorithm to N dimensions. Inf. Comput. 84, 207ā239 (1990)Sakata S., Jensen H.E., HĆøholdt T.: Generalized Berlekamp-Massey decoding of algebraic geometric codes up to half the Feng-Rao bound. IEEE Trans. Inf. Theory 41, 1762ā1768 (1995)Sakata S., Justesen J., Madelung Y., Jensen H.E., HĆøholdt T.: Fast decoding of algebraic geometric codes up to the designed minimum distance. IEEE Trans. Inf. Theory 41, 1672ā1677 (1995)Sathaye A.: On planar curves. Am. J. Math. 99(5), 1105ā1135 (1977)Shannon C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379ā423 and 623ā656 (1948).Skorobogatov A.N., VlÄdut S.G.: On the decoding of algebraic geometric codes. IEEE Trans. Inf. Theory 36, 1051ā1060 (1990)Spivakovsky M.: Valuations in function fields of surfaces. Am. J. Math. 112, 107ā156 (1990)Suzuki M.: Affine plane curves with one place at infinity. Ann. Inst. Fourier 49(2), 375ā404 (1999)Tsfasman S.G., VlÄduÅ£ T.: Zink, modular curves, Shimura curves and Goppa codes, better than VarshamovāGilbert bound. Math. Nachr. 109, 21ā28 (1982)VlÄduÅ£ S.G., Manin Y.I. Linear codes and modular curves. In: Current problems in mathematics, vol. 25, pp. 209ā257. Akad. Nauk SSSR Vseoyuz, Moscow (1984).Zariski O.: The reduction of the singularities of an algebraic surface. Ann. Math. 40, 639ā689 (1939)Zariski O.: Local uniformization on algebraic varieties. Ann. Math. 41, 852ā896 (1940)Zariski O., Samuel P.(1960) Commutative Algebra, vol. II. Springer, Berlin
Design and Analysis of Graph-based Codes Using Algebraic Lifts and Decoding Networks
Error-correcting codes seek to address the problem of transmitting information efficiently and reliably across noisy channels. Among the most competitive codes developed in the last 70 years are low-density parity-check (LDPC) codes, a class of codes whose structure may be represented by sparse bipartite graphs. In addition to having the potential to be capacity-approaching, LDPC codes offer the significant practical advantage of low-complexity graph-based decoding algorithms. Graphical substructures called trapping sets, absorbing sets, and stopping sets characterize failure of these algorithms at high signal-to-noise ratios. This dissertation focuses on code design for and analysis of iterative graph-based message-passing decoders. The main contributions of this work include the following: the unification of spatially-coupled LDPC (SC-LDPC) code constructions under a single algebraic graph lift framework and the analysis of SC-LDPC code construction techniques from the perspective of removing harmful trapping and absorbing sets; analysis of the stopping and absorbing set parameters of hypergraph codes and finite geometry LDPC (FG-LDPC) codes; the introduction of multidimensional decoding networks that encode the behavior of hard-decision message-passing decoders; and the presentation of a novel Iteration Search Algorithm, a list decoder designed to improve the performance of hard-decision decoders.
Adviser: Christine A. Kelle
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
Efficient Multi-Point Local Decoding of Reed-Muller Codes via Interleaved Codex
Reed-Muller codes are among the most important classes of locally correctable
codes. Currently local decoding of Reed-Muller codes is based on decoding on
lines or quadratic curves to recover one single coordinate. To recover multiple
coordinates simultaneously, the naive way is to repeat the local decoding for
recovery of a single coordinate. This decoding algorithm might be more
expensive, i.e., require higher query complexity. In this paper, we focus on
Reed-Muller codes with usual parameter regime, namely, the total degree of
evaluation polynomials is , where is the code alphabet size
(in fact, can be as big as in our setting). By introducing a novel
variation of codex, i.e., interleaved codex (the concept of codex has been used
for arithmetic secret sharing \cite{C11,CCX12}), we are able to locally recover
arbitrarily large number of coordinates of a Reed-Muller code
simultaneously at the cost of querying coordinates. It turns out that
our local decoding of Reed-Muller codes shows ({\it perhaps surprisingly}) that
accessing locations is in fact cheaper than repeating the procedure for
accessing a single location for times. Our estimation of success error
probability is based on error probability bound for -wise linearly
independent variables given in \cite{BR94}
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