Introduction to Coding Theory: Basic Codes and Shannon’S Theorem
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Compilers & Translator Writing Systems
Compilers & Translators Compilers & Translator Writing Systems Prof. R. Eigenmann ECE573, Fall 2005 http://www.ece.purdue.edu/~eigenman/ECE573 ECE573, Fall 2005 1 Compilers are Translators Fortran Machine code C Virtual machine code C++ Transformed source code Java translate Augmented source Text processing language code Low-level commands Command Language Semantic components Natural language ECE573, Fall 2005 2 ECE573, Fall 2005, R. Eigenmann 1 Compilers & Translators Compilers are Increasingly Important Specification languages Increasingly high level user interfaces for ↑ specifying a computer problem/solution High-level languages ↑ Assembly languages The compiler is the translator between these two diverging ends Non-pipelined processors Pipelined processors Increasingly complex machines Speculative processors Worldwide “Grid” ECE573, Fall 2005 3 Assembly code and Assemblers assembly machine Compiler code Assembler code Assemblers are often used at the compiler back-end. Assemblers are low-level translators. They are machine-specific, and perform mostly 1:1 translation between mnemonics and machine code, except: – symbolic names for storage locations • program locations (branch, subroutine calls) • variable names – macros ECE573, Fall 2005 4 ECE573, Fall 2005, R. Eigenmann 2 Compilers & Translators Interpreters “Execute” the source language directly. Interpreters directly produce the result of a computation, whereas compilers produce executable code that can produce this result. Each language construct executes by invoking a subroutine of the interpreter, rather than a machine instruction. Examples of interpreters? ECE573, Fall 2005 5 Properties of Interpreters “execution” is immediate elaborate error checking is possible bookkeeping is possible. E.g. for garbage collection can change program on-the-fly. E.g., switch libraries, dynamic change of data types machine independence. -
Modern Coding Theory: the Statistical Mechanics and Computer Science Point of View
Modern Coding Theory: The Statistical Mechanics and Computer Science Point of View Andrea Montanari1 and R¨udiger Urbanke2 ∗ 1Stanford University, [email protected], 2EPFL, ruediger.urbanke@epfl.ch February 12, 2007 Abstract These are the notes for a set of lectures delivered by the two authors at the Les Houches Summer School on ‘Complex Systems’ in July 2006. They provide an introduction to the basic concepts in modern (probabilistic) coding theory, highlighting connections with statistical mechanics. We also stress common concepts with other disciplines dealing with similar problems that can be generically referred to as ‘large graphical models’. While most of the lectures are devoted to the classical channel coding problem over simple memoryless channels, we present a discussion of more complex channel models. We conclude with an overview of the main open challenges in the field. 1 Introduction and Outline The last few years have witnessed an impressive convergence of interests between disciplines which are a priori well separated: coding and information theory, statistical inference, statistical mechanics (in particular, mean field disordered systems), as well as theoretical computer science. The underlying reason for this convergence is the importance of probabilistic models and/or probabilistic techniques in each of these domains. This has long been obvious in information theory [53], statistical mechanics [10], and statistical inference [45]. In the last few years it has also become apparent in coding theory and theoretical computer science. In the first case, the invention of Turbo codes [7] and the re-invention arXiv:0704.2857v1 [cs.IT] 22 Apr 2007 of Low-Density Parity-Check (LDPC) codes [30, 28] has motivated the use of random constructions for coding information in robust/compact ways [50]. -
Chapter 2 Basics of Scanning And
Chapter 2 Basics of Scanning and Conventional Programming in Java In this chapter, we will introduce you to an initial set of Java features, the equivalent of which you should have seen in your CS-1 class; the separation of problem, representation, algorithm and program – four concepts you have probably seen in your CS-1 class; style rules with which you are probably familiar, and scanning - a general class of problems we see in both computer science and other fields. Each chapter is associated with an animating recorded PowerPoint presentation and a YouTube video created from the presentation. It is meant to be a transcript of the associated presentation that contains little graphics and thus can be read even on a small device. You should refer to the associated material if you feel the need for a different instruction medium. Also associated with each chapter is hyperlinked code examples presented here. References to previously presented code modules are links that can be traversed to remind you of the details. The resources for this chapter are: PowerPoint Presentation YouTube Video Code Examples Algorithms and Representation Four concepts we explicitly or implicitly encounter while programming are problems, representations, algorithms and programs. Programs, of course, are instructions executed by the computer. Problems are what we try to solve when we write programs. Usually we do not go directly from problems to programs. Two intermediate steps are creating algorithms and identifying representations. Algorithms are sequences of steps to solve problems. So are programs. Thus, all programs are algorithms but the reverse is not true. -
A [49 16 6 ]-Linear Code As Product of the [7 4 2 ] Code Due to Aunu and the Hamming [7 4 3 ] Code
General Letters in Mathematics, Vol. 4, No.3 , June 2018, pp.114 -119 Available online at http:// www.refaad.com https://doi.org/10.31559/glm2018.4.3.4 A [49 16 6 ]-Linear Code as Product Of The [7 4 2 ] Code Due to Aunu and the Hamming [7 4 3 ] Code CHUN, Pamson Bentse1, IBRAHIM Alhaji Aminu2, and MAGAMI, Mohe'd Sani3 1 Department Of Mathematics, Plateau State University Bokkos, Jos Nigeria [email protected] 2 Department Of Mathematics, Usmanu Danfodiyo University Sokoto, Sokoto Nigeria [email protected] 3 Department Of Mathematics, Usmanu Danfodiyo University Sokoto, Sokoto Nigeria [email protected] Abstract: The enumeration of the construction due to "Audu and Aminu"(AUNU) Permutation patterns, of a [ 7 4 2 ]- linear code which is an extended code of the [ 6 4 1 ] code and is in one-one correspondence with the known [ 7 4 3 ] - Hamming code has been reported by the Authors. The [ 7 4 2 ] linear code, so constructed was combined with the known Hamming [ 7 4 3 ] code using the ( u|u+v)-construction method to obtain a new hybrid and more practical single [14 8 3 ] error- correcting code. In this paper, we provide an improvement by obtaining a much more practical and applicable double error correcting code whose extended version is a triple error correcting code, by combining the same codes as in [1]. Our goal is achieved through using the product code construction approach with the aid of some proven theorems. Keywords: Cayley tables, AUNU Scheme, Hamming codes, Generator matrix,Tensor product 2010 MSC No: 97H60, 94A05 and 94B05 1. -
Scripting: Higher- Level Programming for the 21St Century
. John K. Ousterhout Sun Microsystems Laboratories Scripting: Higher- Cybersquare Level Programming for the 21st Century Increases in computer speed and changes in the application mix are making scripting languages more and more important for the applications of the future. Scripting languages differ from system programming languages in that they are designed for “gluing” applications together. They use typeless approaches to achieve a higher level of programming and more rapid application development than system programming languages. or the past 15 years, a fundamental change has been ated with system programming languages and glued Foccurring in the way people write computer programs. together with scripting languages. However, several The change is a transition from system programming recent trends, such as faster machines, better script- languages such as C or C++ to scripting languages such ing languages, the increasing importance of graphical as Perl or Tcl. Although many people are participat- user interfaces (GUIs) and component architectures, ing in the change, few realize that the change is occur- and the growth of the Internet, have greatly expanded ring and even fewer know why it is happening. This the applicability of scripting languages. These trends article explains why scripting languages will handle will continue over the next decade, with more and many of the programming tasks in the next century more new applications written entirely in scripting better than system programming languages. languages and system programming -
Coding Theory: Linear Error-Correcting Codes Coding Theory Basic Definitions Error Detection and Correction Finite Fields Anna Dovzhik
Coding Theory: Linear Error- Correcting Codes Anna Dovzhik Outline Coding Theory: Linear Error-Correcting Codes Coding Theory Basic Definitions Error Detection and Correction Finite Fields Anna Dovzhik Linear Codes Hamming Codes Finite Fields Revisited BCH Codes April 23, 2014 Reed-Solomon Codes Conclusion Coding Theory: Linear Error- Correcting 1 Coding Theory Codes Basic Definitions Anna Dovzhik Error Detection and Correction Outline Coding Theory 2 Finite Fields Basic Definitions Error Detection and Correction 3 Finite Fields Linear Codes Linear Codes Hamming Codes Hamming Codes Finite Fields Finite Fields Revisited Revisited BCH Codes BCH Codes Reed-Solomon Codes Reed-Solomon Codes Conclusion 4 Conclusion Definition A q-ary word w = w1w2w3 ::: wn is a vector where wi 2 A. Definition A q-ary block code is a set C over an alphabet A, where each element, or codeword, is a q-ary word of length n. Basic Definitions Coding Theory: Linear Error- Correcting Codes Anna Dovzhik Definition If A = a ; a ;:::; a , then A is a code alphabet of size q. Outline 1 2 q Coding Theory Basic Definitions Error Detection and Correction Finite Fields Linear Codes Hamming Codes Finite Fields Revisited BCH Codes Reed-Solomon Codes Conclusion Definition A q-ary block code is a set C over an alphabet A, where each element, or codeword, is a q-ary word of length n. Basic Definitions Coding Theory: Linear Error- Correcting Codes Anna Dovzhik Definition If A = a ; a ;:::; a , then A is a code alphabet of size q. Outline 1 2 q Coding Theory Basic Definitions Error Detection Definition and Correction Finite Fields A q-ary word w = w1w2w3 ::: wn is a vector where wi 2 A. -
Mceliece Cryptosystem Based on Extended Golay Code
McEliece Cryptosystem Based On Extended Golay Code Amandeep Singh Bhatia∗ and Ajay Kumar Department of Computer Science, Thapar university, India E-mail: ∗[email protected] (Dated: November 16, 2018) With increasing advancements in technology, it is expected that the emergence of a quantum computer will potentially break many of the public-key cryptosystems currently in use. It will negotiate the confidentiality and integrity of communications. In this regard, we have privacy protectors (i.e. Post-Quantum Cryptography), which resists attacks by quantum computers, deals with cryptosystems that run on conventional computers and are secure against attacks by quantum computers. The practice of code-based cryptography is a trade-off between security and efficiency. In this chapter, we have explored The most successful McEliece cryptosystem, based on extended Golay code [24, 12, 8]. We have examined the implications of using an extended Golay code in place of usual Goppa code in McEliece cryptosystem. Further, we have implemented a McEliece cryptosystem based on extended Golay code using MATLAB. The extended Golay code has lots of practical applications. The main advantage of using extended Golay code is that it has codeword of length 24, a minimum Hamming distance of 8 allows us to detect 7-bit errors while correcting for 3 or fewer errors simultaneously and can be transmitted at high data rate. I. INTRODUCTION Over the last three decades, public key cryptosystems (Diffie-Hellman key exchange, the RSA cryptosystem, digital signature algorithm (DSA), and Elliptic curve cryptosystems) has become a crucial component of cyber security. In this regard, security depends on the difficulty of a definite number of theoretic problems (integer factorization or the discrete log problem). -
Modern Coding Theory: the Statistical Mechanics and Computer Science Point of View
Modern Coding Theory: The Statistical Mechanics and Computer Science Point of View Andrea Montanari1 and R¨udiger Urbanke2 ∗ 1Stanford University, [email protected], 2EPFL, ruediger.urbanke@epfl.ch February 12, 2007 Abstract These are the notes for a set of lectures delivered by the two authors at the Les Houches Summer School on ‘Complex Systems’ in July 2006. They provide an introduction to the basic concepts in modern (probabilistic) coding theory, highlighting connections with statistical mechanics. We also stress common concepts with other disciplines dealing with similar problems that can be generically referred to as ‘large graphical models’. While most of the lectures are devoted to the classical channel coding problem over simple memoryless channels, we present a discussion of more complex channel models. We conclude with an overview of the main open challenges in the field. 1 Introduction and Outline The last few years have witnessed an impressive convergence of interests between disciplines which are a priori well separated: coding and information theory, statistical inference, statistical mechanics (in particular, mean field disordered systems), as well as theoretical computer science. The underlying reason for this convergence is the importance of probabilistic models and/or probabilistic techniques in each of these domains. This has long been obvious in information theory [53], statistical mechanics [10], and statistical inference [45]. In the last few years it has also become apparent in coding theory and theoretical computer science. In the first case, the invention of Turbo codes [7] and the re-invention of Low-Density Parity-Check (LDPC) codes [30, 28] has motivated the use of random constructions for coding information in robust/compact ways [50]. -
The Packing Problem in Statistics, Coding Theory and Finite Projective Spaces: Update 2001
The packing problem in statistics, coding theory and finite projective spaces: update 2001 J.W.P. Hirschfeld School of Mathematical Sciences University of Sussex Brighton BN1 9QH United Kingdom http://www.maths.sussex.ac.uk/Staff/JWPH [email protected] L. Storme Department of Pure Mathematics Krijgslaan 281 Ghent University B-9000 Gent Belgium http://cage.rug.ac.be/ ls ∼ [email protected] Abstract This article updates the authors’ 1998 survey [133] on the same theme that was written for the Bose Memorial Conference (Colorado, June 7-11, 1995). That article contained the principal results on the packing problem, up to 1995. Since then, considerable progress has been made on different kinds of subconfigurations. 1 Introduction 1.1 The packing problem The packing problem in statistics, coding theory and finite projective spaces regards the deter- mination of the maximal or minimal sizes of given subconfigurations of finite projective spaces. This problem is not only interesting from a geometrical point of view; it also arises when coding-theoretical problems and problems from the design of experiments are translated into equivalent geometrical problems. The geometrical interest in the packing problem and the links with problems investigated in other research fields have given this problem a central place in Galois geometries, that is, the study of finite projective spaces. In 1983, a historical survey on the packing problem was written by the first author [126] for the 9th British Combinatorial Conference. A new survey article stating the principal results up to 1995 was written by the authors for the Bose Memorial Conference [133]. -
Arxiv:1111.6055V1 [Math.CO] 25 Nov 2011 2 Definitions
Adinkras for Mathematicians Yan X Zhang Massachusetts Institute of Technology November 28, 2011 Abstract Adinkras are graphical tools created for the study of representations in supersym- metry. Besides having inherent interest for physicists, adinkras offer many easy-to-state and accessible mathematical problems of algebraic, combinatorial, and computational nature. We use a more mathematically natural language to survey these topics, suggest new definitions, and present original results. 1 Introduction In a series of papers starting with [8], different subsets of the “DFGHILM collaboration” (Doran, Faux, Gates, Hübsch, Iga, Landweber, Miller) have built and extended the ma- chinery of adinkras. Following the spirit of Feyman diagrams, adinkras are combinatorial objects that encode information about the representation theory of supersymmetry alge- bras. Adinkras have many intricate links with other fields such as graph theory, Clifford theory, and coding theory. Each of these connections provide many problems that can be compactly communicated to a (non-specialist) mathematician. This paper is a humble attempt to bridge the language gap and generate communication. We redevelop the foundations in a self-contained manner in Sections 2 and 4, using different definitions and constructions that we consider to be more mathematically natural for our purposes. Using our new setup, we prove some original results and make new interpretations in Sections 5 and 6. We wish that these purely combinatorial discussions will equip the readers with a mental model that allows them to appreciate (or to solve!) the original representation-theoretic problems in the physics literature. We return to these problems in Section 7 and reconsider some of the foundational questions of the theory. -
Coded Computing Via Binary Linear Codes: Designs and Performance Limits Mahdi Soleymani∗, Mohammad Vahid Jamali∗, and Hessam Mahdavifar, Member, IEEE
1 Coded Computing via Binary Linear Codes: Designs and Performance Limits Mahdi Soleymani∗, Mohammad Vahid Jamali∗, and Hessam Mahdavifar, Member, IEEE Abstract—We consider the problem of coded distributed processing capabilities. A coded computing scheme aims to computing where a large linear computational job, such as a divide the job to k < n tasks and then to introduce n − k matrix multiplication, is divided into k smaller tasks, encoded redundant tasks using an (n; k) code, in order to alleviate the using an (n; k) linear code, and performed over n distributed nodes. The goal is to reduce the average execution time of effect of slower nodes, also referred to as stragglers. In such a the computational job. We provide a connection between the setup, it is often assumed that each node is assigned one task problem of characterizing the average execution time of a coded and hence, the total number of encoded tasks is n equal to the distributed computing system and the problem of analyzing the number of nodes. error probability of codes of length n used over erasure channels. Recently, there has been extensive research activities to Accordingly, we present closed-form expressions for the execution time using binary random linear codes and the best execution leverage coding schemes in order to boost the performance time any linear-coded distributed computing system can achieve. of distributed computing systems [6]–[18]. However, most It is also shown that there exist good binary linear codes that not of the work in the literature focus on the application of only attain (asymptotically) the best performance that any linear maximum distance separable (MDS) codes. -
The Use of Coding Theory in Computational Complexity 1 Introduction
The Use of Co ding Theory in Computational Complexity Joan Feigenbaum ATT Bell Lab oratories Murray Hill NJ USA jfresearchattcom Abstract The interplay of co ding theory and computational complexity theory is a rich source of results and problems This article surveys three of the ma jor themes in this area the use of co des to improve algorithmic eciency the theory of program testing and correcting which is a complexity theoretic analogue of error detection and correction the use of co des to obtain characterizations of traditional complexity classes such as NP and PSPACE these new characterizations are in turn used to show that certain combinatorial optimization problems are as hard to approximate closely as they are to solve exactly Intro duction Complexity theory is the study of ecient computation Faced with a computational problem that can b e mo delled formally a complexity theorist seeks rst to nd a solution that is provably ecient and if such a solution is not found to prove that none exists Co ding theory which provides techniques for robust representation of information is valuable b oth in designing ecient solutions and in proving that ecient solutions do not exist This article surveys the use of co des in complexity theory The improved upp er b ounds obtained with co ding techniques include b ounds on the numb er of random bits used by probabilistic algo rithms and the communication complexity of cryptographic proto cols In these constructions co des are used to design small sample spaces that approximate the b ehavior