Error Correcting Codes
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Superregular Matrices and Applications to Convolutional Codes
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repositório Institucional da Universidade de Aveiro Superregular matrices and applications to convolutional codes P. J. Almeidaa, D. Nappa, R.Pinto∗,a aCIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro, Portugal. Abstract The main results of this paper are twofold: the first one is a matrix theoretical result. We say that a matrix is superregular if all of its minors that are not trivially zero are nonzero. Given a a × b, a ≥ b, superregular matrix over a field, we show that if all of its rows are nonzero then any linear combination of its columns, with nonzero coefficients, has at least a − b + 1 nonzero entries. Secondly, we make use of this result to construct convolutional codes that attain the maximum possible distance for some fixed parameters of the code, namely, the rate and the Forney indices. These results answer some open questions on distances and constructions of convolutional codes posted in the literature [6, 9]. Key words: convolutional code, Forney indices, optimal code, superregular matrix 2000MSC: 94B10, 15B33, 15B05 1. Introduction Several notions of superregular matrices (or totally positive) have appeared in different areas of mathematics and engineering having in common the specification of some properties regarding their minors [2, 3, 5, 11, 14]. In the context of coding theory these matrices have entries in a finite field F and are important because they can be used to generate linear codes with good distance properties. -
Octonion Multiplication and Heawood's
CONFLUENTES MATHEMATICI Bruno SÉVENNEC Octonion multiplication and Heawood’s map Tome 5, no 2 (2013), p. 71-76. <http://cml.cedram.org/item?id=CML_2013__5_2_71_0> © Les auteurs et Confluentes Mathematici, 2013. Tous droits réservés. L’accès aux articles de la revue « Confluentes Mathematici » (http://cml.cedram.org/), implique l’accord avec les condi- tions générales d’utilisation (http://cml.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation á fin strictement personnelle du copiste est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Confluentes Math. 5, 2 (2013) 71-76 OCTONION MULTIPLICATION AND HEAWOOD’S MAP BRUNO SÉVENNEC Abstract. In this note, the octonion multiplication table is recovered from a regular tesse- lation of the equilateral two timensional torus by seven hexagons, also known as Heawood’s map. Almost any article or book dealing with Cayley-Graves algebra O of octonions (to be recalled shortly) has a picture like the following Figure 0.1 representing the so-called ‘Fano plane’, which will be denoted by Π, together with some cyclic ordering on each of its ‘lines’. The Fano plane is a set of seven points, in which seven three-point subsets called ‘lines’ are specified, such that any two points are contained in a unique line, and any two lines intersect in a unique point, giving a so-called (combinatorial) projective plane [8,7]. -
Reed-Solomon Code and Its Application 1 Equivalence
IERG 6120 Coding Theory for Storage Systems Lecture 5 - 27/09/2016 Reed-Solomon Code and Its Application Lecturer: Kenneth Shum Scribe: Xishi Wang 1 Equivalence of Codes Two linear codes are said to be equivalent if one of them can be obtained from the other by means of a sequence of transformations of the following types: (i) a permutation of the positions of the code; (ii) multiplication of symbols in a fixed position by a non-zero scalar in F . Note that these transformations can be applied to all code symbols. 2 Reed-Solomon Code In RS code each symbol in the codewords is the evaluation of a polynomial at one point α, namely, 2 1 3 6 α 7 6 2 7 6 α 7 f(α) = c0 c1 c2 ··· ck−1 6 7 : 6 . 7 4 . 5 αk−1 The whole codeword is given by n such evaluations at distinct points α1; ··· ; αn, 2 1 1 ··· 1 3 6 α1 α2 ··· αn 7 6 2 2 2 7 6 α1 α2 ··· αn 7 T f(α1) f(α2) ··· f(αn) = c0 c1 c2 ··· ck−1 6 7 = c G: 6 . .. 7 4 . 5 k−1 k−1 k−1 α1 α2 ··· αn G is the generator matrix of RSq(n; k; d) code. The order of writing down the field elements α1 to αn is not important as far as the code structure is concerned, as any permutation of the αi's give an equivalent code. i j=1;:::;n The generator matrix G is a Vandermonde matrix, which is of the form [aj]i=0;:::;k−1. -
An Introduction to Pythagorean Arithmetic
AN INTRODUCTION TO PYTHAGOREAN ARITHMETIC by JASON SCOTT NICHOLSON B.Sc. (Pure Mathematics), University of Calgary, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming tc^ the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1996 © Jason S. Nicholson, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my i department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Dale //W 39, If96. DE-6 (2/88) Abstract This thesis provides a look at some aspects of Pythagorean Arithmetic. The topic is intro• duced by looking at the historical context in which the Pythagoreans nourished, that is at the arithmetic known to the ancient Egyptians and Babylonians. The view of mathematics that the Pythagoreans held is introduced via a look at the extraordinary life of Pythagoras and a description of the mystical mathematical doctrine that he taught. His disciples, the Pythagore• ans, and their school and history are briefly mentioned. Also, the lives and works of some of the authors of the main sources for Pythagorean arithmetic and thought, namely Euclid and the Neo-Pythagoreans Nicomachus of Gerasa, Theon of Smyrna, and Proclus of Lycia, are looked i at in more detail. -
Multiplication Fact Strategies Assessment Directions and Analysis
Multiplication Fact Strategies Wichita Public Schools Curriculum and Instructional Design Mathematics Revised 2014 KCCRS version Table of Contents Introduction Page Research Connections (Strategies) 3 Making Meaning for Operations 7 Assessment 9 Tools 13 Doubles 23 Fives 31 Zeroes and Ones 35 Strategy Focus Review 41 Tens 45 Nines 48 Squared Numbers 54 Strategy Focus Review 59 Double and Double Again 64 Double and One More Set 69 Half and Then Double 74 Strategy Focus Review 80 Related Equations (fact families) 82 Practice and Review 92 Wichita Public Schools 2014 2 Research Connections Where Do Fact Strategies Fit In? Adapted from Randall Charles Fact strategies are considered a crucial second phase in a three-phase program for teaching students basic math facts. The first phase is concept learning. Here, the goal is for students to understand the meanings of multiplication and division. In this phase, students focus on actions (i.e. “groups of”, “equal parts”, “building arrays”) that relate to multiplication and division concepts. An important instructional bridge that is often neglected between concept learning and memorization is the second phase, fact strategies. There are two goals in this phase. First, students need to recognize there are clusters of multiplication and division facts that relate in certain ways. Second, students need to understand those relationships. These lessons are designed to assist with the second phase of this process. If you have students that are not ready, you will need to address the first phase of concept learning. The third phase is memorization of the basic facts. Here the goal is for students to master products and quotients so they can recall them efficiently and accurately, and retain them over time. -
Linear Block Codes Vector Spaces
Linear Block Codes Vector Spaces Let GF(q) be fixed, normally q=2 • The set Vn consisting of all n-dimensional vectors (a n-1, a n-2,•••a0), ai ∈ GF(q), 0 ≤i ≤n-1 forms an n-dimensional vector space. • In Vn, we can also define the inner product of any two vectors a = (a n-1, a n-2,•••a0) and b = (b n-1, b n-2,•••b0) n−1 i ab= ∑ abi i i=0 • If a•b = 0, then a and b are said to be orthogonal. Linear Block codes • Let C ⊂ Vn be a block code consisting of M codewords. • C is said to be linear if a linear combination of two codewords C1 and C2, a 1C1+a 2C2, is still a codeword, that is, C forms a subspace of Vn. a 1, ∈ a2 GF(q) • The dimension of the linear block code (a subspace of Vn) is denoted as k. M=qk. Dual codes Let C ⊂Vn be a linear code with dimension K. ┴ The dual code C of C consists of all vectors in Vn that are orthogonal to every vector in C. The dual code C┴ is also a linear code; its dimension is equal to n-k. Generator Matrix Assume q=2 Let C be a binary linear code with dimension K. Let {g1,g2,…gk} be a basis for C, where gi = (g i(n-1) , g i(n-2) , … gi0 ), 1 ≤i≤K. Then any codeword Ci in C can be represented uniquely as a linear combination of {g1,g2,…gk}. -
Some Mathematics
Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981 You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links. C Some Mathematics C.1 Finite field theory Most linear codes are expressed in the language of Galois theory Why are Galois fields an appropriate language for linear codes? First, a defi- nition and some examples. AfieldF is a set F = {0,F} such that 1. F forms an Abelian group under an addition operation ‘+’, with + 01 · 01 0 being the identity; [Abelian means all elements commute, i.e., 0 01 0 00 satisfy a + b = b + a.] 1 10 1 01 2. F forms an Abelian group under a multiplication operation ‘·’; multiplication of any element by 0 yields 0; Table C.1. Addition and 3. these operations satisfy the distributive rule (a + b) · c = a · c + b · c. multiplication tables for GF (2). For example, the real numbers form a field, with ‘+’ and ‘·’denoting ordinary addition and multiplication. + 01AB 0 01AB AGaloisfieldGF q q ( ) is a field with a finite number of elements . 1 10BA A unique Galois field exists for any q = pm,wherep is a prime number A AB 01 and m is a positive integer; there are no other finite fields. B BA10 · 01AB GF (2). The addition and multiplication tables for GF (2) are shown in ta- 0 00 0 0 ble C.1. These are the rules of addition and multiplication modulo 2. 1 01AB A AB GF (p). -
On a Class of Matrices with Low Displacement Rank M
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 325 (2001) 161–176 www.elsevier.com/locate/laa On a class of matrices with low displacement rank M. Barnabei∗, L.B. Montefusco Department of Mathematics, University of Bologna, Piazza di Porta S. Donato, 5 40127 Bologna, Italy Received 5 June 2000; accepted 5 September 2000 Submitted by L. Elsner Abstract AmatrixA such that, for some matrices U and V, the matrix AU − VAor the matrix A − VAU has a rank which is small compared with the order of the matrix is called a matrix with displacement structure. In this paper the authors single out a new class of matrices with dis- placement structure—namely, finite sections of recursive matrices—which includes the class of finite Hurwitz matrices, widely used in computer graphics. For such matrices it is possible to give an explicit evaluation of the displacement rank and an algebraic description, and hence a stable numerical evaluation, of the corresponding generators. The generalized Schur algorithm can therefore be used for the fast factorization of some classes of such matrices, and exhibits a backward stable behavior. © 2001 Elsevier Science Inc. All rights reserved. AMS classification: 65F05; 65F30; 15A23; 15A57 Keywords: Recursive matrices; Displacement structure; Fast factorization algorithm; Stability 1. Introduction It is well known that the notion of matrix with displacement structure is used in the literature to define classes of matrices for which fast O(n2) factorization algorithms can be given (see, e.g., [9]). -
Chapter 2: Linear Algebra User's Manual
Preprint typeset in JHEP style - HYPER VERSION Chapter 2: Linear Algebra User's Manual Gregory W. Moore Abstract: An overview of some of the finer points of linear algebra usually omitted in physics courses. May 3, 2021 -TOC- Contents 1. Introduction 5 2. Basic Definitions Of Algebraic Structures: Rings, Fields, Modules, Vec- tor Spaces, And Algebras 6 2.1 Rings 6 2.2 Fields 7 2.2.1 Finite Fields 8 2.3 Modules 8 2.4 Vector Spaces 9 2.5 Algebras 10 3. Linear Transformations 14 4. Basis And Dimension 16 4.1 Linear Independence 16 4.2 Free Modules 16 4.3 Vector Spaces 17 4.4 Linear Operators And Matrices 20 4.5 Determinant And Trace 23 5. New Vector Spaces from Old Ones 24 5.1 Direct sum 24 5.2 Quotient Space 28 5.3 Tensor Product 30 5.4 Dual Space 34 6. Tensor spaces 38 6.1 Totally Symmetric And Antisymmetric Tensors 39 6.2 Algebraic structures associated with tensors 44 6.2.1 An Approach To Noncommutative Geometry 47 7. Kernel, Image, and Cokernel 47 7.1 The index of a linear operator 50 8. A Taste of Homological Algebra 51 8.1 The Euler-Poincar´eprinciple 54 8.2 Chain maps and chain homotopies 55 8.3 Exact sequences of complexes 56 8.4 Left- and right-exactness 56 { 1 { 9. Relations Between Real, Complex, And Quaternionic Vector Spaces 59 9.1 Complex structure on a real vector space 59 9.2 Real Structure On A Complex Vector Space 64 9.2.1 Complex Conjugate Of A Complex Vector Space 66 9.2.2 Complexification 67 9.3 The Quaternions 69 9.4 Quaternionic Structure On A Real Vector Space 79 9.5 Quaternionic Structure On Complex Vector Space 79 9.5.1 Complex Structure On Quaternionic Vector Space 81 9.5.2 Summary 81 9.6 Spaces Of Real, Complex, Quaternionic Structures 81 10. -
A Standard Generator/Parity Check Matrix for Codes from the Cayley Tables Due to the Non-Associative (123)-Avoiding Patterns of AUNU Numbers
Universal Journal of Applied Mathematics 4(2): 39-41, 2016 http://www.hrpub.org DOI: 10.13189/ujam.2016.040202 A Standard Generator/Parity Check Matrix for Codes from the Cayley Tables Due to the Non-associative (123)-Avoiding Patterns of AUNU Numbers Ibrahim A.A1, Chun P.B2,*, Abubakar S.I3, Garba A.I1, Mustafa.A1 1Department of Mathematics, Usmanu Danfodiyo University Sokoto, Nigeria 2Department of Mathematics, Plateau State University Bokkos, Nigeria 3Department of Mathematics, Sokoto State University Sokoto, Nigeria Copyright©2016 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 international License. Abstract In this paper, we aim at utilizing the Cayley Eulerian graphs due to AUNU patterns had been reported in tables demonstrated by the Authors[1] in the construction of [2] and [3]. This special class of the (132) and a Generator/Parity check Matrix in standard form for a Code (123)-avoiding class of permutation patterns which were first say C Our goal is achieved first by converting the Cayley reported [4], where some group and graph theoretic tables in [1] using Mod2 arithmetic into a Matrix with entries properties were identified, had enjoyed a wide range of from the binary field. Echelon Row operations are then applications in various areas of applied Mathematics. In[1], performed (carried out) on the matrix in line with existing we described how the non-associative and non-commutative algorithms and propositions to obtain a matrix say G whose properties of the patterns can be established using their rows spans C and a matrix say H whose rows spans , the Cayley tables where a binary operation was defined to act on dual code of C, where G and H are given as, G = (Ik| X ) and the (132) and (123)-avoiding patterns of the AUNU numbers T H= ( -X |In-k ). -
Lec16.Ps (Mpage)
Potential Problems with Large Blocks 2 CSC310 – Information Theory Sam Roweis • Using very large blocks could potentially cause some serious practical problems with storage/retrieval of codewords. • In particular, if we are encoding blocks of K bits, our code will have 2K codewords. For K ≈ 1000 this is a huge number! Lecture 16: • How could we even store all the codewords? • How could we retrieve (look up) the N bit codeword corresponding to a given K bit message? Equivalent Codes & Systematic Forms • How could we check if a given block of N bits is a valid codeword or a forbidden encoding? • Last class, we saw how to solve all these problems by representing the codes mathematically and using the magic of linear algebra. November 9, 2005 • The valid codewords formed a subspace in the vector space of the N finite field Z2 . Reminder: Dealing with Long Blocks 1 Generator and Parity Check Matrices 3 • Recall that Shannon’s second theorem tells us that for any noisy • Using aritmetic mod 2, we could represent a code using either a set channel, there is some code which allows us to achieve error free of basis functions or a set of constraint (check) equations, transmission at a rate up to the capacity. represented a generator matrix G or a parity check matrix H. • However, this might require us to encode our message in very long • Every linear combination basis vectors is a valid codeword & all blocks. Why? valid codewords are spanned by the basis; similarly all valid • Intuitively it is because we need to add just the right fraction of codewords satisfy every check equation & any bitstring which redundancy; too little and we won’t be able to correct the erorrs, satisfies all equations is a valid codeword. -
Estimating the Generator Matrix from Discret Data Set
Estimating the Generator Matrix from Discret Data Set Romayssa Touaimia (Algeria) Department Probability and Statistics, Mathematical Institute Eötvös Loránd University, Budapest, Hungary A thesis submitted for the degree of MSc in Mathematics 2019. May Declaration I herewith declare that I have produced this thesis without the prohibited assistance of third parties and without making use of aids other than those specified; notions taken over directly or indirectly from other sources have been identified as such. This paper has not previously been presented in identical or similar form to any other Hungarian or foreign examination board. The thesis work was conducted under the supervision of Mr. Pröhle Tamás at Eötvös Loránd University. Budapest, May, 2019 Acknowledgements First and foremost, I would like to thank my supervision and advisor Mr. Prőhle Tamás for introducing me to this topic, for his help through the difficulties that I faced, his patience with me, his helpful suggestions and his assistance in every step through the process. I am grateful for the many discussions that we had. I would like to thank my parents whom were always there for me, sup- ported me and believed in me. Abstract Continuous Markov chain is determined by its transition rates which are the entries of the generator matrix. This generator matrix is used to fully describe the process. In this work, we will first start with an introduction about the impor- tance of the application of the Markov Chains in real life, then we will give some definitions that help us to understand the discrete and continuous time Markov chains, then we will see the relation between transition matrix and the generator matrix.