LINEAR CODES WITH NON-UNIFORM ERROR CORRECTION CAPABILITY
|
|
- Martina Mason
- 5 years ago
- Views:
Transcription
1 LINEAR CODES WITH NON-UNIFORM ERROR CORRECTION CAPABILITY By Margaret Ann Bernard The University of the West Indies and Bhu Dev Sharma Xavier University of Louisiana, New Orleans ABSTRACT This paper introduces a class of linear codes which are non-uniform error correcting, i.e. they have the capability of correcting different errors in different codewords. A technique for specifying error characteristics in terms of algebraic inequalities, rather than the traditional spheres of radius e, is used. A construction is given for deriving these codes from known linear block codes. This is accomplished by a new method called parity sectioned reduction. In this method, the parity check matrix of a uniform error correcting linear code is reduced by dropping some rows and columns and the error range inequalities are modified. 1. INTRODUCTION The linear codes studied in coding literature for correcting random errors are such that the codes can correct uniformly up to e random errors in every code word. However, the situation may arise where certain words have a greater requirement for error control than others. A constant length code which corrects non-uniform errors i.e. different number of errors in different code words, may be more suitable. Some work has already appeared on non-uniform error correction. In [5], (see also [1] ), the authors examined perfect codes and showed that by a process of 'sectioning', non-uniform error correcting codes can be produced that remain 'perfect' in the sense that their error ranges remain disjoint and exhaust the whole space. The idea arises also in [3] where purely combinatorial results on sphere packings of different radii have been considered. The codes widely studied in the literature are linear codes in general and specifically minimum distance specified codes like BCH codes [6]. If non-uniform error correcting codes are to be obtained, it is natural to consider obtaining them from linear codes and other well-defined codes. In this paper, we develop a systematic method to produce linear codes which are non-uniform error correcting from known linear codes. The approach focuses on the errors to be corrected; this is done by studying the error ranges, representing them as algebraic inequalities. This technique arose out of a study made by the authors on error correcting codes with variable word lengths [2,4]. Since there is no well defined code
2 space for a variable length code, a technique was developed for representing error ranges in inequality form, which have meaning in spaces of any dimension. The errors to be corrected do not necessarily have to form a sphere of radius e around the code word but may form an asymmetric figure corresponding to the error characteristics. A non-uniform error correcting linear code is produced by 'sectioning ' a uniform error correcting linear code at a parity check position. The new parity check matrix is obtained by reducing the parity check matrix of the uniform error correcting code in a particular fashion. The error ranges are represented by modified inequalities. In section 2, we give some definitions and concepts. In section 3, we discuss sectioning of linear codes at parity check positions; this produces the non-uniform error correcting codes. In section 4, we consider the effect of sectioning linear codes at information positions; what we obtain are shortened linear codes and their coset codes. 2. DEFINITIONS AND CONCEPTS Let C be an e random error correcting, (n,k) linear code over GF(q) with codewords {c 1, c 2,...., c qk }. The error range, E( c i ), of a codeword c i, i = 1, 2,....., qk consists of the set of n-tuples {x = x 1 x 2... x n } over GF(q) such that w ( c i - x ) e, where w (. ) is the Hamming weight of the vector in (. ). Traditionally in coding theory, the error range of a codeword c i has been represented as a sphere of radius e around c i. In this paper, an alternative representation of error ranges is used. Algebraic inequalities are used to define and determine the error ranges of codewords. If c i = ( c i,1, c i,2,.., c i,n ), i = 1, 2,....., qk is a codeword of C, the error range inequality of c i is x 1 - c i,1 + x 2 - c i, x n - c i,n e where x j {0,1,..., q-1}, j = 1, 2,..., n and x j - c i,j, j = 1, 2,...., n, is the Hamming distance between the digits xj and c i, j. For the binary case, x j - c i,j is simply the absolute difference between xj and c i, j. Each set of values for x 1, x 2,... x n which is a solution to the inequality represents a vector which is at a distance e or less from c i. The set of all solutions to the inequality is the error range of c i. This algebraic inequality thus forms a convenient representation of the error range. It is an alternative to the geometric representation of the error range as a sphere. In later sections of this paper, we use a process called sectioning of a code; we now give a definition of a sectioned code and take it up later for further study. Definition : Let C be an e error correcting (n,k) linear code over GF(q) with code words { c 1, c 2,...., c qk }; let the error range inequalities of the codewords c i, i = 1, 2,....., qk be given by n x j - c i,j e i = 1, 2,....., qk. j = 1
3 For any integer g e, an (n-g) sectioned code of C is a code C,with codeword length n-g, which is obtained by deleting some g positions from each code word of C and by fixing the values of the corresponding g variables in the error range inequalities of the code words of C. Some of the new range inequalities formed may have no valid solutions; the codewords of C corresponding to these inequalities are simply dropped in forming C. In general, the error range inequalities of the codewords of a sectioned code do not all have the same number of solutions; the sectioned code is, in general, non-uniform error correcting. Next, we give a theorem which is a generalization of the well known, minimum distance 2e+1 criterion as it applies to non-uniform error correcting codes; it will be used in later sections. THEOREM 2.1 (Sharma & Bernard [5] ) A code C with code word length n can correct e 1 random errors in m 1 code words, e 2 errors in m 2 code words and so on, up to e g errors in m g code words, if and only if we can partition C into g subsets C 1, C 2,..., C g of sizes m 1, m 2,..., m g respectively, such that for each c C i and c' Cj, i, j = 1,2,..., g, d ( c, c' ) e i + ej + 1. Sectioning of a code may be done at information or parity check positions. We will first discuss the sectioning of linear codes at parity check positions. 3. SECTIONING LINEAR CODES AT PARITY CHECK POSITIONS In this section we examine the effect of sectioning linear codes at parity check positions. What we obtain are classes of non-uniform error correcting linear codes. The method used, which we refer to as 'parity sectioned reduction', induces a reduction of the parity check matrix that has not been previously considered in the literature. We first introduce the ideas by means of two examples; following these, the general result is given in Theorem 3.1. In the first example, we discuss sectioning of a single error correcting code; this produces non-uniform error correcting codes of smaller codeword length which are capable of correcting single errors in some codewords and zero errors in other codewords. In the second example, we discuss sectioning of a double error correcting BCH code; this produces non-uniform error correcting codes of smaller codeword length which are capable of correcting two, one and zero errors.
4 EXAMPLE 3.1 Consider the binary, single error correcting, (6,3) linear systematic code C = { c 1, c 2,...., c 2 3 }. The parity check matrix in systematic form is as follows: H 3,6 = [ A I 3 ] = The error range inequalities of the codewords c i, i = 1, 2,....., 2 3 are 6 x j - c i,j 1 i = 1, 2,....., 2 3. [3.1] Since this code is single error correcting (e = 1), we may select at most one position for sectioning (i.e. g e ). We may select any one of the parity check positions for parity sectioned reduction, say c i,6, i = 1, 2,....., 2 3. Since this is a binary code, in the error range inequalities, x 6 may have one of two possible values, namely 0 or 1. Let us consider first x 6 = 0. The error range inequalities of 3.1 may be written as: 5 x j - c i,j c i,6 1 i = 1, 2,....., 2 3. [3.2] For those codewords of C with c i,6 = 0, the constant on the right hand side of the inequalities of 3.2 will be unchanged. For those codewords of C with c i,6 = 1, the constant on the right hand side of the inequalities will be reduced by 1. The sectioned code C' produced by the sectioning has codeword length 5 and is capable of correcting e 1 = 1 errors in those code words formed from words of C with c i,6 = 0, and e 2 = 1-1 = 0 errors in those code words formed from words of C with c i,6 = 1. The code C' consists of two disjoint subsets C 1 and C 2 of code words such that if c 1,c 2 C 1 d ( c 1,c 2 ) 3 if c 1,c 2 C 2 d ( c 1,c 2 ) 3 if c 1 C 1 and c 2 C 2 d ( c 1,c 2 ) 2 This satisfies the distance criteria of Theorem 2.1 for e 1 = 1 and e 2 = 0. The parity check matrix H' of the new code C' can be obtained by an interesting reduction of H. We are sectioning at the third of the three parity check positions; H' is obtained by deleting the third column of I 3 (i.e. the last column of H) and the third row of I 3 (i.e. the third row of H) to give H' = The code C' obtained from H' is a linear (5,3) code capable of correcting e 1 = 1 errors in the 2 2 code words of C 1 (those code words of C'with c i,2 + c i,3 = 0 ) and e 2 = 0 errors in the 2 2 code words of C 2 (those code words of C'with c i,2 + c i,3 = 1 ).
5 In the preceeding discussion, we have considered the situation for x 6 = 0. We now consider the case when x 6 = 1. The range inequalities of 3.1 would be 5 x j - c i,j c i,6 1 i = 1, 2,....., 2 3. What we obtain is a sectioned code with the same two disjoint subsets C 1 and C 2 as in C' but now with the words of C 1 capable of correcting 0 errors and those of C 2 capable of correcting single error. The above discussion was for sectioning at c i,6. The result would be similar if we sectioned at any of the parity check positions. We would obtain a non-uniform error correcting code with the codeword length reduced to 5 but with the information bits unchanged. The method introduced for reducing the parity check matrix H n-k,n = [ A I n-k ] is as follows: If we section at the pth parity check position, we delete the pth column and row of I n-k (deleting the pth row of I n-k is to be interpreted as deleting the pth row of H). In Example 3.1, we used a code which is single error correcting and obtained sectioned codes with single and zero error correction capability. In the second example, we will use a double error correcting BCH code; we will obtain non-uniform error correcting sectioned codes capable of correcting double, single and zero errors in different codewords. EXAMPLE 3.2 Consider the binary (15,7) double error correcting BCH code generated by g(x) = ( x 4 + x + 1 ) ( x 4 + x 3 + x + 1 ) with parity check matrix H = 1 α α 2 α 3... α 14 1 α 3 α 6 α 9... α 12 where α is a root of x 4 + x + 1 and the minimum polynomial of α 3 is x 4 + x 3 + x 2 + x+1 The error range inequalities of the codewords c i, i = 1, 2,....., 2 7 are 15 x j - c i,j 2 i = 1, 2,....., 2 7. We now give the parity check matrix in systematic form for ease of visualizing the ideas following: H 8,15 = [ A I 8 ] =
6 We may select any parity check position for parity sectioned reduction, say the 8th parity check position, c i,15 ; let us also set x 15 = 0 in the error range inequalities. We form the parity check matrix of a (14,7) non-uniform error correcting sectioned code by deleting the last (15th) column and last (8th) row of H (i.e. the 8th column and 8th row of I 8 ). We obtain two disjoint subsets C 1 and C 2 of code words. C 1 has 2 6 code words, namely those for which the dropped parity check equation was c i,4 + c i,6 + c i,7 =0 The error range inequalities of the codewords of C 1 are: 14 x j - c i,j 2 i = 1, 2,....., 2 6. C 2 also has 2 6 code words, namely those for which the dropped parity check equation was c i,4 + c i,6 + c i,7 = 1. The error range inequalities of the codewords of C 2 are: 14 x j - c i,j 1 i = 1, 2,....., 2 6. We may proceed further by selecting any two parity check positions for dropping, say the 7th and 8th parity check positions, c i,14 and c i,15. Let us select also x 14 = x 15 = 0 in the inequalities. The parity check matrix of the (13,7) sectioned code is obtained by dropping the 14th and 15th columns and the 7th and 8th rows of H (i.e. the 7th and 8th columns and rows of I 8 ). We obtain three disjoint subsets C 1, C 2 and C 3 of code words. C 1 has 2 5 code words, with error range inequalities 13 x j - c i,j 2 i = 1, 2,....., 2 5. C 2 has code words, with error range inequalities 13 x j - c i,j 1 i = 1, 2,....., 2 5. C 3 has 2 5 code words, with error range inequalities 13 x j - c i,j 0 i = 1, 2,....., 2 5. The distance criteria of Theorem 2.1 are clearly satisfied. The code is a (13,7) linear code capable of correcting e 1 = 2 errors in the code words of C 1, e 2 = 1 errors in the code words of C 2, and e 3 = 0 errors in the code words of C 3. The two examples discussed illustrate a process for deriving non-uniform error correcting codes which we have called 'parity sectioned reduction' of the parity check matrix and the error range inequalities of a linear code. We now formally define parity sectioned reduction for the binary case: Definition : Let C be a binary e error correcting (n,k) linear systematic code with parity check matrix H n-k,n = [ A I n-k ] and error range inequalities n x j - c i,j e i = 1, 2,....., 2k. j = 1
7 By g-parity sectioned reduction of the code C, we mean the following operations on H and the error range inequalities: 1. select some g ( e ) parity check positions for sectioning; if the code is sectioned at the pth check position, then delete the pth column and row of I n-k. A reduced matrix H n-k-g,n-g = [ A' I n-k-g ] is obtained. 2. in each code word of C, delete the g parity check digits; in the error range inequalities, assign values from (0,1) to the variables corresponding to these g positions. Next, we will state in Theorem 3.1, the method discussed in this section for deriving nonuniform error correcting linear codes; but first, we state a Lemma that is needed in the proof of Theorem 3.1. LEMMA 3.1 Let C be a q-ary (n,k) linear code. The number of code words of C which have given constant values in some g ( k ) positions, is q k-g. PROOF This Lemma can be proved in a straightforward manner using coset decomposition with respect to that subgroup of the code which has all zeros in the given positions. Consider the set C' of code words of C which have 0's in g positions c i 1, c i,... c i ; g < k. The vector (0, 0,..., 0) belongs to C', and straightforwardly C' is a subgroup of C. Next, consider the coset decomposition of C with respect to its subgroup C'. There are clearly q g cosets. Thus C' has q (k-g) vectors. Now, take any coset; it has q(k-g) vectors all having same fixed values in positions c i 1, c i,... c i This proves the result. THEOREM 3.1 Let C be a binary, e-error correcting (n,k) linear, systematic code with parity check matrix H n-k,n = [ A I n-k ]. g-parity sectioned reduction of H gives a code C' which is non-uniform error correcting (n-g,k) linear code having code words in g+1 sets {C 0 ', C 1 ',.... C g ' } with j g 2 k-g code words in C j ', j = 0, 1,... g, such that in the word c C j ', the code C' can correct up to e-j errors. PROOF Let C have 2 k code words { c 1, c 2,...., c 2 k } correcting randomly up to e errors. The error range inequalities for code words c i are n x j - c i,j e i = 1, 2,....., 2k. j = 1 Without loss of generality, let the columns to be deleted in parity sectioned reduction be the last g columns of H. Obviously, C being a systematic code, the deleted g positions are check positions, as required. We may write the error range inequalities as:
8 n-g n x j - c i,j + x j - c i,j e i = 1, 2,....., 2k. j = 1 j = n-g+1 In the sectioning, we are at liberty to set the values of the g variables x n-g+1, x n-g+2,...,x n of the range inequalities to either 0 or 1. Let us choose x n-g+1 = x n-g+2 =.... = x n = 0. We consider now the g-bit portions c i,n-g+1, c i,n-g+2,... c i,n, i = 1,2,... 2 k of the code words of C. The code words which have j ( j = 0, 1,... g ) non-zero values in the last g bits can be selected in j g ways and each of these, according to Lemma 3.1, occurs 2 k-g times. Thus the number of code words with j non-zero values in the last g bits is g j 2 k-g. When the g bits are dropped, the right hand side of the inequalities would be reduced by j for those code words which had j non-zero values. The parity sectioned reduction therefore gives an (n-g,k) code C' = { C 0 ', C 1 ',..., C g ' } in which the error range of each code word in the subset C j, j = 0, 1,..., g, contains all vectors at a distance e-j or less from it. It is easy to see that the ranges remain disjoint and the distance criteria of Theorem 2.1 are satisfied. Hence the Theorem. 4.SECTIONING LINEAR CODES AT INFORMATION POSITIONS In this section, we examine the effect of sectioning linear codes at information positions. The situation is quite different from that obtained in sectioning at parity check positions. If an information position is dropped, we naturally would expect to get a shortened code. The reduction of the parity check matrix is simply to delete the columns corresponding to the sectioned positions. For the sake of completeness, we look at one example, mainly to see what happens with the range inequalities. Let us consider the (6,3) linear code C of Example 3.1. We may select any of the information positions for sectioning, say c i,1, i = 1,2,..., 2 3. Also, in the range inequalities, x 1 may take values 0 or 1. Let us select x 1 = 0. For the 2 2 code words of C which have c i,1 = 0, the right hand side of the inequalities will be unaffected; the shortened code, C" will continue to correct single errors in those code words. Those code words of C which have c i,1 = 1 will now not satisfy the reduced parity check equations and hence have no corresponding codewords in the shortened code. Hence, the sectioned code, C" is a shortened (5,2)code with uniform single error correction capability. The range inequalities are: 6 x j - c i,j 1 i = 1, 2,....., 2 2. j = 2 If, in sectioning the code C, we made the other choice, x 1 = 1, the code we obtain is also uniform single error correcting; it is a coset code of C" formed from words of C in which c i,1 = 1. We may select any of the other information positions for sectioning. In all cases, when we section at an information position, we do not get a non-uniform error correcting code; rather, we get the shortened code and a coset code of that shortened code.
9 5. CONCLUDING REMARKS The method of parity sectioned reduction of the H matrix, introduced for constructing non-uniform error correcting codes, has not been considered previously in the literature while several of the known modifications of H have produced very interesting and useful codes. Algebraic inequalities were used as an essential tool for defining and determining the error ranges of code words. Traditionally, error ranges have been studied in terms of spheres and sphere packings. The range inequalities provide an alternative representation of error ranges that allow determination of the error ranges in a different manner. Finally, one may be tempted to examine the uniform error correcting linear codes, without sectioning, for non-uniform error correction. However, this does not work, at least for the Hamming codes because it is not possible to partition the code words in two or more different sets wherein the distance criterion of Theorem 2.1, for different values of e i 's, could work. The sectioning of the BCH, RS and other well known codes may provide interesting situations. REFERENCES 1. M.A.Bernard "Error correcting Codes with Variable Lengths and Non-uniform Errors", Ph.D. Thesis, The University of the West Indies, Trinidad, M.A.Bernard and B.D.Sharma "A Lower Bound on Average Code word Length of Variable Length Error correcting Codes", IEEE IT Vol. 36, No. 6, pp , B.Montaron and G. Cohen "Codes Parfaits Binaires A Plusieurs Rayons", Revue Du Cethedec NS 79-2, pp.35-58, B.D.Sharma and M.A.Bernard "A Search for Perfect Codes of Variable Word Lengths", Journal of Computing and Information, Vol.1, No.1, pp.45-68, B.D.Sharma and M.A.Bernard "Combinatorial Results on Non-uniform Error correcting Codes", presented at Fourth Carbondale Combinatorics Conference, Carbondale (Ill. ), Nov. 2-4, F.J.Macwilliams and N.T.A.Sloane "The Theory of Error-correcting Codes", North Holland Pub. Co., 1978.
Construction C : an inter-level coded version of Construction C
Construction C : an inter-level coded version of Construction C arxiv:1709.06640v2 [cs.it] 27 Dec 2017 Abstract Besides all the attention given to lattice constructions, it is common to find some very
More informationThe Probabilistic Method
The Probabilistic Method Po-Shen Loh June 2010 1 Warm-up 1. (Russia 1996/4 In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees
More informationEC500. Design of Secure and Reliable Hardware. Lecture 9. Mark Karpovsky
EC500 Design of Secure and Reliable Hardware Lecture 9 Mark Karpovsky 1 1 Arithmetical Codes 1.1 Detection and Correction of errors in arithmetical channels (adders, multipliers, etc) Let = 0,1,,2 1 and
More informationOn Rainbow Cycles in Edge Colored Complete Graphs. S. Akbari, O. Etesami, H. Mahini, M. Mahmoody. Abstract
On Rainbow Cycles in Edge Colored Complete Graphs S. Akbari, O. Etesami, H. Mahini, M. Mahmoody Abstract In this paper we consider optimal edge colored complete graphs. We show that in any optimal edge
More informationON THE STRONGLY REGULAR GRAPH OF PARAMETERS
ON THE STRONGLY REGULAR GRAPH OF PARAMETERS (99, 14, 1, 2) SUZY LOU AND MAX MURIN Abstract. In an attempt to find a strongly regular graph of parameters (99, 14, 1, 2) or to disprove its existence, we
More informationA Vizing-like theorem for union vertex-distinguishing edge coloring
A Vizing-like theorem for union vertex-distinguishing edge coloring Nicolas Bousquet, Antoine Dailly, Éric Duchêne, Hamamache Kheddouci, Aline Parreau Abstract We introduce a variant of the vertex-distinguishing
More informationModule 7. Independent sets, coverings. and matchings. Contents
Module 7 Independent sets, coverings Contents and matchings 7.1 Introduction.......................... 152 7.2 Independent sets and coverings: basic equations..... 152 7.3 Matchings in bipartite graphs................
More informationThe strong chromatic number of a graph
The strong chromatic number of a graph Noga Alon Abstract It is shown that there is an absolute constant c with the following property: For any two graphs G 1 = (V, E 1 ) and G 2 = (V, E 2 ) on the same
More informationOn Covering a Graph Optimally with Induced Subgraphs
On Covering a Graph Optimally with Induced Subgraphs Shripad Thite April 1, 006 Abstract We consider the problem of covering a graph with a given number of induced subgraphs so that the maximum number
More informationMath 170- Graph Theory Notes
1 Math 170- Graph Theory Notes Michael Levet December 3, 2018 Notation: Let n be a positive integer. Denote [n] to be the set {1, 2,..., n}. So for example, [3] = {1, 2, 3}. To quote Bud Brown, Graph theory
More informationNotes for Lecture 24
U.C. Berkeley CS170: Intro to CS Theory Handout N24 Professor Luca Trevisan December 4, 2001 Notes for Lecture 24 1 Some NP-complete Numerical Problems 1.1 Subset Sum The Subset Sum problem is defined
More information2386 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 6, JUNE 2006
2386 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 6, JUNE 2006 The Encoding Complexity of Network Coding Michael Langberg, Member, IEEE, Alexander Sprintson, Member, IEEE, and Jehoshua Bruck,
More informationMatching Algorithms. Proof. If a bipartite graph has a perfect matching, then it is easy to see that the right hand side is a necessary condition.
18.433 Combinatorial Optimization Matching Algorithms September 9,14,16 Lecturer: Santosh Vempala Given a graph G = (V, E), a matching M is a set of edges with the property that no two of the edges have
More informationOn the Max Coloring Problem
On the Max Coloring Problem Leah Epstein Asaf Levin May 22, 2010 Abstract We consider max coloring on hereditary graph classes. The problem is defined as follows. Given a graph G = (V, E) and positive
More informationThe Encoding Complexity of Network Coding
The Encoding Complexity of Network Coding Michael Langberg Alexander Sprintson Jehoshua Bruck California Institute of Technology Email: mikel,spalex,bruck @caltech.edu Abstract In the multicast network
More informationLinear Block Codes. Allen B. MacKenzie Notes for February 4, 9, & 11, Some Definitions
Linear Block Codes Allen B. MacKenzie Notes for February 4, 9, & 11, 2015 This handout covers our in-class study of Chapter 3 of your textbook. We ll introduce some notation and then discuss the generator
More informationOn the Relationships between Zero Forcing Numbers and Certain Graph Coverings
On the Relationships between Zero Forcing Numbers and Certain Graph Coverings Fatemeh Alinaghipour Taklimi, Shaun Fallat 1,, Karen Meagher 2 Department of Mathematics and Statistics, University of Regina,
More information31.6 Powers of an element
31.6 Powers of an element Just as we often consider the multiples of a given element, modulo, we consider the sequence of powers of, modulo, where :,,,,. modulo Indexing from 0, the 0th value in this sequence
More informationEDGE-COLOURED GRAPHS AND SWITCHING WITH S m, A m AND D m
EDGE-COLOURED GRAPHS AND SWITCHING WITH S m, A m AND D m GARY MACGILLIVRAY BEN TREMBLAY Abstract. We consider homomorphisms and vertex colourings of m-edge-coloured graphs that have a switching operation
More informationOn the Computational Complexity of Nash Equilibria for (0, 1) Bimatrix Games
On the Computational Complexity of Nash Equilibria for (0, 1) Bimatrix Games Bruno Codenotti Daniel Štefankovič Abstract The computational complexity of finding a Nash equilibrium in a nonzero sum bimatrix
More informationSize of a problem instance: Bigger instances take
2.1 Integer Programming and Combinatorial Optimization Slide set 2: Computational Complexity Katta G. Murty Lecture slides Aim: To study efficiency of various algo. for solving problems, and to classify
More informationWinning Positions in Simplicial Nim
Winning Positions in Simplicial Nim David Horrocks Department of Mathematics and Statistics University of Prince Edward Island Charlottetown, Prince Edward Island, Canada, C1A 4P3 dhorrocks@upei.ca Submitted:
More information3 No-Wait Job Shops with Variable Processing Times
3 No-Wait Job Shops with Variable Processing Times In this chapter we assume that, on top of the classical no-wait job shop setting, we are given a set of processing times for each operation. We may select
More informationMT5821 Advanced Combinatorics
MT5821 Advanced Combinatorics 4 Graph colouring and symmetry There are two colourings of a 4-cycle with two colours (red and blue): one pair of opposite vertices should be red, the other pair blue. There
More informationREDUCING GRAPH COLORING TO CLIQUE SEARCH
Asia Pacific Journal of Mathematics, Vol. 3, No. 1 (2016), 64-85 ISSN 2357-2205 REDUCING GRAPH COLORING TO CLIQUE SEARCH SÁNDOR SZABÓ AND BOGDÁN ZAVÁLNIJ Institute of Mathematics and Informatics, University
More informationVertex Magic Total Labelings of Complete Graphs 1
Vertex Magic Total Labelings of Complete Graphs 1 Krishnappa. H. K. and Kishore Kothapalli and V. Ch. Venkaiah Centre for Security, Theory, and Algorithmic Research International Institute of Information
More informationMathematical and Algorithmic Foundations Linear Programming and Matchings
Adavnced Algorithms Lectures Mathematical and Algorithmic Foundations Linear Programming and Matchings Paul G. Spirakis Department of Computer Science University of Patras and Liverpool Paul G. Spirakis
More informationGraphs connected with block ciphers
Graphs connected with block ciphers OTOKAR GROŠEK Slovak University of Technology Department of Applied Informatics Ilkovičova 3 812 19 Bratislava SLOVAKIA PAVOL ZAJAC Slovak University of Technology Department
More informationCSC Linear Programming and Combinatorial Optimization Lecture 12: Semidefinite Programming(SDP) Relaxation
CSC411 - Linear Programming and Combinatorial Optimization Lecture 1: Semidefinite Programming(SDP) Relaxation Notes taken by Xinwei Gui May 1, 007 Summary: This lecture introduces the semidefinite programming(sdp)
More informationGraph Algorithms Matching
Chapter 5 Graph Algorithms Matching Algorithm Theory WS 2012/13 Fabian Kuhn Circulation: Demands and Lower Bounds Given: Directed network, with Edge capacities 0and lower bounds l for Node demands for
More informationLATIN SQUARES AND TRANSVERSAL DESIGNS
LATIN SQUARES AND TRANSVERSAL DESIGNS *Shirin Babaei Department of Mathematics, University of Zanjan, Zanjan, Iran *Author for Correspondence ABSTRACT We employ a new construction to show that if and if
More informationSAT-CNF Is N P-complete
SAT-CNF Is N P-complete Rod Howell Kansas State University November 9, 2000 The purpose of this paper is to give a detailed presentation of an N P- completeness proof using the definition of N P given
More informationThe complement of PATH is in NL
340 The complement of PATH is in NL Let c be the number of nodes in graph G that are reachable from s We assume that c is provided as an input to M Given G, s, t, and c the machine M operates as follows:
More informationVertex 3-colorability of claw-free graphs
Algorithmic Operations Research Vol.2 (27) 5 2 Vertex 3-colorability of claw-free graphs Marcin Kamiński a Vadim Lozin a a RUTCOR - Rutgers University Center for Operations Research, 64 Bartholomew Road,
More informationComputing the Minimum Hamming Distance for Z 2 Z 4 -Linear Codes
Computing the Minimum Hamming Distance for Z 2 Z 4 -Linear Codes Marta Pujol and Mercè Villanueva Combinatorics, Coding and Security Group (CCSG) Universitat Autònoma de Barcelona (UAB) VIII JMDA, Almería
More informationRevisiting the bijection between planar maps and well labeled trees
Revisiting the bijection between planar maps and well labeled trees Daniel Cosmin Porumbel September 1, 2014 Abstract The bijection between planar graphs and well labeled trees was published by Cori and
More informationParameterized Complexity of Independence and Domination on Geometric Graphs
Parameterized Complexity of Independence and Domination on Geometric Graphs Dániel Marx Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany. dmarx@informatik.hu-berlin.de
More informationA geometric non-existence proof of an extremal additive code
A geometric non-existence proof of an extremal additive code Jürgen Bierbrauer Department of Mathematical Sciences Michigan Technological University Stefano Marcugini and Fernanda Pambianco Dipartimento
More informationAn Eternal Domination Problem in Grids
Theory and Applications of Graphs Volume Issue 1 Article 2 2017 An Eternal Domination Problem in Grids William Klostermeyer University of North Florida, klostermeyer@hotmail.com Margaret-Ellen Messinger
More informationPoint-Set Topology 1. TOPOLOGICAL SPACES AND CONTINUOUS FUNCTIONS
Point-Set Topology 1. TOPOLOGICAL SPACES AND CONTINUOUS FUNCTIONS Definition 1.1. Let X be a set and T a subset of the power set P(X) of X. Then T is a topology on X if and only if all of the following
More informationMOST attention in the literature of network codes has
3862 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 8, AUGUST 2010 Efficient Network Code Design for Cyclic Networks Elona Erez, Member, IEEE, and Meir Feder, Fellow, IEEE Abstract This paper introduces
More informationAbstract. A graph G is perfect if for every induced subgraph H of G, the chromatic number of H is equal to the size of the largest clique of H.
Abstract We discuss a class of graphs called perfect graphs. After defining them and getting intuition with a few simple examples (and one less simple example), we present a proof of the Weak Perfect Graph
More informationComputability Theory
CS:4330 Theory of Computation Spring 2018 Computability Theory Other NP-Complete Problems Haniel Barbosa Readings for this lecture Chapter 7 of [Sipser 1996], 3rd edition. Sections 7.4 and 7.5. The 3SAT
More informationSmall Survey on Perfect Graphs
Small Survey on Perfect Graphs Michele Alberti ENS Lyon December 8, 2010 Abstract This is a small survey on the exciting world of Perfect Graphs. We will see when a graph is perfect and which are families
More informationNegative Numbers in Combinatorics: Geometrical and Algebraic Perspectives
Negative Numbers in Combinatorics: Geometrical and Algebraic Perspectives James Propp (UMass Lowell) June 29, 2012 Slides for this talk are on-line at http://jamespropp.org/msri-up12.pdf 1 / 99 I. Equal
More informationThe (extended) coset leader and list weight enumerator
The (extended) coset leader and list weight enumerator Relinde Jurrius (joint work with Ruud Pellikaan) Vrije Universiteit Brussel Fq11 July 24, 2013 Relinde Jurrius (VUB) Coset leader weight enumerator
More informationTreewidth and graph minors
Treewidth and graph minors Lectures 9 and 10, December 29, 2011, January 5, 2012 We shall touch upon the theory of Graph Minors by Robertson and Seymour. This theory gives a very general condition under
More informationPLANAR GRAPH BIPARTIZATION IN LINEAR TIME
PLANAR GRAPH BIPARTIZATION IN LINEAR TIME SAMUEL FIORINI, NADIA HARDY, BRUCE REED, AND ADRIAN VETTA Abstract. For each constant k, we present a linear time algorithm that, given a planar graph G, either
More informationChordal deletion is fixed-parameter tractable
Chordal deletion is fixed-parameter tractable Dániel Marx Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany. dmarx@informatik.hu-berlin.de Abstract. It
More informationComputation with No Memory, and Rearrangeable Multicast Networks
Discrete Mathematics and Theoretical Computer Science DMTCS vol. 16:1, 2014, 121 142 Computation with No Memory, and Rearrangeable Multicast Networks Serge Burckel 1 Emeric Gioan 2 Emmanuel Thomé 3 1 ERMIT,
More informationLine Graphs and Circulants
Line Graphs and Circulants Jason Brown and Richard Hoshino Department of Mathematics and Statistics Dalhousie University Halifax, Nova Scotia, Canada B3H 3J5 Abstract The line graph of G, denoted L(G),
More information1. Lecture notes on bipartite matching February 4th,
1. Lecture notes on bipartite matching February 4th, 2015 6 1.1.1 Hall s Theorem Hall s theorem gives a necessary and sufficient condition for a bipartite graph to have a matching which saturates (or matches)
More informationMonochromatic loose-cycle partitions in hypergraphs
Monochromatic loose-cycle partitions in hypergraphs András Gyárfás Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Budapest, P.O. Box 27 Budapest, H-364, Hungary gyarfas.andras@renyi.mta.hu
More informationOn the Balanced Case of the Brualdi-Shen Conjecture on 4-Cycle Decompositions of Eulerian Bipartite Tournaments
Electronic Journal of Graph Theory and Applications 3 (2) (2015), 191 196 On the Balanced Case of the Brualdi-Shen Conjecture on 4-Cycle Decompositions of Eulerian Bipartite Tournaments Rafael Del Valle
More informationNon-zero disjoint cycles in highly connected group labelled graphs
Non-zero disjoint cycles in highly connected group labelled graphs Ken-ichi Kawarabayashi Paul Wollan Abstract Let G = (V, E) be an oriented graph whose edges are labelled by the elements of a group Γ.
More informationAlgebraic method for Shortest Paths problems
Lecture 1 (06.03.2013) Author: Jaros law B lasiok Algebraic method for Shortest Paths problems 1 Introduction In the following lecture we will see algebraic algorithms for various shortest-paths problems.
More informationChain Packings and Odd Subtree Packings. Garth Isaak Department of Mathematics and Computer Science Dartmouth College, Hanover, NH
Chain Packings and Odd Subtree Packings Garth Isaak Department of Mathematics and Computer Science Dartmouth College, Hanover, NH 1992 Abstract A chain packing H in a graph is a subgraph satisfying given
More informationExtended and generalized weight enumerators
Extended and generalized weight enumerators Relinde Jurrius Ruud Pellikaan Eindhoven University of Technology, The Netherlands International Workshop on Coding and Cryptography, 2009 1/23 Outline Previous
More informationThe Borsuk-Ulam theorem- A Combinatorial Proof
The Borsuk-Ulam theorem- A Combinatorial Proof Shreejit Bandyopadhyay April 14, 2015 1 Introduction The Borsuk-Ulam theorem is perhaps among the results in algebraic topology having the greatest number
More informationACO Comprehensive Exam October 12 and 13, Computability, Complexity and Algorithms
1. Computability, Complexity and Algorithms Given a simple directed graph G = (V, E), a cycle cover is a set of vertex-disjoint directed cycles that cover all vertices of the graph. 1. Show that there
More information1. Lecture notes on bipartite matching
Massachusetts Institute of Technology 18.453: Combinatorial Optimization Michel X. Goemans February 5, 2017 1. Lecture notes on bipartite matching Matching problems are among the fundamental problems in
More informationLecture 4: 3SAT and Latin Squares. 1 Partial Latin Squares Completable in Polynomial Time
NP and Latin Squares Instructor: Padraic Bartlett Lecture 4: 3SAT and Latin Squares Week 4 Mathcamp 2014 This talk s focus is on the computational complexity of completing partial Latin squares. Our first
More informationA graph is finite if its vertex set and edge set are finite. We call a graph with just one vertex trivial and all other graphs nontrivial.
2301-670 Graph theory 1.1 What is a graph? 1 st semester 2550 1 1.1. What is a graph? 1.1.2. Definition. A graph G is a triple (V(G), E(G), ψ G ) consisting of V(G) of vertices, a set E(G), disjoint from
More informationOn Soft Topological Linear Spaces
Republic of Iraq Ministry of Higher Education and Scientific Research University of AL-Qadisiyah College of Computer Science and Formation Technology Department of Mathematics On Soft Topological Linear
More informationCHAPTER 8. Copyright Cengage Learning. All rights reserved.
CHAPTER 8 RELATIONS Copyright Cengage Learning. All rights reserved. SECTION 8.3 Equivalence Relations Copyright Cengage Learning. All rights reserved. The Relation Induced by a Partition 3 The Relation
More informationNP-completeness of generalized multi-skolem sequences
Discrete Applied Mathematics 155 (2007) 2061 2068 www.elsevier.com/locate/dam NP-completeness of generalized multi-skolem sequences Gustav Nordh 1 Department of Computer and Information Science, Linköpings
More informationStar Decompositions of the Complete Split Graph
University of Dayton ecommons Honors Theses University Honors Program 4-016 Star Decompositions of the Complete Split Graph Adam C. Volk Follow this and additional works at: https://ecommons.udayton.edu/uhp_theses
More informationComparative Study of Domination Numbers of Butterfly Graph BF(n)
Comparative Study of Domination Numbers of Butterfly Graph BF(n) Indrani Kelkar 1 and B. Maheswari 2 1. Department of Mathematics, Vignan s Institute of Information Technology, Visakhapatnam - 530046,
More informationOn graph decompositions modulo k
On graph decompositions modulo k A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB, England. Abstract. We prove that, for
More information18.S34 (FALL 2007) PROBLEMS ON HIDDEN INDEPENDENCE AND UNIFORMITY
18.S34 (FALL 2007) PROBLEMS ON HIDDEN INDEPENDENCE AND UNIFORMITY All the problems below (with the possible exception of the last one), when looked at the right way, can be solved by elegant arguments
More informationInformation Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay
Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 11 Coding Strategies and Introduction to Huffman Coding The Fundamental
More informationDM545 Linear and Integer Programming. Lecture 2. The Simplex Method. Marco Chiarandini
DM545 Linear and Integer Programming Lecture 2 The Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. 2. 3. 4. Standard Form Basic Feasible Solutions
More informationABSTRACT ALGEBRA FINAL PROJECT: GROUP CODES AND COSET DECODING
ABSTRACT ALGEBRA FINAL PROJECT: GROUP CODES AND COSET DECODING 1. Preface: Stumbling Blocks and the Learning Process Initially, this topic was a bit intimidating to me. It seemed highly technical at first,
More informationHashing. Yufei Tao. Department of Computer Science and Engineering Chinese University of Hong Kong
Department of Computer Science and Engineering Chinese University of Hong Kong In this lecture, we will revisit the dictionary search problem, where we want to locate an integer v in a set of size n or
More informationPANCYCLICITY WHEN EACH CYCLE CONTAINS k CHORDS
Discussiones Mathematicae Graph Theory xx (xxxx) 1 13 doi:10.7151/dmgt.2106 PANCYCLICITY WHEN EACH CYCLE CONTAINS k CHORDS Vladislav Taranchuk Department of Mathematics and Statistics California State
More informationOnline Coloring Known Graphs
Online Coloring Known Graphs Magnús M. Halldórsson Science Institute University of Iceland IS-107 Reykjavik, Iceland mmh@hi.is, www.hi.is/ mmh. Submitted: September 13, 1999; Accepted: February 24, 2000.
More informationarxiv: v1 [cs.it] 9 Feb 2009
On the minimum distance graph of an extended Preparata code C. Fernández-Córdoba K. T. Phelps arxiv:0902.1351v1 [cs.it] 9 Feb 2009 Abstract The minimum distance graph of an extended Preparata code P(m)
More informationTHE BASIC THEORY OF PERSISTENT HOMOLOGY
THE BASIC THEORY OF PERSISTENT HOMOLOGY KAIRUI GLEN WANG Abstract Persistent homology has widespread applications in computer vision and image analysis This paper first motivates the use of persistent
More informationLecture 15: The subspace topology, Closed sets
Lecture 15: The subspace topology, Closed sets 1 The Subspace Topology Definition 1.1. Let (X, T) be a topological space with topology T. subset of X, the collection If Y is a T Y = {Y U U T} is a topology
More informationParameterized graph separation problems
Parameterized graph separation problems Dániel Marx Department of Computer Science and Information Theory, Budapest University of Technology and Economics Budapest, H-1521, Hungary, dmarx@cs.bme.hu Abstract.
More informationError Correcting Codes
Error Correcting Codes 2. The Hamming Codes Priti Shankar Priti Shankar is with the Department of Computer Science and Automation at the Indian Institute of Science, Bangalore. Her interests are in Theoretical
More informationOn the construction of nested orthogonal arrays
isid/ms/2010/06 September 10, 2010 http://wwwisidacin/ statmath/eprints On the construction of nested orthogonal arrays Aloke Dey Indian Statistical Institute, Delhi Centre 7, SJSS Marg, New Delhi 110
More informationDeficient Quartic Spline Interpolation
International Journal of Computational Science and Mathematics. ISSN 0974-3189 Volume 3, Number 2 (2011), pp. 227-236 International Research Publication House http://www.irphouse.com Deficient Quartic
More informationarxiv: v3 [cs.ds] 18 Apr 2011
A tight bound on the worst-case number of comparisons for Floyd s heap construction algorithm Ioannis K. Paparrizos School of Computer and Communication Sciences Ècole Polytechnique Fèdèrale de Lausanne
More informationFUTURE communication networks are expected to support
1146 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 13, NO 5, OCTOBER 2005 A Scalable Approach to the Partition of QoS Requirements in Unicast and Multicast Ariel Orda, Senior Member, IEEE, and Alexander Sprintson,
More informationFlexible Coloring. Xiaozhou Li a, Atri Rudra b, Ram Swaminathan a. Abstract
Flexible Coloring Xiaozhou Li a, Atri Rudra b, Ram Swaminathan a a firstname.lastname@hp.com, HP Labs, 1501 Page Mill Road, Palo Alto, CA 94304 b atri@buffalo.edu, Computer Sc. & Engg. dept., SUNY Buffalo,
More informationPart II. Graph Theory. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 53 Paper 3, Section II 15H Define the Ramsey numbers R(s, t) for integers s, t 2. Show that R(s, t) exists for all s,
More informationByzantine Consensus in Directed Graphs
Byzantine Consensus in Directed Graphs Lewis Tseng 1,3, and Nitin Vaidya 2,3 1 Department of Computer Science, 2 Department of Electrical and Computer Engineering, and 3 Coordinated Science Laboratory
More informationHamming Codes. s 0 s 1 s 2 Error bit No error has occurred c c d3 [E1] c0. Topics in Computer Mathematics
Hamming Codes Hamming codes belong to the class of codes known as Linear Block Codes. We will discuss the generation of single error correction Hamming codes and give several mathematical descriptions
More informationAlgebraic Graph Theory- Adjacency Matrix and Spectrum
Algebraic Graph Theory- Adjacency Matrix and Spectrum Michael Levet December 24, 2013 Introduction This tutorial will introduce the adjacency matrix, as well as spectral graph theory. For those familiar
More informationHW Graph Theory SOLUTIONS (hbovik) - Q
1, Diestel 9.3: An arithmetic progression is an increasing sequence of numbers of the form a, a+d, a+ d, a + 3d.... Van der Waerden s theorem says that no matter how we partition the natural numbers into
More informationTheorem 2.9: nearest addition algorithm
There are severe limits on our ability to compute near-optimal tours It is NP-complete to decide whether a given undirected =(,)has a Hamiltonian cycle An approximation algorithm for the TSP can be used
More informationON SWELL COLORED COMPLETE GRAPHS
Acta Math. Univ. Comenianae Vol. LXIII, (1994), pp. 303 308 303 ON SWELL COLORED COMPLETE GRAPHS C. WARD and S. SZABÓ Abstract. An edge-colored graph is said to be swell-colored if each triangle contains
More informationLOW-DENSITY PARITY-CHECK (LDPC) codes [1] can
208 IEEE TRANSACTIONS ON MAGNETICS, VOL 42, NO 2, FEBRUARY 2006 Structured LDPC Codes for High-Density Recording: Large Girth and Low Error Floor J Lu and J M F Moura Department of Electrical and Computer
More informationDisjoint directed cycles
Disjoint directed cycles Noga Alon Abstract It is shown that there exists a positive ɛ so that for any integer k, every directed graph with minimum outdegree at least k contains at least ɛk vertex disjoint
More informationCSC 310, Fall 2011 Solutions to Theory Assignment #1
CSC 310, Fall 2011 Solutions to Theory Assignment #1 Question 1 (15 marks): Consider a source with an alphabet of three symbols, a 1,a 2,a 3, with probabilities p 1,p 2,p 3. Suppose we use a code in which
More informationA TESSELLATION FOR ALGEBRAIC SURFACES IN CP 3
A TESSELLATION FOR ALGEBRAIC SURFACES IN CP 3 ANDREW J. HANSON AND JI-PING SHA In this paper we present a systematic and explicit algorithm for tessellating the algebraic surfaces (real 4-manifolds) F
More informationSuperconcentrators of depth 2 and 3; odd levels help (rarely)
Superconcentrators of depth 2 and 3; odd levels help (rarely) Noga Alon Bellcore, Morristown, NJ, 07960, USA and Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv
More informationBipartite Perfect Matching in O(n log n) Randomized Time. Nikhil Bhargava and Elliot Marx
Bipartite Perfect Matching in O(n log n) Randomized Time Nikhil Bhargava and Elliot Marx Background Matching in bipartite graphs is a problem that has many distinct applications. Many problems can be reduced
More informationInterleaving Schemes on Circulant Graphs with Two Offsets
Interleaving Schemes on Circulant raphs with Two Offsets Aleksandrs Slivkins Department of Computer Science Cornell University Ithaca, NY 14853 slivkins@cs.cornell.edu Jehoshua Bruck Department of Electrical
More information