DOCSLIB.ORG

Discrete Mathematics on a Class of Binary Linear Completely Transitive Codes with Arbitrary Covering Radius$

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Discrete Mathematics on a Class of Binary Linear Completely Transitive Codes with Arbitrary Covering Radius$ View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 309 (2009) 5011–5016 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc On a class of binary linear completely transitive codes with arbitrary covering radiusI J. Rifà a,∗, V.A. Zinoviev b a Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain b Institute for Problems of Information Transmission of the Russian Academy of Sciences, Bol'shoi Karetnyi per. 19, GSP-4, Moscow, 101447, Russia article info a b s t r a c t Article history: An infinite class of new binary linear completely transitive (and so, completely regular) Received 30 October 2007 codes is given. The covering radius of these codes is growing with the length of the code. In Received in revised form 6 November 2008 particular, for any integer ρ ≥ 2, there exist two codes in the constructed class with d D 3, Accepted 5 March 2009 2 ρ 2 ρC1 Available online 29 March 2009 covering radius ρ and lengths 2 and 2 , respectively. The corresponding distance- transitive graphs, which can be defined as coset graphs of these completely transitive codes Keywords: are described. Completely regular code ' 2009 Elsevier B.V. All rights reserved. Completely transitive code Covering radius Distance-regular graph Distance-transitive graph Intersection numbers Outer distance Uniformly packed code 1. Introduction n Let F be the finite field of two elements. A binary .n; N; d/ code C is a subset of F of length n, cardinality N, and minimum n k distance d. When C is a k-dimensional linear subspace of F we will denote it by Tn; k; dU and notice that, in this case, N D 2 . n The support of v D .v1; : : : ; vn/ 2 F is supp.v/ D fj j vj 6D 0g. Say that a vector v covers a vector z if the condition zi 6D 0 implies zi D vi. For a binary .n; N; d/ code C, following Delsarte [7], define the outer distance s D s.C/ of C as the number of nonzero ? D ? ? components ηi , i 1;:::; n of the vector (η0 ; : : : ; ηn / obtained by the MacWilliams transform of the (average) distance distribution η.C/, where η.C/ D (η0; : : : ; ηn/ and 1 ηj D · jf.u; v/ V u; v 2 C; d.u; v/ D jgj: N Hence, if C is a linear Tn; k; dU code then s.C/ is the number of different nonzero weights of the codewords in the dual Tn; n − k; d?U code C?. A linear Tn; k; dU code C can be given by its parity check matrix H of size .n − k/ × n. Code C is the set of all vectors c of t t n length n such that Hc D 0, where c means the transpose vector of c. For any x 2 F we denote by H.x/ the syndrome of vector x, so H.x/ D Hxt. Denote by wt.x/ the (Hamming) weight of the vector x. I This work has been partially supported by the Spanish MEC Grants MTM2006-03250, TSI2006-14005-C02-01 and PCI2006-A7-0616 as well as by the Russian fund of fundamental research (the number of the project, 06 - 01 - 00226). ∗ Corresponding author. E-mail address: [email protected] (J. Rifà). 0012-365X/$ – see front matter ' 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2009.03.004 5012 J. Rifà, V.A. Zinoviev / Discrete Mathematics 309 (2009) 5011–5016 n Given any vector v 2 F , its distance to the code C is d.v; C/ D minx2C fd.v; x/g and the covering radius of the code C is D n f g ρ.C/ maxv2F d.v; C/ : Let D D C C x be a coset of C. The weight wt.D/ of D is the minimum weight of the codewords in D. For a binary code C with covering radius ρ define n C.i/ D fx 2 F V d.x; C/ D ig; i D 1; 2; : : : ; ρ: (1) n n For any vector x 2 F denote by W .x/ the sphere of radius one centered in x, i.e., W .x/ D fy 2 F V d.x; y/ D 1g. We say that two vectors x and y are neighbors if y 2 W .x/. Definition 1 ([8]). A code C is completely regular, if, for all l ≥ 0, every vector x 2 C.l/ has the same number cl of neighbors in C.l − 1/ and the same number bl of neighbors in C.l C 1/. Also, define al D .q − 1/n − bl − cl and notice that c0 D bρ D 0. Define by fb0;:::; bρ−1I c1;:::; cρ g the intersection array of C. For a binary code C let Perm.C/ be its permutation stabilizer group, hence the group of all the coordinate permutations θ such that any vector v D .v1; v2; : : : ; vn/ 2 C is transformed in θ.v/ D .vθ.1/; vθ.2/; : : : ; vθ.n// 2 C. For any θ 2 Perm.C/ and any translate D D C C x of C define the action of θ on D as θ.D/ D C C θ.x/. Definition 2 ([11]). Let C be a binary linear code with covering radius ρ. The code C is said to be completely transitive when n the set fC C x V x 2 F g of all different cosets of C is partitioned under the action of Perm.C/ into exactly ρ C 1 orbits. Since two cosets in the same orbit should have the same weight distribution, it is clear, that any completely transitive code is completely regular too. Classification of completely regular codes is a hard open problem of algebraic coding theory, which is extremely important for distance-regular graphs and association schemes (see [6–8] and references therein). Many completely regular codes come from perfect codes [4,10] and it has been conjectured for a long time [8] that if C is a completely regular code and jCj > 2, then we have e ≤ 3. For completely transitive codes, the problem of existence is solved in [3,5] in the sense that for e ≥ 4 such nontrivial codes do not exist. Solé [11] (see also [1]) describes a construction of an infinite family of completely regular codes with growing covering radius. Taking s copies of perfect 1-error correcting codes of length n we obtain a completely regular code of length s · n and with covering radius ρ D s. In the literature, there are no other infinite families of completely transitive codes with growing covering radius. Our purpose in this paper is to construct a class of binary linear completely regular and completely transitive codes for which the covering radius is growing with the length of the code. These codes are defined by their parity check matrices, which have a very simple structure. The paper is organized as follows. In Section 2 we give the main construction of the completely transitive codes, which have special parity check matrices. The corresponding distance-regular graphs, which can be defined as coset graphs of these completely regular codes are shortly described in Section 3. 2. Main construction m For a given natural number m where m ≥ 3 define F` as the set of all binary vectors of length m and weight `. .m;`/ × m m Definition 3. Denote by H the binary matrix of size m ` , whose columns are exactly all vectors from F` (i.e., each m .m;`/ .m;`/ vector from F` occurs once as a column of H ). Now define the binary linear code C , whose parity check matrix is the matrix H.m;`/. Lemma 1. For any natural numbers m and `, 2 ≤ ` ≤ m − 1 the binary linear Tn; k; dU code C D C.m;`/ has parameters: m for even `: n D ; k D n − m C 1; d ≥ 3; ` and m for odd `: n D ; k D n − m; d ≥ 4: ` Proof. The proof is immediate. .m;l/ .m;l/ Let Perm.C / be the permutation stabilizer group of the code C . Denote by Pi;j the m × m permutation matrix which transposes coordinates i and j, so Pi;j is the m × m identity matrix with the ith column and jth column transposed. .m;l/ .m;l/ Note that Pi;jH coincides with the matrix H after swapping the ith and jth rows. J. Rifà, V.A. Zinoviev / Discrete Mathematics 309 (2009) 5011–5016 5013 .m;`/ .m;l/ n Lemma 2. For any two rows ith and jth in H there exists a permutation θi;j 2 Perm.C /, such that for any x 2 F : .m;l/ .m;l/ H (θi;j.x// D .Pi;jH /.x/: n .m;l/ Proof. Let ri; rj 2 F be, respectively, the ith row and jth row vectors in the matrix H . Vectors ri and rj disagree in − − − − − 2 m 1 − 2 m 2 D 2 m 1 m 2 coordinates and share exactly m 2 nonzero coordinates. The column vectors, where `−1 `−2 `−1 `−2 `−2 ri; rj disagree, can be taken in p pairs and, in each pair, the two columns have exactly the same coordinates in all the rows different from the ith and the jth. Let is1, js1 be the two columns of the first pair, and so on, is2, js2; :::; isp, jsp, where − − p D m l m 2 .
Load more

Recommended publications
  • 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].
    [Show full text]
  • 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].
    [Show full text]
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • Algorithmic Coding Theory ¡
    ALGORITHMIC CODING THEORY Atri Rudra State University of New York at Buffalo 1 Introduction Error-correcting codes (or just codes) are clever ways of representing data so that one can recover the original information even if parts of it are corrupted. The basic idea is to judiciously introduce redundancy so that the original information can be recovered when parts of the (redundant) data have been corrupted. Perhaps the most natural and common application of error correcting codes is for communication. For example, when packets are transmitted over the Internet, some of the packets get corrupted or dropped. To deal with this, multiple layers of the TCP/IP stack use a form of error correction called CRC Checksum [Pe- terson and Davis, 1996]. Codes are used when transmitting data over the telephone line or via cell phones. They are also used in deep space communication and in satellite broadcast. Codes also have applications in areas not directly related to communication. For example, codes are used heavily in data storage. CDs and DVDs can work even in the presence of scratches precisely because they use codes. Codes are used in Redundant Array of Inexpensive Disks (RAID) [Chen et al., 1994] and error correcting memory [Chen and Hsiao, 1984]. Codes are also deployed in other applications such as paper bar codes, for example, the bar ¡ To appear as a book chapter in the CRC Handbook on Algorithms and Complexity Theory edited by M. Atallah and M. Blanton. Please send your comments to [email protected] 1 code used by UPS called MaxiCode [Chandler et al., 1989].
    [Show full text]
  • An Introduction to Algorithmic Coding Theory
    An Introduction to Algorithmic Coding Theory M. Amin Shokrollahi Bell Laboratories 1 Part 1: Codes 1-1 A puzzle What do the following problems have in common? 2 Problem 1: Information Transmission G W D A S U N C L R V MESSAGE J K E MASSAGE ?? F T X M I H Z B Y 3 Problem 2: Football Pool Problem Bayern M¨unchen : 1. FC Kaiserslautern 0 Homburg : Hannover 96 1 1860 M¨unchen : FC Karlsruhe 2 1. FC K¨oln : Hertha BSC 0 Wolfsburg : Stuttgart 0 Bremen : Unterhaching 2 Frankfurt : Rostock 0 Bielefeld : Duisburg 1 Feiburg : Schalke 04 2 4 Problem 3: In-Memory Database Systems Database Corrupted! User memory malloc malloc malloc malloc Code memory 5 Problem 4: Bulk Data Distribution 6 What do they have in common? They all can be solved using (algorithmic) coding theory 7 Basic Idea of Coding Adding redundancy to be able to correct! MESSAGE MESSAGE MESSAGE MASSAGE Objectives: • Add as little redundancy as possible; • correct as many errors as possible. 8 Codes A code of block-length n over the alphabet GF (q) is a set of vectors in GF (q)n. If the set forms a vector space over GF (q), then the code is called linear. [n, k]q-code 9 Encoding Problem ? ? ? source ? ? ? Codewords Efficiency! 10 Linear Codes: Generator Matrix Generator matrix source codeword k n k x n O(n2) 11 Linear Codes: Parity Check Matrix 0 Parity check matrix codeword O(n2) after O(n3) preprocessing. 12 The Decoding Problem: Maximum Likelihood Decoding abcd 7→ abcd | abcd | abcd Received: a x c d | a b z d | abcd.
    [Show full text]
  • Voting Rules As Error-Correcting Codes ∗ Ariel D
    Artificial Intelligence 231 (2016) 1–16 Contents lists available at ScienceDirect Artificial Intelligence www.elsevier.com/locate/artint ✩ Voting rules as error-correcting codes ∗ Ariel D. Procaccia, Nisarg Shah , Yair Zick Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA a r t i c l e i n f o a b s t r a c t Article history: We present the first model of optimal voting under adversarial noise. From this viewpoint, Received 21 January 2015 voting rules are seen as error-correcting codes: their goal is to correct errors in the input Received in revised form 13 October 2015 rankings and recover a ranking that is close to the ground truth. We derive worst-case Accepted 20 October 2015 bounds on the relation between the average accuracy of the input votes, and the accuracy Available online 27 October 2015 of the output ranking. Empirical results from real data show that our approach produces Keywords: significantly more accurate rankings than alternative approaches. © Social choice 2015 Elsevier B.V. All rights reserved. Voting Ground truth Adversarial noise Error-correcting codes 1. Introduction Social choice theory develops and analyzes methods for aggregating the opinions of individuals into a collective decision. The prevalent approach is motivated by situations in which opinions are subjective, such as political elections, and focuses on the design of voting rules that satisfy normative properties [1]. An alternative approach, which was proposed by the marquis de Condorcet in the 18th Century, had confounded schol- ars for centuries (due to Condorcet’s ambiguous writing) until it was finally elucidated by Young [34].
    [Show full text]
  • Adinkras for Mathematicians
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 366, Number 6, June 2014, Pages 3325–3355 S 0002-9947(2014)06031-5 Article electronically published on February 6, 2014 ADINKRAS FOR MATHEMATICIANS YAN X. ZHANG Abstract. Adinkras are graphical tools created to study representations of supersymmetry algebras. Besides having inherent interest for physicists, the study of adinkras has already shown non-trivial connections with coding the- ory and Clifford algebras. Furthermore, adinkras offer many easy-to-state and accessible mathematical problems of algebraic, combinatorial, and computa- tional nature. We survey these topics for a mathematical audience, make new connections to other areas (homological algebra and poset theory), and solve some of these said problems, including the enumeration of all hyper- cube adinkras up through dimension 5 and the enumeration of odd dashings of adinkras for any dimension. 1. Introduction In a series of papers starting with [11], different subsets of the “DFGHILM collaboration” (Doran, Faux, Gates, H¨ubsch, Iga, Landweber, Miller) have built and extended the machinery of adinkras. Following the ubiquitous spirit of visual diagrams in physics, adinkras are combinatorial objects that encode information about the representation theory of supersymmetry algebras. 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. In short, adinkras are chromotopologies (a class of edge-colored bipartite graphs) with two additional structures, a dashing condition on the edges and a ranking condition on the vertices.
    [Show full text]
  • Group Algebra and Coding Theory
    Group algebras and Coding Theory Marinˆes Guerreiro ∗† September 7, 2015 Abstract Group algebras have been used in the context of Coding Theory since the beginning of the latter, but not in its full power. The work of Ferraz and Polcino Milies entitled Idempotents in group algebras and minimal abelian codes (Finite Fields and their Applications, 13, (2007) 382-393) gave origin to many thesis and papers linking these two subjects. In these works, the techniques of group algebras are mainly brought into play for the computing of the idempotents that generate the minimal codes and the minimum weight of such codes. In this paper I summarize the main results of the work done by doctorate students and research partners of Polcino Milies and Ferraz. Keywords: idempotents, group algebra, coding theory. Dedicated to Prof. C´esar Polcino Milies on the occasion of his seventieth birthday. arXiv:1412.8042v2 [math.RA] 4 Sep 2015 1 Introduction The origins of Information Theory and Error Correcting Codes Theory are in the papers by Shannon [64] and Hamming [35], where they settled the theoretical foundations for such theories. ∗M. Guerreiro is with Departamento de Matem´atica, Universidade Federal de Vi¸cosa, CEP 36570-000 - Vi¸cosa-MG (Brasil), supported by CAPES, PROCAD 915/2010 and FAPEMIG, APQ CEX 00438-08 and PCE-00151-12(Brazil). E-mail: [email protected]. †This work was presented at Groups, Rings and Group Rings 2014 on the occasion of the 70th Birthday of C´esar Polcino Milies, Ubatuba-SP (Brasil) 21-26 July, 2014. 1 For a non empty finite set A, called alphabet, a code C of length n is n simply a proper subset of A and an n-tuple (a0, a1,...,an−1) ∈ C is called a word of the code C.
    [Show full text]
  • Topology Types of Adinkras and the Corresponding Representations of N-Extended Supersymmetry Arxiv:0806.0050V1 [Hep-Th] 31
    nvriyo ahntnMteaisDprmn|eprieUiest aua cecsDvso|adCleeMteaisDepartment Mathematics College Division|Bard Sciences Natural University Department|Pepperdine Mathematics Washington of University x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x o pt 2supercharges. 32 to up for x x x x x x hre,a ela h aia oe,crepnigt iia supermultiplets, minimal to corresponding codes, maximal the as well as charges, x x x x x x xii h nmrto fteeeuvlnecassfralcssu o2 super- 26 to up cases all for classes equivalence these of enumeration the exhibit x x x x x x fAika n hrfr uemlilt.Uigrslsfo oigter we theory coding from results Using supermultiplets. therefore and Adinkras of x x x x x x h lsicto fsc oe rvdsamasfrdsrbn qiaec classes equivalence describing for means a provides codes such of classification the x x x x orsodpeieyt obyee iaylna ro-orcigcds othat so codes, error-correcting linear binary even doubly to precisely correspond x x x x x x Z 2 eepanhwteeqoin groups quotient these how explain . We ) utetgop r ugop f( of subgroups are groups quotient arXiv:0806.0050v1 [hep-th]31May2008 x x N x x x x dmninlcbs hr the where cubes, -dimensional N of quotients and diagrams Adinkra between x x x x x x hoei oooia nains npriua,w eosrt relationship a demonstrate we particular, In invariants. topological theoretic x x x x x x etne olln uesmer,wihrle ngraph- on relies which supersymmetry, worldline -extended N of supermultiplets x x x x epeetfrhrpors oadacmlt lsicto ceefrdescribing for scheme classification complete a toward progress further present
    [Show full text]
  • Curriculum Vitae
    Crystal Hoyt March 30, 2014 Curriculum Vitae Research Interests Algebraic and combinatorial representation theory Education · University of California, Berkeley, USA Ph.D., Mathematics, May 2007. Dissertation: \Kac-Moody superalgebras of finite growth". Advisor: Vera Serganova. · Texas A & M University, College Station, USA B.S., Mathematics, May 2001, summa cum laude. Employment · Technion - Israel Institute of Technology Aly Kaufman Postdoctoral Fellow October 2011 - February 2014 Hosts: Shlomi Gelaki, Tobias Hartnick Adjunct Lecturer: Spring 2014 · Bar-Ilan University, Israel Postdoctoral Researcher October 2010 - September 2011 Host: Jonathan Beck · The Weizmann Institute of Science, Israel Postdoctoral Fellow July 2007 - August 2009, December 2009 - June 2010 Hosts: Maria Gorelik, Anthony Joseph Adjunct Lecturer: Spring 2011; Winter 2013 · Nara Women's University, Japan JSPS Postdoctoral fellow Japanese Society for the Promotion of Science September 2009 - November 2009 Host: Tomoyuki Arakawa Visiting Positions · University of Michigan, Ann Arbor, 7.7 - 20.7.2013; 20.1 - 31.1.2014 · RIMS: Research Institute for Mathematical Sciences, Kyoto University, Japan, 27.6 - 12.7.2011 · Nara Women's University, Japan, 8.11 - 22.11.2008 1 Crystal Hoyt March 30, 2014 Curriculum Vitae Refereed Journals · C. Hoyt, Good gradings of basic Lie superalgebras, Israel Journal of Math- ematics, 192 no. 1 (2012), 251-280. · C. Hoyt, Regular Kac-Moody superalgebras and integrable highest weight modules, Journal of Algebra, 324 no. 12 (2010), 3308-3354. · C. Hoyt and S. Reif, Simplicity of vacuum modules over affine Lie super- algebras, Journal of Algebra, 321 no. 10 (2009), 2861-2874. · C. Hoyt and V. Serganova, Classification of finite-growth general Kac- Moody superalgebras, Communications in Algebra, 35 no.
    [Show full text]