DOCSLIB.ORG

A Z4-Linear Code of High Minimum Lee Distance Derived from a Hyperoval

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Z4-Linear Code of High Minimum Lee Distance Derived from a Hyperoval Advances in Mathematics of Communications doi:10.3934/amc.2011.5.275 Volume 5, No. 2, 2011, 275{286 A Z4-LINEAR CODE OF HIGH MINIMUM LEE DISTANCE DERIVED FROM A HYPEROVAL Michael Kiermaier and Johannes Zwanzger Institut f¨urMathematik Universit¨atBayreuth 95440 Bayreuth, Germany (Communicated by Patrick Sol´e) Abstract. In this paper we present a new non-free Z4-linear code of length 29 and size 128 whose minimum Lee distance is 28. Its Gray image is a nonlinear binary code with parameters (58; 27; 28), having twice as many codewords as the biggest linear binary codes of equal length and minimum distance. The code also improves the known lower bound on the maximal size of binary block codes of length 58 and minimum distance 28. Originally the code was found by a heuristic computer search. We give a geometric construction based on a hyperoval in the projective Hjelmslev plane over Z4 which allows an easy computation of the symmetrized weight enumerator and the automorphism group. Furthermore, a generalization of this construction to all Galois rings of characteristic 4 is discussed. 1. Introduction A bit more than 40 years ago in papers of Nordstrom, Robinson, Preparata and Kerdock [16, 17, 12], some nonlinear binary codes with better minimum distance than any linear code of equal size and length were constructed. Later it was discov- ered [14,5] that they can be considered as images of linear codes over Z4 under the so-called Gray map. This led to an increased interest and a lot of investigation in linear codes over Z4, also a theory for linear codes over more general rings like finite chain rings and Frobenius rings was developed. However, apart from the Preparata series, not many examples are known where the Gray image of a Z4-linear code provably outperforms all linear binary codes of the same length and size. ^ Recently, a new example of such a Z4-linear code C was found by a heuristic ap- proach similar to [22]. A subsequent analysis of the code revealed a clear geometric structure. The Lee weight enumerator of C^ is 28 32 36 44 LeeC^ = 1 + 105X + 7X + 14X + X : So the Gray image of C^ has the parameters (58; 27; 28), while the optimal minimum distance for a linear binary [58; 7]-code is 27 [19]. Furthermore, C^ surpasses the hitherto largest known nonlinear binary block code of length 58 and minimum distance 28 [20, 13] by 4 codewords. 2000 Mathematics Subject Classification: Primary: 94B05; Secondary: 51C05, 51E21. Key words and phrases: Coding theory, Gray image, minimum distance, Hjelmslev geometry, hyperoval. The first author is supported by DFG grant WA 1666/4-1. 275 c 2011 AIMS-SDU 276 Michael Kiermaier and Johannes Zwanzger This paper is organized as follows: In Section2, we give a brief introduction to the theory of Z4-linear codes. A more detailed overview can be found in [11, Chapter 12]. In Section3, the required basics about the projective Hjelmslev plane over Z4 are presented. For a deeper discussion of projective Hjelmslev geometries and the connection to linear codes over finite chain rings, the reader is referred to [7,9]. In Section4, a geometric construction of C^ based on a hyperoval in the projective Hjelmslev plane over Z4 is given, which allows an easy computation of the symmetrized weight enumerator and the automorphism group of C^. The question for a generalization of the construction is discussed in Section5. 2. Z4-linear codes n A submodule C of the Z4-module Z4 is called Z4-linear code of length n. There is a unique pair of non-negative integers (r1; r2) such that ∼ r1 r2 C = Z4 ⊕ 2Z4 r1 r2 as Z4-modules. The partition 2 1 ` 2r1 + r2 is called the shape of C. Let ∗ ∗ Z4 = f1; 3g be the unit group of Z4, Rad(Z4) = Z4 n Z4 = f0; 2g the radical of Z4 and S = f0; 1g ⊂ Z4, which is a set of representatives of the factor ring Z4=Rad(Z4). The code C is free as a Z4-module if and only if r2 = 0. In the case r1 = 0, C n n is contained in 2Z4 , which is isomorphic to F2 as F2-modules. This shows that Z4-linear codes with r1 = 0 are just binary linear codes, and for a proper Z4-linear code we may assume r1 > 0. (r1+r2)×n A matrix Γ 2 Z4 is called a generator matrix of C if its rows generate C as Z4-module, and exactly r1 rows contain at least one unit, while in the remaining r2 rows all entries are from Rad(Z4). Without loss of generality we may assume that this will be the case for the last r2 rows of Γ. Then t r1 r2 C = fv Γ: v 2 Z4 × S g r1 r2 and for every c 2 C there is a unique information vector v 2 Z4 × S such that c = vtΓ. 2.1. Weight and metric. The Lee weight 8 0 7! 0 <> wLee : Z4 ! R; 1; 3 7! 1 :>2 7! 2 n gives rise to a weight function on Z4 by summing up the weights of all the compo- nents; as there is no danger of confusion, it will also be denoted by wLee. It is easy to check that the function n n 0 0 dLee : Z4 × Z4 ! R; (c; c ) 7! wLee(c − c ) n 0 is a metric on Z4 , the Lee metric. dLee(c; c ) is also called the Lee distance between 0 0 0 c and c and the value dmin(C) := minfdLee(c; c ): c 6= c 2 Cg the minimum Lee distance of C. The Lee weight enumerator of C is given by X wLee(c) LeeC = X 2 Z[X]: c2C Like in classical coding theory over finite fields, one of the main goals is to find a Z4-linear code for given parameters n, r1 and r2 whose minimum Lee distance is as Advances in Mathematics of Communications Volume 5, No. 2 (2011), 275{286 A Z4-linear code of high minimum Lee distance 277 large as possible. As a special case of linear codes over finite chain rings, a table of Z4-linear codes of high minimum Lee distance can be found in [21]. Furthermore, for a codeword c = (c0; c1; : : : ; cn−1) we define the numbers a0 = ∗ jfi : ci = 0gj, a1 = jfi : ci = 2gj and a2 = jfi : ci 2 Z4gj. The monomial a0 a1 a2 wsym(c) = X0 X1 X2 2 Z[X0;X1;X2] will be called the symmetrized weight of c. Furthermore, the symmetrized weight P enumerator of C is given by symC = c2C wsym(c). The symmetrized weight carries more information than the Lee weight: Given the symmetrized weight s of a codeword c, the Lee weight of c can be computed by evaluating s at X0 = 0, X1 = 2 and X2 = 1. Similarly, the Lee weight enumerator can be derived from the symmetrized weight enumerator. We want to mention that the indexing of the polynomial variables of the sym- metrized weight differs from [11] as well as from [7], where yet another indexing is used. The reason for this is that on the one hand, we feel the index order should re- spect the linear order on the ideals of Z4, while the zero element should be counted by X0. 2.2. Transformation into a binary code. Obviously the function 8 >0 7! 00 > 2 <1 7! 01 γ : Z4 ! F2; >2 7! 11 > :3 7! 10 2 is an isometry between the metric spaces (Z4; dLee) and (F2; dHam), where dHam is 2 the Hamming metric on F2. γ is called the Gray map on Z4. Like above, it is canonically extendible to n 2n γ : Z4 ! F2 : n Hence, every block code C ⊂ Z4 can be transformed via γ into a binary code γ(C) of the same size and weight distribution and of twice the length. However, Z4-linearity of C usually does not result in F2-linearity of γ(C). n 2.3. Automorphisms. The linear isometries of the metric space (Z4 ; dLee) are ex- n n n actly the monomial transformations Z4 ! Z4 . Thus each linear isometry ι : Z4 ! n n Z4 can be written as (σ; v) 2 Sn × {±1g , where σ denotes the permutation of the coordinates and for all i 2 f0; : : : ; n − 1g, the ith component of all codewords is multiplied by vi. σ will be called the permutation part of ι, and for a subgroup G of the monomial group, the permutation group consisting of all permutation parts of the elements in G will be called permutation part of G, accordingly. Given a Z4- linear code C of length n, the stabilizer group of C in the group of linear isometries is called the automorphism group Aut(C) of C. The group of all linear isometries acts on the set of all Z4-linear codes of length n. Two Z4-linear codes of length n are called isomorphic if and only if they are contained in the same orbit. The kernel of this group action is cyclic of order 2 and generated by the linear isometry (id; (−1;:::; −1)t). To get a faithful group action, we look at the factor group I of the group of linear isometries modulo this kernel. The notion of the permutation part will be carried forward to elements and subgroups of I, since it does not depend on the choice of the representatives. Given a Z4-linear code C of length n, we call the stabilizer group of C with respect to the group action of I the projective automorphism group of C and denote it by PAut(C).
Load more

Recommended publications
  • Coding Theory, Information Theory and Cryptology : Proceedings of the EIDMA Winter Meeting, Veldhoven, December 19-21, 1994
    Coding theory, information theory and cryptology : proceedings of the EIDMA winter meeting, Veldhoven, December 19-21, 1994 Citation for published version (APA): Tilborg, van, H. C. A., & Willems, F. M. J. (Eds.) (1994). Coding theory, information theory and cryptology : proceedings of the EIDMA winter meeting, Veldhoven, December 19-21, 1994. Euler Institute of Discrete Mathematics and its Applications. Document status and date: Published: 01/01/1994 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
    [Show full text]
  • Open Thesis-Cklee-Jul21.Pdf
    The Pennsylvania State University The Graduate School College of Engineering EFFICIENT INFORMATION ACCESS FOR LOCATION-BASED SERVICES IN MOBILE ENVIRONMENTS A Dissertation in Computer Science and Engineering by Chi Keung Lee c 2009 Chi Keung Lee Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2009 The dissertation of Chi Keung Lee was reviewed and approved∗ by the following: Wang-Chien Lee Associate Professor of Computer Science and Engineering Dissertation Advisor Chair of Committee Donna Peuquet Professor of Geography Piotr Berman Associate Professor of Computer Science and Engineering Daniel Kifer Assistant Professor of Computer Science and Engineering Raj Acharya Professor of Computer Science and Engineering Head of Computer Science and Engineering ∗Signatures are on file in the Graduate School. Abstract The demand for pervasive access of location-related information (e.g., local traffic, restau- rant locations, navigation maps, weather conditions, pollution index, etc.) fosters a tremen- dous application base of Location Based Services (LBSs). Without loss of generality, we model location-related information as spatial objects and the accesses of such information as Location-Dependent Spatial Queries (LDSQs) in LBS systems. In this thesis, we address the efficiency issue of LDSQ processing through several innovative techniques. First, we develop a new client-side data caching model, called Complementary Space Caching (CSC), to effectively shorten data access latency over wireless channels. This is mo- tivated by an observation that mobile clients, with only a subset of objects cached, do not have sufficient knowledge to assert whether or not certain object access from the remote server are necessary for given LDSQs.
    [Show full text]
  • Checking Whether a Word Is Hamming-Isometric in Linear Time
    Checking whether a word is Hamming-isometric in linear time Marie-Pierre B´eal ID ,∗ Maxime Crochemore ID † July 26, 2021 Abstract duced Fibonacci cubes in which nodes are on a binary alphabet and avoid the factor 11. The notion of d- A finite word f is Hamming-isometric if for any two ary n-cubes has subsequently be extended to define word u and v of the same length avoiding f, u can the generalized Fibonacci cube [10, 11, 19] ; it is the 2 be transformed into v by changing one by one all the subgraph Qn(f) of a 2-ary n-cube whose nodes avoid letters on which u differs from v, in such a way that some factor f. In this framework, a binary word f is 2 all of the new words obtained in this process also said to be Lee-isometric when, for any n 1, Qn(f) 2 ≥ avoid f. Words which are not Hamming-isometric can be isometrically embedded into Qn, that is, the 2 have been characterized as words having a border distance between two words u and v vertices of Qn(f) 2 2 with two mismatches. We derive from this charac- is the same in Qn(f) and in Qn. terization a linear-time algorithm to check whether On a binary alphabet, the definition of a Lee- a word is Hamming-isometric. It is based on pat- isometric word can be equivalently given by ignor- tern matching algorithms with k mismatches. Lee- ing hypercubes and adopting a point of view closer isometric words over a four-letter alphabet have been to combinatorics on words.
    [Show full text]
  • Modern Techniques for Constraint Solving the Casper Experience
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by New University of Lisbon's Repository MARCO VARGAS CORREIA MODERN TECHNIQUES FOR CONSTRAINT SOLVING THE CASPER EXPERIENCE Dissertação apresentada para obtenção do Grau de Doutor em Engenharia Informática, pela Universidade Nova de Lisboa, Faculdade de Ciências e Tecnologia. UNIÃO EUROPEIA Fundo Social Europeu LISBOA 2010 ii Acknowledgments First I would like to thank Pedro Barahona, the supervisor of this work. Besides his valuable scientific advice, his general optimism kept me motivated throughout the long, and some- times convoluted, years of my PhD. Additionally, he managed to find financial support for my participation on international conferences, workshops, and summer schools, and even for the extension of my grant for an extra year and an half. This was essential to keep me focused on the scientific work, and ultimately responsible for the existence of this dissertation. Designing, developing, and maintaining a constraint solver requires a lot of effort. I have been lucky to count on the help and advice of colleagues contributing directly or indirectly to the design and implementation of CASPER, to whom I am truly indebted: Francisco Azevedo introduced me to constraint solving over set variables, and was the first to point out the potential of incremental propagation. The set constraint solver integrating the work presented in this dissertation is based on his own solver CARDINAL, which he has thoroughly explained to me. Jorge Cruz was very helpful for clarifying many issues about real valued arithmetic, and propagation of constraints over real valued variables in general.
    [Show full text]
  • Download Book
    Lecture Notes in Computer Science 3279 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen University of Dortmund, Germany Madhu Sudan Massachusetts Institute of Technology, MA, USA Demetri Terzopoulos New York University, NY, USA Doug Tygar University of California, Berkeley, CA, USA Moshe Y. Vardi Rice University, Houston, TX, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany Geoffrey M. Voelker Scott Shenker (Eds.) Peer-to-Peer Systems III Third International Workshop, IPTPS 2004 La Jolla, CA, USA, February 26-27, 2004 Revised Selected Papers 13 Volume Editors Geoffrey M. Voelker University of California, San Diego Department of Computer Science and Engineering 9500 Gilman Dr., MC 0114, La Jolla, CA 92093-0114, USA E-mail: [email protected] Scott Shenker University of California, Berkeley Computer Science Division, EECS Department 683 Soda Hall, 1776, Berkeley, CA 94720, USA E-mail: [email protected] Library of Congress Control Number: Applied for CR Subject Classification (1998): C.2.4, C.2, H.3, H.4, D.4, F.2.2, E.1, D.2 ISSN 0302-9743 ISBN 3-540-24252-X Springer Berlin Heidelberg New York This work is subject to copyright.
    [Show full text]
  • Anonymous Opt-Out and Secure Computation in Data Mining
    ANONYMOUS OPT-OUT AND SECURE COMPUTATION IN DATA MINING Samuel Steven Shepard A Thesis Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE, MASTER OF ARTS December 2007 Committee: Ray Kresman, Advisor Larry Dunning So-Hsiang Chou Tong Sun ii ABSTRACT Ray Kresman, Advisor Privacy preserving data mining seeks to allow users to share data while ensuring individual and corpo- rate privacy concerns are addressed. Recently algorithms have been introduced to maintain privacy even when all but two parties collude. However, exogenous information and unwanted statistical disclosure can weaken collusion requirements and allow for approximation of sensitive information. Our work builds upon previous algorithms, putting cycle-partitioned secure sum into the mathematical framework of edge-disjoint Hamiltonian cycles and providing an additional anonymous “opt-out” procedure to help prevent unwanted statistical disclosure. iii Table of Contents CHAPTER 1: DATA MINING 1 1.1 The Basics . 1 1.2 Association Rule Mining Algorithms . 2 1.3 Distributed Data Mining . 3 1.4 Privacy-preserving Data Mining . 4 1.5 Secure Sum . 5 CHAPTER 2: HAMILTONIAN CYCLES 7 2.1 Graph Theory: An Introduction . 7 2.2 Generating Edge-disjoint Hamiltonian Cycles . 8 2.2.1 Fully-Connected . 8 2.2.2 Edge-disjoint Hamiltonian Cycles in Torus . 11 CHAPTER 3: COLLUSION RESISTANCE WITH OPT-OUT 14 3.1 Cycle-partitioned Secure Sum . 15 3.2 Problems with CPSS: Statistical Disclosure . 21 3.3 Anonymous Opt-Out . 22 3.4 Anonymous ID Assignment . 26 3.5 CPSS with Opt-out .
    [Show full text]
  • A Comparative Analysis of String Metric Algorithms
    International Journal of Latest Trends in Engineering and Technology Special Issue SACAIM 2017, pp. 138-142 e-ISSN:2278-621X A COMPARATIVE ANALYSIS OF STRING METRIC ALGORITHMS BabithaD’Cunha1, RoshiniKarishmaKarkada2 & Mrs.SuchethaVijaykumar3 Abstract: String metric is an important attribute for analysing a string pattern. A string metric provides a number indicating algorithm-specific indication distance. String metric includes many algorithms. One such algorithm is Levenshtein distance is a measure of the similarity between two strings, which we will refer to as the source string and the target string. Hamming distance algorithm is to find the distance between two strings of same length and the number of positions in which the corresponding symbols are different. Lee distance algorithm is used to state the number and length of disjoint paths between two nodes in a torus. We hereby propose to do a comparative study of all the above three algorithms by taking factors such as time and volume of the string to be searched and do an analysis of the same. Key Words: String Metric, Lee distance, Hamming distance,Levenshtein distance. 1. INTRODUCTION A string metric (also known as a string similarity metric or string distance function) is a metric that measures distance ("inverse similarity") between two text strings for approximate string matching. String matching is a technique to discover pattern from the specified input string. String matching algorithms are used to find the matches between the pattern and specified string. Various string matching algorithms are used to solve the string matching problems like wide window pattern matching, approximate string matching, polymorphic string matching, string matching with minimum mismatches, prefix matching, suffix matching, similarity measure etc.
    [Show full text]
  • A Study of Metrics of Distance and Correlation Between Ranked Lists
    A Study of Metrics of Distance and Correlation Between Ranked Lists for Compositionality Detection Christina Lioma and Niels Dalum Hansen Department of Computer Science, University of Copenhagen, Denmark Abstract Compositionality in language refers to how much the meaning of some phrase can be decomposed into the meaning of its constituents and the way these constituents are combined. Based on the premise that substitution by synonyms is meaning-preserving, compositionality can be approximated as the semantic similarity between a phrase and a version of that phrase where words have been replaced by their synonyms. Different ways of representing such phrases exist (e.g., vectors [1] or language models [2]), and the choice of representation affects the measurement of semantic similarity. We propose a new compositionality detection method that represents phrases as ranked lists of term weights. Our method approximatesthe semantic similarity between two ranked list representations using a range of well-known distance and correlation metrics. In contrast to most state-of-the-art approaches in compositionality detection, our method is completely unsupervised. Experiments with a publicly available dataset of 1048 human-annotatedphrases shows that, compared to strong supervised baselines, our approach provides superior measurement of compositionality using any of the dis- tance and correlation metrics considered. Keywords: Compositionality Detection, Metrics of Distance and Correlation 1. Introduction Compositionality in natural language describes the extent to which the meaning of a phrase can be decomposed into the meaning of its constituents and the way these arXiv:1703.03640v1 [cs.CL] 10 Mar 2017 constituents are combined. Automatic compositionality detection has long received attention (see, for instance, [1], [3], [4]).
    [Show full text]
  • Correcting Charge-Constrained Errors in the Rank-Modulation Scheme Anxiao (Andrew) Jiang, Member, IEEE, Moshe Schwartz, Member, IEEE, and Jehoshua Bruck, Fellow, IEEE
    2112 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 5, MAY 2010 Correcting Charge-Constrained Errors in the Rank-Modulation Scheme Anxiao (Andrew) Jiang, Member, IEEE, Moshe Schwartz, Member, IEEE, and Jehoshua Bruck, Fellow, IEEE Abstract—We investigate error-correcting codes for a the good performance [7]. To expand its storage capacity, research rank-modulation scheme with an application to flash memory on multilevel cells with large values of is actively underway. n devices. In this scheme, a set of cells stores information in the For flash memories, writing is more time- and energy-con- permutation induced by the different charge levels of the indi- vidual cells. The resulting scheme eliminates the need for discrete suming than reading [7]. The main factor is the iterative cell-pro- cell levels, overcomes overshoot errors when programming cells (a gramming procedure designed to avoid over-programming [2] serious problem that reduces the writing speed), and mitigates the (raising the cell’s charge level above its target level). In flash problem of asymmetric errors. In this paper, we study the proper- memories, cells are organized into blocks, where each block has ties of error-correcting codes for charge-constrained errors in the a large number of cells [7]. Cells can be programmed rank-modulation scheme. In this error model the number of errors corresponds to the minimal number of adjacent transpositions re- individually, but to decrease the state of a cell, the whole block quired to change a given stored permutation to another erroneous has to be erased to the lowest state and then re-programmed.
    [Show full text]
  • On Codes Over the Rings Fq + Ufq + Vfq + Uvfq
    The Islamic University of Gaza Deanery of Higher Studies Fuculty of Science Departemnt of Mathematics On Codes Over The Rings Fq + uFq + vFq + uvFq M.Sc. Thesis Presented by Ibrahim M. Yaghi Supervised by Dr. Mohammed M. AL-AShker Submitted in Partial Fulfillment of the Requirements for M.Sc. Degree 2013 Abstract Codes over finite rings have been studied in the early 1970's [4]. A great deal of attention has been given to codes over finite rings from 1990 [20], because of their new role in algebraic coding theory and their successful applications. In previous studies, some authors determined the structure of the ring F2 + uF2 + vF2 + 2 2 uvF2, where u = v = 0 and uv = vu, also they obtained linear codes, self dual codes, cyclic codes, and consta-cyclic codes over this ring as in [24],[25],[26],[27]. In this thesis, we aim to generalize the previous studies from the ring F2+uF2+vF2+uvF2 2 2 to the ring Fq + uFq + vFq + uvFq, where q is a power of the prime p and u = v = 0 uv = vu, so we obtain the linear codes over R = Fq + uFq + vFq + uvFq, then we investigate self dual codes over R and we find that it can be generalized only when q is a power of the prime 2, also we obtain cyclic and consta-cyclic codes over R, and we generalize the Gray map used for codes over F2 + uF2 + vF2 + uvF2, finally we obtain another gray map for codes over R. i Acknowledgements After Almighty Allah, I am grateful to my parents for their subsidization, patience, and love.
    [Show full text]
  • Needleman–Wunsch Algorithm
    Not logged in Talk Contributions Create account Log in Article Talk Read Edit View history Search Wikipedia Wiki Loves Love: Documenting festivals and celebrations of love on Commons. Help Wikimedia and win prizes by sending photos. Main page Contents Featured content Current events Needleman–Wunsch algorithm Random article From Wikipedia, the free encyclopedia Donate to Wikipedia Wikipedia store This article may be too technical for most readers to understand. Please help improve it Interaction to make it understandable to non-experts, without removing the technical details. (September 2013) (Learn how and when to remove this template message) Help About Wikipedia The Needleman–Wunsch algorithm is an algorithm used in bioinformatics to align Class Sequence Community portal protein or nucleotide sequences. It was one of the first applications of dynamic alignment Recent changes Worst-case performance Contact page programming to compare biological sequences. The algorithm was developed by Saul B. Needleman and Christian D. Wunsch and published in 1970.[1] The Worst-case space complexity Tools algorithm essentially divides a large problem (e.g. the full sequence) into a series of What links here smaller problems and uses the solutions to the smaller problems to reconstruct a solution to the larger problem.[2] It is also Related changes sometimes referred to as the optimal matching algorithm and the global alignment technique. The Needleman–Wunsch algorithm Upload file is still widely used for optimal global alignment, particularly when
    [Show full text]
  • Codes, Metrics, and Applications
    Codes, metrics, and applications Alexander Barg University of Maryland IITP RAS, Moscow ISIT 2016 Problems: § Distance distribution of codes § Duality of linear codes § Ordered linear codes and matroid invariants § Channel models matched to the ordered metrics; applications to polar codes, wiretap channels § Association schemes and bounds on the size of codes § Further extensions Acknowledgment: NSF grants CCF1217245 “Ordered metrics and their applications” CCF1422955, CCF0916919, CCF0807411 Talk summary Coding theory in a class of metric spaces: combinatorial and information-theoretic results and applications. § Channel models matched to the ordered metrics; applications to polar codes, wiretap channels § Association schemes and bounds on the size of codes § Further extensions Acknowledgment: NSF grants CCF1217245 “Ordered metrics and their applications” CCF1422955, CCF0916919, CCF0807411 Talk summary Coding theory in a class of metric spaces: combinatorial and information-theoretic results and applications. Problems: § Distance distribution of codes § Duality of linear codes § Ordered linear codes and matroid invariants Acknowledgment: NSF grants CCF1217245 “Ordered metrics and their applications” CCF1422955, CCF0916919, CCF0807411 Talk summary Coding theory in a class of metric spaces: combinatorial and information-theoretic results and applications. Problems: § Distance distribution of codes § Duality of linear codes § Ordered linear codes and matroid invariants § Channel models matched to the ordered metrics; applications to polar
    [Show full text]