DOCSLIB.ORG

Goppa and Srivastava Codes Over Finite Rings

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Goppa and Srivastava Codes Over Finite Rings “main” — 2005/10/10 — 16:27 — page 231 — #1 Volume 24, N. 2, pp. 231–244, 2005 Copyright © 2005 SBMAC ISSN 0101-8205 www.scielo.br/cam Goppa and Srivastava codes over finite rings ANTONIO APARECIDO DE ANDRADE1 and REGINALDO PALAZZO JR.2∗ 1Department of Mathematics, Ibilce, Unesp, 15054-000 São José do Rio Preto, SP, Brazil 2Department of Telematics, Feec, Unicamp, 13083-852 Campinas, SP, Brazil E-mails: [email protected] / [email protected] Abstract. Goppa and Srivastava codes over arbitrary local finite commutative rings with identity are constructed in terms of parity-cleck matrices. An efficient decoding procedure, based on the modified Berlekamp-Massey algorithm, is proposed for Goppa codes. Mathematical subject classification: 11T71, 94B05, 94B40. Key words: Goppa code, Srivastava code. 1 Introduction Goppa codes form a subclass of alternant codes and they are described in terms of a polynomial called Goppa polynomial. The most famous subclasses of alternant codes are BCH codes and Goppa codes, the former for their simple and easily ins- trumented decoding algorithm, and the latter for meeting the Gilbert-Varshamov bound. However, most of the work regarding construction and decoding of Goppa codes has been done considering codes over finite fields. On the other hand, linear codes over rings have recently generated a great deal of interest. Linear codes over local finite commutative rings with identity have been dis- cussed in papers by Andrade [1], [2], [3] where it was extended the notion of Hamming, Reed-Solomon, BCH and alternant codes over these rings. In this paper we describe a construction technique of Goppa and Srivastava codes over local finite commutative rings. The core of the construction technique #609/04. Received: 3/VI/04. Accepted: 11/V/05. “main” — 2005/10/10 — 16:27 — page 232 — #2 232 GOPPA AND SRIVASTAVA CODES OVER FINITE RINGS mimics that of Goppa codes over a finite field, and is addressed, in this paper, from the point of view of specifying a cyclic subgroup of the group of units of an extension ring of finite rings. The decoding algorithm for Goppa codes consists of four major steps: (1) calculation of the syndromes, (2) calculation of the elementary symmetric functions by modified Berlekamp-Massey algorithm, (3) calculation of the error-location numbers, and (4) calculation of the error magnitudes. This paper is organized as follows. In Section 2, we describe a construction of Goppa codes over local finite commutative rings and an efficient decoding procedure. In Section 3, we describe a construction of Srivastava codes over local finite commutative rings. Finally, in Section 5, the concluding remarks are drawn. 2 Goppa Codes In this section we describe a construction technique of Goppa codes over arbitrary local finite commutative rings in terms of parity-check matrices, which isvery similar to the one proposed by Goppa [4] over finite fields. First, we review basic facts from the Galois theory of local finite commutative rings. Throughout this paper denotes a local finite commutative ring with identity, A m maximal ideal and residue field K A GF(p ), for some prime p, M = M ≡ m will be a positive integer, and x denotes the ring of polynomials in the A[ ] variable x over . The natural projection x K x is denoted by μ, where A A[ ] → [ ] μ(a(x)) a(x). = Let f (x) be a monic polynomial of degree h in x such that μ( f (x)) is A[ ] irreducible in K x . Then f (x) is also irreducible in x [5, Theorem XIII.7]. [ ] A[ ] Let be the ring x / f (x) . Then is a local finite commutative ring with R A[ ] h i R identity and it is called a Galois extension of of degree h. Its residue field is A mh 1 / 1 GF(p ), where 1 is the unique maximal ideal of , and K R M M R = ≡ mh K1∗ is the multiplicative group of K1, whose order is p 1. − Let ∗ denote the multiplicative group of units of . It follows that ∗ is R R R an abelian group, and therefore it can be expressed as a direct product of cyclic groups. We are interested in the maximal cyclic subgroup of ∗, hereafter R s denoted by s, whose elements are the roots of x 1 for some positive integer G − Comp. Appl. Math., Vol. 24, N. 2, 2005 “main” — 2005/10/10 — 16:27 — page 233 — #3 ANTONIO APARECIDO DE ANDRADE and REGINALDO PALAZZO JR. 233 s such that gcd(s, p) 1. There is only one maximal cyclic subgroup of ∗ = R having order relatively prime to p [5, Theorem XVIII.2]. This cyclic group has order s pmh 1. = − The Goppa codes are specified in terms of a polynomial g(z) called Goppa polynomial. In contrast to cyclic codes, where it is difficult to estimate the minimum Hamming distance d from the generator polynomial, Goppa codes have the property that d deg(g(z)) 1. ≥ r + Let g(z) g0 g1z gr z be a polynomial with coefficients in and = + + ∙ ∙ ∙ + R g 0. Let η (α , α , , α ) be a vector consisting of distinct elements of r 6= = 1 2 ∙ ∙ ∙ n s such that g(αi ) are units from for i 1, 2, , n. G R = ∙ ∙ ∙ Definition 2.1. A shortened Goppa code (η, ω, g) of length n s over C ≤ A has parity-check matrix 1 1 g(α1)− g(αn)− 1 ∙ ∙ ∙ 1 α1g(α1)− αn g(αn)− H . ∙ ∙ ∙ . , (1) = . .. . r 1 1 r 1 1 α − g(α1)− α − g(αn)− 1 ∙ ∙ ∙ n 1 1 1 where ω (g(α− ), g(α− ), , g(α− )). The polynomial g(z) is called = 1 2 ∙ ∙ ∙ n Goppa polynomial. Definition 2.2. Let (η, ω, g) be a Goppa code. C • If g(z) is irreducible then (η, ω, g) is called an irreducible Goppa code. C • If, for all c (c1, c2, , cn) (η, ω, g), it is true that c0 = ∙ ∙ ∙ ∈ C = (cn, cn 1, , c1) (η, ω, g), then (η, ω, g) is called a reversible − ∙ ∙ ∙ ∈ C C Goppa code. • If g(z) (z α)r then (η, ω, g) is called a commutative Goppa code. = − C • If g(z) has no multiple zeros then (η, ω, g) is called a separable Goppa C code. Comp. Appl. Math., Vol. 24, N. 2, 2005 “main” — 2005/10/10 — 16:27 — page 234 — #4 234 GOPPA AND SRIVASTAVA CODES OVER FINITE RINGS Remark 2.1. Let (η, ω, g) be a Goppa code. C 1. We have (η, ω, g) is a linear code. C 2. A parity-check matrix with elements from is then obtained by replacing A each entry of H by the corresponding column vector of length h from . A rl 3. For a Goppa code with polynomial gl (z) (z βl ) , where βl s, we = − ∈ G have rl rl rl (α1 βl )− (α2 βl )− (αn βl )− − r − r ∙ ∙ ∙ − r α1(α1 βl ) l α2(α2 βl ) l αn(αn βl ) l − − − − ∙ ∙ ∙ − − Hl . = . .. rl 1 r rl 1 r rl 1 r α − (α1 βl ) l α − (α2 βl ) l αn − (αn βl ) l 1 − − 2 − − ∙ ∙ ∙ − − which is row-equivalent to 1 1 1 r r r (α1 βl ) l (α2 βl ) l (αn βl ) l − − ∙ ∙ ∙ − (α1 βl ) (α2 βl ) (αn βl ) − r − r − r (α1 βl ) l (α2 βl ) l (αn βl ) l Hl − − ∙ ∙ ∙ − = . .. . r 1 r 1 r 1 (α1 βl ) l − (α2 βl ) l − (αn βl ) l − − r − r − r (α1 βl ) l (α2 βl ) l ∙ ∙ ∙ (αn βl ) l − − − 1 1 1 r r r (α1 βl ) l (α2 βl ) l (αn βl ) l −1 −1 ∙ ∙ ∙ −1 (r 1) (r 1) (r 1) (α1 βl ) l (α2 βl ) l (αn βl ) l − − − − ∙ ∙ ∙ − − . = . .. . 1 1 1 (α1 βl ) (α2 βl ) ∙ ∙ ∙ (αn βl ) − − − k rl k Consequently, if g(z) l 1(z βl ) i 1 gl (z), then the Goppa = = − = = code is the intersection of Goppa codes with Goppa polynomial gl (z) Q Q = rl (z βl ) , for l 1, 2, , k, and its parity-check matrix is given by − = ∙ ∙ ∙ H1 H2 H . = . . H k Comp. Appl. Math., Vol. 24, N. 2, 2005 “main” — 2005/10/10 — 16:27 — page 235 — #5 ANTONIO APARECIDO DE ANDRADE and REGINALDO PALAZZO JR. 235 4. Alternant codes are a special case of Goppa codes [3, Definition 2.1]. It is possible to obtain an estimate of the minimum Hamming distance d of (η, ω, g) directly from the Goppa polynomial g(z). The next theorem provides C such an estimate. Theorem 2.1. The code (η, ω, g) has minimum Hamming distance d r 1. C ≥ + Proof. We have (η, ω, g) is an alternant code (n, η, ω) with η (α1, C C 1 1 1 = α , , α ) and ω (g(α )− , g(α )− , , g(α )− ) [3, Definition 2.1]. By 2 ∙ ∙ ∙ n = 1 2 ∙ ∙ ∙ n [3, Theorem 2.1] it follows (η, ω, g) has minimum distance d r 1. C ≥ + x 3 Example 2.1. Let Z2 i and A[ ] , where f (x) x x 1 is A = [ ] R = f (x) = + + irreducible over and i 2 1. If α is a rooth ofi f (x), then α generates a cyclic A 3 = − 4 2 3 2 group s of order s 2 1 7. Let η (α, α , 1, α ), g(z) z z 1 and G 1 4 =1 − =1 2 1= 3 5 6 = + + ω (g(α)− , g(α )− , g(1)− , g(α )− ) (α , α , 1, α ). Since deg(g(z)) = = = 3, it follows that α3 α5 1 α6 H α4 α2 1 α , = α5 α6 1 α3 is the parity-check matrix of a Goppa code (η, ω,g) over with length 4 and C A minimum Hamming distance at least 4.
Load more

Recommended publications
  • Parity Alternating Permutations Starting with an Odd Integer
    Parity alternating permutations starting with an odd integer Frether Getachew Kebede1a, Fanja Rakotondrajaob aDepartment of Mathematics, College of Natural and Computational Sciences, Addis Ababa University, P.O.Box 1176, Addis Ababa, Ethiopia; e-mail: [email protected] bD´epartement de Math´ematiques et Informatique, BP 907 Universit´ed’Antananarivo, 101 Antananarivo, Madagascar; e-mail: [email protected] Abstract A Parity Alternating Permutation of the set [n] = 1, 2,...,n is a permutation with { } even and odd entries alternatively. We deal with parity alternating permutations having an odd entry in the first position, PAPs. We study the numbers that count the PAPs with even as well as odd parity. We also study a subclass of PAPs being derangements as well, Parity Alternating Derangements (PADs). Moreover, by considering the parity of these PADs we look into their statistical property of excedance. Keywords: parity, parity alternating permutation, parity alternating derangement, excedance 2020 MSC: 05A05, 05A15, 05A19 arXiv:2101.09125v1 [math.CO] 22 Jan 2021 1. Introduction and preliminaries A permutation π is a bijection from the set [n]= 1, 2,...,n to itself and we will write it { } in standard representation as π = π(1) π(2) π(n), or as the product of disjoint cycles. ··· The parity of a permutation π is defined as the parity of the number of transpositions (cycles of length two) in any representation of π as a product of transpositions. One 1Corresponding author. n c way of determining the parity of π is by obtaining the sign of ( 1) − , where c is the − number of cycles in the cycle representation of π.
    [Show full text]
  • On the Parity of the Number of Multiplicative Partitions and Related Problems
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 11, November 2012, Pages 3793–3803 S 0002-9939(2012)11254-7 Article electronically published on March 15, 2012 ON THE PARITY OF THE NUMBER OF MULTIPLICATIVE PARTITIONS AND RELATED PROBLEMS PAUL POLLACK (Communicated by Ken Ono) Abstract. Let f(N) be the number of unordered factorizations of N,where a factorization is a way of writing N as a product of integers all larger than 1. For example, the factorizations of 30 are 2 · 3 · 5, 5 · 6, 3 · 10, 2 · 15, 30, so that f(30) = 5. The function f(N), as a multiplicative analogue of the (additive) partition function p(N), was first proposed by MacMahon, and its study was pursued by Oppenheim, Szekeres and Tur´an, and others. Recently, Zaharescu and Zaki showed that f(N) is even a positive propor- tion of the time and odd a positive proportion of the time. Here we show that for any arithmetic progression a mod m,thesetofN for which f(N) ≡ a(mod m) possesses an asymptotic density. Moreover, the density is positive as long as there is at least one such N. For the case investigated by Zaharescu and Zaki, we show that f is odd more than 50 percent of the time (in fact, about 57 percent). 1. Introduction Let f(N) be the number of unordered factorizations of N,whereafactorization of N is a way of writing N as a product of integers larger than 1. For example, f(12) = 4, corresponding to 2 · 6, 2 · 2 · 3, 3 · 4, 12.
    [Show full text]
  • The ℓ-Parity Conjecture Over the Constant Quadratic Extension
    THE `-PARITY CONJECTURE OVER THE CONSTANT QUADRATIC EXTENSION KĘSTUTIS ČESNAVIČIUS Abstract. For a prime ` and an abelian variety A over a global field K, the `-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton-Dyer, the Z`-corank of the `8-Selmer group and the analytic rank agree modulo 2. Assuming that char K ¡ 0, we prove that the `-parity conjecture holds for the base change of A to the constant quadratic extension if ` is odd, coprime to char K, and does not divide the degree of every polarization of A. The techniques involved in the proof include the étale cohomological interpretation of Selmer groups, the Grothendieck–Ogg–Shafarevich formula, and the study of the behavior of local root numbers in unramified extensions. 1. Introduction 1.1. The `-parity conjecture. The Birch and Swinnerton-Dyer conjecture (BSD) predicts that the completed L-function LpA; sq of an abelian variety A over a global field K extends meromorphically to the whole complex plane, vanishes to the order rk A at s “ 1, and satisfies the functional equation LpA; 2 ´ sq “ wpAqLpA; sq; (1.1.1) where rk A and wpAq are the Mordell–Weil rank and the global root number of A. The vanishing assertion combines with (1.1.1) to give “BSD modulo 2,” namely, the parity conjecture: ? p´1qrk A “ wpAq: Selmer groups tend to be easier to study than Mordell–Weil groups, so, fixing a prime `, one sets 8 rk` A :“ dimQ` HompSel` A; Q`{Z`q bZ` Q`; where Sel 8 A : lim Sel n A ` “ ÝÑ ` is the `8-Selmer group, notes that the conjectured finiteness of the Shafarevich–Tate group XpAq implies rk` A “ rk A, and instead of the parity conjecture considers the `-parity conjecture: ? p´1qrk` A “ wpAq: (1.1.2) 1.2.
    [Show full text]
  • Some Problems of Erdős on the Sum-Of-Divisors Function
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 3, Pages 1–26 (April 5, 2016) http://dx.doi.org/10.1090/btran/10 SOME PROBLEMS OF ERDOS˝ ON THE SUM-OF-DIVISORS FUNCTION PAUL POLLACK AND CARL POMERANCE For Richard Guy on his 99th birthday. May his sequence be unbounded. Abstract. Let σ(n) denote the sum of all of the positive divisors of n,and let s(n)=σ(n) − n denote the sum of the proper divisors of n. The functions σ(·)ands(·) were favorite subjects of investigation by the late Paul Erd˝os. Here we revisit three themes from Erd˝os’s work on these functions. First, we improve the upper and lower bounds for the counting function of numbers n with n deficient but s(n) abundant, or vice versa. Second, we describe a heuristic argument suggesting the precise asymptotic density of n not in the range of the function s(·); these are the so-called nonaliquot numbers. Finally, we prove new results on the distribution of friendly k-sets, where a friendly σ(n) k-set is a collection of k distinct integers which share the same value of n . 1. Introduction Let σ(n) be the sum of the natural number divisors of n,sothatσ(n)= d|n d. Let s(n) be the sum of only the proper divisors of n,sothats(n)=σ(n) − n. Interest in these functions dates back to the ancient Greeks, who classified numbers as deficient, perfect,orabundant according to whether s(n) <n, s(n)=n,or s(n) >n, respectively.
    [Show full text]
  • Gender Equality: Glossary of Terms and Concepts
    GENDER EQUALITY: GLOSSARY OF TERMS AND CONCEPTS GENDER EQUALITY Glossary of Terms and Concepts UNICEF Regional Office for South Asia November 2017 Rui Nomoto GENDER EQUALITY: GLOSSARY OF TERMS AND CONCEPTS GLOSSARY freedoms in the political, economic, social, a cultural, civil or any other field” [United Nations, 1979. ‘Convention on the Elimination of all forms of Discrimination Against Women,’ Article 1]. AA-HA! Accelerated Action for the Health of Adolescents Discrimination can stem from both law (de jure) or A global partnership, led by WHO and of which from practice (de facto). The CEDAW Convention UNICEF is a partner, that offers guidance in the recognizes and addresses both forms of country context on adolescent health and discrimination, whether contained in laws, development and puts a spotlight on adolescent policies, procedures or practice. health in regional and global health agendas. • de jure discrimination Adolescence e.g., in some countries, a woman is not The second decade of life, from the ages of 10- allowed to leave the country or hold a job 19. Young adolescence is the age of 10-14 and without the consent of her husband. late adolescence age 15-19. This period between childhood and adulthood is a pivotal opportunity to • de facto discrimination consolidate any loss/gain made in early e.g., a man and woman may hold the childhood. All too often adolescents - especially same job position and perform the same girls - are endangered by violence, limited by a duties, but their benefits may differ. lack of quality education and unable to access critical health services.i UNICEF focuses on helping adolescents navigate risks and vulnerabilities and take advantage of e opportunities.
    [Show full text]
  • Introduction to Groups, Rings and Fields
    Introduction to Groups, Rings and Fields HT and TT 2011 H. A. Priestley 0. Familiar algebraic systems: review and a look ahead. GRF is an ALGEBRA course, and specifically a course about algebraic structures. This introduc- tory section revisits ideas met in the early part of Analysis I and in Linear Algebra I, to set the scene and provide motivation. 0.1 Familiar number systems Consider the traditional number systems N = 0, 1, 2,... the natural numbers { } Z = m n m, n N the integers { − | ∈ } Q = m/n m, n Z, n = 0 the rational numbers { | ∈ } R the real numbers C the complex numbers for which we have N Z Q R C. ⊂ ⊂ ⊂ ⊂ These come equipped with the familiar arithmetic operations of sum and product. The real numbers: Analysis I built on the real numbers. Right at the start of that course you were given a set of assumptions about R, falling under three headings: (1) Algebraic properties (laws of arithmetic), (2) order properties, (3) Completeness Axiom; summarised as saying the real numbers form a complete ordered field. (1) The algebraic properties of R You were told in Analysis I: Addition: for each pair of real numbers a and b there exists a unique real number a + b such that + is a commutative and associative operation; • there exists in R a zero, 0, for addition: a +0=0+ a = a for all a R; • ∈ for each a R there exists an additive inverse a R such that a+( a)=( a)+a = 0. • ∈ − ∈ − − Multiplication: for each pair of real numbers a and b there exists a unique real number a b such that · is a commutative and associative operation; • · there exists in R an identity, 1, for multiplication: a 1 = 1 a = a for all a R; • · · ∈ for each a R with a = 0 there exists an additive inverse a−1 R such that a a−1 = • a−1 a = 1.∈ ∈ · · Addition and multiplication together: forall a,b,c R, we have the distributive law a (b + c)= a b + a c.
    [Show full text]
  • Matrix Characterization of MDS Linear Codes Over Modules
    LINEAR ALGEBRA AND ITS APPLICATIONS ELSEVIER Linear Algebra and its Applications 277 (1998) 57 61 Matrix characterization of MDS linear codes over modules Xue-Dong Dong, Cheong Boon Soh *, Erry Gunawan Block $2, School of Electrical and Electronic' Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singupore Received 23 June 1997; accepted 13 August 1997 Submitted by R. Brualdi Abstract Let R be a commutative ring with identity, N be an R-module, and M = (a,/),.×~. be a matrix over R. A linear code C of length n over N is defined to be a submodule of N '~. It is shown that a linear code C(k, r) with parity check matrix (-MI/,.) is maximum dis- tance separable (MDS) iff the determinant of every h × h submatrix, h = 1,2,..., rain{k, r}, of M is not an annihilator of any nonzero element of N. This characterization is used to derive some results for group codes over abelian groups. © 1998 Elsevier Science Inc. All rights reserved. Keywords. Linear codes; MDS codes; Codes over rings; Group codes: Codes over modules I. Introduction Recently, Zain and Rajan [1] gave a good algebraic characterization of max- imum distance separable (MDS) group codes over cyclic groups. They pointed out that the algebraic approach can be extended to the general case of group codes over abelian groups using the complex theory of determinants of matri- ces over noncommutative rings [1]. In this short paper, the matrix characteriza- tions of MDS codes over finite fields and MDS group codes over cyclic groups are extended to linear codes with systematic parity check matrices over *Corresponding author.
    [Show full text]
  • Fairness Definitions Explained
    2018 ACM/IEEE International Workshop on Software Fairness Fairness Definitions Explained Sahil Verma Julia Rubin Indian Institute of Technology Kanpur, India University of British Columbia, Canada [email protected] [email protected] ABSTRACT training data containing observations whose categories are known. Algorithm fairness has started to attract the attention of researchers We collect and clarify most prominent fairness definitions for clas- in AI, Software Engineering and Law communities, with more than sification used in the literature, illustrating them on a common, twenty different notions of fairness proposed in the last few years. unifying example – the German Credit Dataset [18]. This dataset Yet, there is no clear agreement on which definition to apply in is commonly used in fairness literature. It contains information each situation. Moreover, the detailed differences between multiple about 1000 loan applicants and includes 20 attributes describing definitions are difficult to grasp. To address this issue, thispaper each applicant, e.g., credit history, purpose of the loan, loan amount collects the most prominent definitions of fairness for the algo- requested, marital status, gender, age, job, and housing status. It rithmic classification problem, explains the rationale behind these also contains an additional attribute that describes the classification definitions, and demonstrates each of them on a single unifying outcome – whether an applicant has a good or a bad credit score. case-study. Our analysis intuitively explains why the same case When illustrating the definitions, we checked whether the clas- can be considered fair according to some definitions and unfair sifier that uses this dataset exhibits gender-related bias.
    [Show full text]
  • Interferometry with Independent Bose-Einstein Ondensates: Parity As an EPR/Bell Quantum Variable Franck Laloë, William Mullin
    Interferometry with independent Bose-Einstein ondensates: parity as an EPR/Bell quantum variable Franck Laloë, William Mullin To cite this version: Franck Laloë, William Mullin. Interferometry with independent Bose-Einstein ondensates: parity as an EPR/Bell quantum variable. 2008. hal-00345342v1 HAL Id: hal-00345342 https://hal.archives-ouvertes.fr/hal-00345342v1 Preprint submitted on 8 Dec 2008 (v1), last revised 14 Sep 2009 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Interferometry with independent Bose-Einstein condensates: parity as an EPR/Bell quantum variable F. Lalo¨ea and W. J. Mullinb aLaboratoire Kastler Brossel, ENS, UPMC, CNRS ; 24 rue Lhomond, 75005 Paris, France bDepartment of Physics, University of Massachusetts, Amherst, Massachusetts 01003 USA December 8, 2008 Abstract When independent Bose-Einstein condensates (BEC), described quantum mechanically by Fock (number) states, are sent into interferometers, the measurement of the output port at which the particles are detected provides a binary measurement, with two possible results ±1. With two inter- ferometers and two BEC’s, the parity (product of all results obtained at each interferometer) has all the features of an Einstein-Podolsky-Rosen quantity, with perfect correlations predicted by quantum mechanics when the settings (phase shifts of the interferometers) are the same.
    [Show full text]
  • Parity Check Matrices and Product Representations of Squares
    Parity check matrices and product representations of squares Assaf Naor Jacques VerstraÄete Microsoft Research University of Waterloo [email protected] [email protected] Abstract Let NF(n; k; r) denote the maximum number of columns in an n-row matrix with entries in a ¯nite ¯eld F in which each column has at most r nonzero entries and every k columns are linearly independent over F. We obtain near-optimal upper bounds for NF(n; k; r) in the case r cr 2 + k 4 k > r. Namely, we show that NF(n; k; r) ¿ n where c ¼ 3 for large k. Our method is based on a novel reduction of the problem to the extremal problem for cycles in graphs, and yields a fast algorithm for ¯nding short linear dependences. We present additional applications of this method to problems in extremal hypergraph theory and combinatorial number theory. \It is certainly odd to have an instruction in an algorithm asking you to play with some numbers to ¯nd a subset with product a square . Why should we expect to ¯nd such a subsequence, and, if it exists, how can we ¯nd it e±ciently?" Carl Pomerance [32]. 1 Introduction Since the mid-1930s it has been well established that there is a tight connection between combi- natorial number theory and extremal combinatorics (see [33, 12] and the references therein). The basic paradigm is that for certain number theoretical problems one can construct a combinatorial object (e.g. a graph or a hypergraph), and prove that it cannot contain certain con¯gurations (e.g.
    [Show full text]
  • Carmichael Numbers in the Sequence {2 K + 1}N≥1
    n Carmichael numbers in the sequence f2 k + 1gn¸1 Javier Cilleruelo Instituto de Ciencias Matem¶aticas(CSIC-UAM-UC3M-UCM) and Departamento de Matem¶aticas Universidad Aut¶onomade Madrid 28049, Madrid, Espa~na [email protected] Florian Luca Instituto de Matem¶aticas Universidad Nacional Autonoma de M¶exico C.P. 58089, Morelia, Michoac¶an,M¶exico fl[email protected] Amalia Pizarro-Madariaga Departamento de Matem¶aticas Uinversidad de Valparaiso, Chile [email protected] May 22, 2012 1 Introduction A Carmichael number is a positive integer N which is composite and the con- gruence aN ´ a (mod N) holds for all integers a. The smallest Carmichael number is N = 561 and was found by Carmichael in 1910 in [6]. It is well{ known that there are in¯nitely many Carmichael numbers (see [1]). Here, we let k be any odd positive integer and study the presence of Carmichael numbers in the sequence of general term 2nk + 1. Since it is known [15] that the sequence 2n + 1 does not contain Carmichael numbers, we will assume that k ¸ 3 through the paper. We have the following result. For a positive integer m let ¿(m) be the number of positive divisors of m. We also write !(m) for the number of distinct prime factors of m. For a positive real number x we write log x for its natural logarithm. 1 Theorem 1. Let k ¸ 3 be an odd integer. If N = 2nk + 1 is Carmichael, then 6 2 2 n < 22£10 ¿(k) (log k) !(k): (1) The proof of Theorem 1 uses a quantitative version of the Subspace Theorem as well as lower bounds for linear forms in logarithms of algebraic numbers.
    [Show full text]
  • The Odd-Even Effect in Multiplication: Parity Rule Or Familiarity with Even Numbers?
    Memory & Cognition 2000, 28 (3), 358-365 The odd-even effect in multiplication: Parity rule or familiarity with even numbers? AUETTELOCHYandXA~ERSERON Universite Catholique de Louvain, Louvain-La-Neuve, Belgium MARGARETE DELAZER UniversiUitklinikjilr Neurologie, Innsbruck, Austria and BRIAN BUTTERWORTH University College ofLondon, London, England This study questions the evidence that a parity rule is used during the verification of multiplication. Previous studies reported that products are rejected faster when they violate the expected parity, which was attributed to the use of a rule (Krueger, 1986; Lemaire & Fayol, 1995). This experiment tested an alternative explanation of this effect: the familiarity hypothesis. Fifty subjects participated in a verifi­ cation task with contrasting types of problems (even X even, odd X odd, mixed). Some aspects of our results constitute evidence against the use of the parity rule: False even answers were rejected slowly, even when the two operands were odd. Wesuggest that the odd-even effect in verification of multipli­ cation could not be due to the use of the parity rule, but rather to a familiarity with even numbers (three quarters of products are indeed even). Besides the issues concerning the representation for­ feet has been observed for both addition (Krueger & mat in which parity information is coded and the way this Hallford, 1984) and multiplication (Krueger, 1986), and information is retrieved (Campbell & Clark, 1992; Clark it has been attributed to the use ofa rule. Inproduct ver­ & Campbell, 1991; Dehaene, Bossini, & Giraux, 1993; ification, subjects likewise use the rule "the true product McCloskey, Aliminosa, & Sokol, 1991; McCloskey, Cara­ must be even ifany multiplier is even; otherwise it must mazza, & Basili, 1985), the role of parity in arithmetic be odd" (Krueger, 1986).
    [Show full text]