Robust locally testable codes and products of codes

We continue the investigation of locally testable codes, i.e., error‐correcting codes for which membership of a given word in the code can be tested probabilistically by examining it in very few locations. We give two general results on local testability: First, motivated by the recently proposed notion of robust probabilistically checkable proofs, we introduce the notion of robust local testability of codes. We relate this notion to a product of codes introduced by Tanner and show a very simple composition lemma for this notion. Next, we show that codes built by tensor products can be tested robustly and somewhat locally by applying a variant of a test and proof technique introduced by Raz and Safra in the context of testing low‐degree multivariate polynomials (which are a special case of tensor codes). Combining these two results gives us a generic construction of codes of inverse polynomial rate that are testable with poly‐logarithmically many queries. We note that these locally testable tensor codes can be obtained from any linear error correcting code with good distance. Previous results on local testability, albeit much stronger quantitatively, rely heavily on algebraic properties of the underlying codes. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

A combination of testability and decodability by tensor products

Ben‐Sasson and Sudan (RSA 2006) showed that taking the repeated tensor product of linear codes with very large distance results in codes that are locally testable. Due to the large distance requirement the associated tensor products could be applied only over sufficiently large fields. Then Meir (SICOMP 2009) used this result to present a combinatorial construction of locally testable codes with largest known rate. As a consequence, this construction was obtained over sufficiently large fields. In this paper we improve the result of Ben‐Sasson and Sudan and show that for any linear codes the associated tensor products are locally testable. Consequently, the construction of Meir can be taken over any field, including the binary field. Moreover, a combination of our result with the result of Spielman (IEEE IT, 1996) implies a construction of linear codes (over any field) that combine the following properties:

• have constant rate and constant relative distance;
• have blocklength n and are testable with n^? queries, for any constant ? > 0;
• linear time encodable and linear‐time decodable from a constant fraction of errors.

Furthermore, a combination of our result with the result of Guruswami et al. (STOC 2009) implies a similar corollary for list‐decodable codes. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 572–598, 2015     

Testing algebraic geometric codes

Property testing was initially studied from various motivations in 1990’s. A code C  GF (r)n is locally testable if there is a randomized algorithm which can distinguish with high possibility the codewords from a vector essentially far from the code by only accessing a very small (typically constant) number of the vector’s coordinates. The problem of testing codes was firstly studied by Blum, Luby and Rubinfeld and closely related to probabilistically checkable proofs (PCPs). How to characterize locally te...     

Orthogonal designs,self‐dual codes,and the Leech lattice

Symmetric designs and Hadamard matrices are used to construct binary and ternary self‐dual codes. Orthogonal designs are shown to be useful in construction of self‐dual codes over large fields. In this paper, we first introduce a new array of order 12, which is suitable for any set of four amicable circulant matrices. We apply some orthogonal designs of order 12 to construct new self‐dual codes over large finite fields, which lead us to the odd Leech lattice by Construction A. © 2005 Wiley Periodicals, Inc. J Combin Designs 13: 184–194, 2005.     

Some Hadamard matrices of order 32 and their binary codes

It is known that all doubly‐even self‐dual codes of lengths 8 or 16, and the extended Golay code, can be constructed from some binary Hadamard matrix of orders 8, 16, and 24, respectively. In this note, we demonstrate that every extremal doubly‐even self‐dual [32,16,8] code can be constructed from some binary Hadamard matrix of order 32. © 2004 Wiley Periodicals, Inc.     

Combinatorial pseudo-triangulations

We prove that a planar graph is generically rigid in the plane if and only if it can be embedded as a pseudo-triangulation. This generalizes the main result of [Haas et al. Planar minimally rigid graphs and pseudo-triangulations, Comput. Geom. 31(1–2) (2005) 31–61] which treats the minimally generically rigid case.The proof uses the concept of combinatorial pseudo-triangulation, CPT, in the plane and has two main steps: showing that a certain “generalized Laman property” is a necessary and sufficient condition for a CPT to be “stretchable”, and showing that all generically rigid plane graphs admit a CPT assignment with that property.Additionally, we propose the study of CPTs on closed surfaces.     

Symmetric LDPC codes and local testing

Coding theoretic and complexity theoretic considerations naturally lead to the question of generating symmetric, sparse, redundant linear systems. This paper provides a new way of construction with better parameters and new lower bounds.Low Density Parity Check (LDPC) codes are linear codes defined by short constraints (a property essential for local testing of a code). Some of the best (theoretically and practically) used codes are LDPC. Symmetric codes are those in which all coordinates “look the same,” namely there is some transitive group acting on the coordinates which preserves the code. Some of the most commonly used locally testable codes (especially in PCPs and other proof systems), including all “low-degree” codes, are symmetric. Requiring that a symmetric binary code of length n has large (linear or near-linear) distance seems to suggest a “con ict” between 1/rate and density (constraint length). In known constructions, if one is constant, then the other is almost the worst possible - n/poly(logn).Our main positive result simultaneously achieves symmetric low density, constant rate codes generated by a single constraint. We present an explicit construction of a symmetric and transitive binary code of length n, near-linear distance n/(log logn)², of constant rate and with constraints of length (logn)⁴. The construction is in the spirit of Tanner codes, namely the codewords are indexed by the edges of a sparse regular expander graph. The main novelty is in our construction of a transitive (non Abelian!) group acting on these edges which preserves the code. Our construction is one instantiation of a framework we call Cayley Codes developed here, that may be viewed as extending zig-zag product to symmetric codes.Our main negative result is that the parameters obtained above cannot be significantly improved, as long as the acting group is solvable (like the one we use). More specifically, we show that in constant rate and linear distance codes (aka “good” codes) invariant under solvable groups, the density (length of generating constraints) cannot go down to a constant, and is bounded below by (log^(Ω(?)) n)(an Ω(?) iterated logarithm) if the group has a derived series of length ?. This negative result precludes natural local tests with constantly many queries for such solvable “good” codes.     

On Hadamard matrices of order 2(p+1) with an automorphism of odd prime order p

In this paper, we investigate Hadamard matrices of order 2(p + 1) with an automorphism of odd prime order p. In particular, the classification of such Hadamard matrices for the cases p = 19 and 23 is given. Self‐dual codes related to such Hadamard matrices are also investigated. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 367–380, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10052     

An efficiently computed lower bound on the number of recombinations in phylogenetic networks: Theory and empirical study

Phylogenetic networks are models of sequence evolution that go beyond trees, allowing biological operations that are not tree-like. One of the most important biological operations is recombination between two sequences. An established problem [J. Hein, Reconstructing evolution of sequences subject to recombination using parsimony, Math. Biosci. 98 (1990) 185–200; J. Hein, A heuristic method to reconstruct the history of sequences subject to recombination, J. Molecular Evoluation 36 (1993) 396–405; Y. Song, J. Hein, Parsimonious reconstruction of sequence evolution and haplotype blocks: finding the minimum number of recombination events, in: Proceedings of 2003 Workshop on Algorithms in Bioinformatics, Berlin, Germany, 2003, Lecture Notes in Computer Science, Springer, Berlin; Y. Song, J. Hein, On the minimum number of recombination events in the evolutionary history of DNA sequences, J. Math. Biol. 48 (2003) 160–186; L. Wang, K. Zhang, L. Zhang, Perfect phylogenetic networks with recombination, J. Comput. Biol. 8 (2001) 69–78; S.R. Myers, R.C. Griffiths, Bounds on the minimum number of recombination events in a sample history, Genetics 163 (2003) 375–394; V. Bafna, V. Bansal, Improved recombination lower bounds for haplotype data, in: Proceedings of RECOMB, 2005; Y. Song, Y. Wu, D. Gusfield, Efficient computation of close lower and upper bounds on the minimum number of needed recombinations in the evolution of biological sequences, Bioinformatics 21 (2005) i413–i422. Bioinformatics (Suppl. 1), Proceedings of ISMB, 2005, D. Gusfield, S. Eddhu, C. Langley, Optimal, efficient reconstruction of phylogenetic networks with constrained recombination, J. Bioinform. Comput. Biol. 2(1) (2004) 173–213; D. Gusfield, Optimal, efficient reconstruction of root-unknown phylogenetic networks with constrained and structured recombination, J. Comput. Systems Sci. 70 (2005) 381–398] is to find a phylogenetic network that derives an input set of sequences, minimizing the number of recombinations used. No efficient, general algorithm is known for this problem. Several papers consider the problem of computing a lower bound on the number of recombinations needed. In this paper we establish a new, efficiently computed lower bound. This result is useful in methods to estimate the number of needed recombinations, and also to prove the optimality of algorithms for constructing phylogenetic networks under certain conditions [D. Gusfield, S. Eddhu, C. Langley, Optimal, efficient reconstruction of phylogenetic networks with constrained recombination, J. Bioinform. Comput. Biol. 2(1) (2004) 173–213; D. Gusfield, Optimal, efficient reconstruction of root-unknown phylogenetic networks with constrained and structured recombination, J. Comput. Systems Sci. 70 (2005) 381–398; D. Gusfield, Optimal, efficient reconstruction of root-unknown phylogenetic networks with constrained recombination, Technical Report, Department of Computer Science, University of California, Davis, CA, 2004]. The lower bound is based on a structural, combinatorial insight, using only the site conflicts and incompatibilities, and hence it is fundamental and applicable to many biological phenomena other than recombination, for example, when gene conversions or recurrent or back mutations or cross-species hybridizations cause the phylogenetic history to deviate from a tree structure. In addition to establishing the bound, we examine its use in more complex lower bound methods, and compare the bounds obtained to those obtained by other established lower bound methods.     

SERK2v2: A new second‐order stabilized explicit Runge‐Kutta method for stiff problems

Traditionally, explicit numerical algorithms have not been used with stiff ordinary differential equations (ODEs) due to their stability. Implicit schemes are usually very expensive when used to solve systems of ODEs with very large dimension. Stabilized Runge‐Kutta methods (also called Runge–Kutta–Chebyshev methods) were proposed to try to avoid these difficulties. The Runge–Kutta methods are explicit methods with extended stability domains, usually along the negative real axis. They can easily be applied to large problem classes with low memory demand, they do not require algebra routines or the solution of large and complicated systems of nonlinear equations, and they are especially suited for discretizations using the method of lines of two and three dimensional parabolic partial differential equations. In Martín‐Vaquero and Janssen [Comput Phys Commun 180 (2009), 1802–1810], we showed that previous codes based on stabilized Runge–Kutta algorithms have some difficulties in solving problems with very large eigenvalues and we derived a new code, SERK2, based on sixth‐order polynomials. Here, we develop a new method based on second‐order polynomials with up to 250 stages and good stability properties. These methods are efficient numerical integrators of very stiff ODEs. Numerical experiments with both smooth and nonsmooth data support the efficiency and accuracy of the new algorithms when compared to other well‐known second‐order methods such as RKC and ROCK2. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Convergence analysis of the streamline diffusion and discontinuous Galerkin methods for the Vlasov‐Fokker‐Planck system

We prove stability estimates and derive optimal convergence rates for the streamline diffusion and discontinuous Galerkin finite element methods for discretization of the multi‐dimensional Vlasov‐Fokker‐Planck system. The focus is on the theoretical aspects, where we deal with construction and convergence analysis of the discretization schemes. Some related special cases are implemented in M. Asadzadeh [Appl Comput Meth 1(2) (2002), 158–175] and M. Asadzadeh and A. Sopasakis [Comput Meth Appl Mech Eng 191(41–42) (2002), 4641–4661]. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Gold codes,Hadamard partitions and the security of CDMA systems

Let V = {1, 2, . . . , M} and let be a set of Hadamard matrices with the property that the magnitude of the dot product of any two rows of distinct matrices is bounded above. A Hadamard partition is any partition of the set of all rows of the matrices H _i into Hadamard matrices. Such partitions have an application to the security of quasi-synchronous code-division multiple-access radio systems when loosely synchronized (LS) codes are used as spreading codes. A new generation of LS code can be used for each information bit to be spread. For each generation, a Hadamard matrix from some partition is selected for use in the code construction. This code evolution increases security against eavesdropping and jamming. One security aspect requires that the number of Hadamard partitions be large. Thus the number of partitions is studied here. If a Kerdock code construction is used for the set of matrices, the Hadamard partition constructed is shown to be unique. It is also shown here that this is not the case if a Gold (or Gold-like) code construction is used. In this case the number of Hadamard partitions can be enumerated, and is very large.      

Constructing self-orthogonal and Hermitian self-orthogonal codes via weighing matrices and orbit matrices

We define the notion of an orbit matrix with respect to standard weighing matrices, and with respect to types of weighing matrices with entries in a finite field. In the latter case we primarily restrict our attention the fields of order 2, 3 and 4. We construct self-orthogonal and Hermitian self-orthogonal linear codes over finite fields from these types of weighing matrices and their orbit matrices respectively. We demonstrate that this approach applies to several combinatorial structures such as Hadamard matrices and balanced generalized weighing matrices. As a case study we construct self-orthogonal codes from some weighing matrices belonging to some well known infinite families, such as the Paley conference matrices, and weighing matrices constructed from ternary periodic Golay pairs.     

Local and parallel finite element algorithms for the Steklov eigenvalue problem

Based on the work of Xu and Zhou [Math Comput 69 (2000) 881–909], we propose and analyze in this article local and parallel finite element algorithms for the Steklov eigenvalue problem. We also prove a local error estimate which is suitable for the case that the locally refined region contains singular points lying on the boundary of domain, which is an improvement of the existing results. Numerical experiments are reported finally to validate our theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 399–417, 2016     

Skew Hadamard designs and their codes

Skew Hadamard designs (4n – 1, 2n – 1, n – 1) are associated to order 4n skew Hadamard matrices in the natural way. We study the codes spanned by their incidence matrices A and by I + A and show that they are self-dual after extension (resp. extension and augmentation) over fields of characteristic dividing n. Quadratic Residues codes are obtained in the case of the Paley matrix. Results on the p-rank of skew Hadamard designs are rederived in that way. Codes from skew Hadamard designs are classified. An optimal self-dual code over GF(5) is rediscovered in length 20. Six new inequivalent [56, 28, 16] self-dual codes over GF(7) are obtained from skew Hadamard matrices of order 56, improving the only known quadratic double circulant code of length 56 over GF(7).     

The writhe of permutations and random framed knots

We introduce and study the writhe of a permutation, a circular variant of the well‐known inversion number. This simple permutation statistics has several interpretations, which lead to some interesting properties. For a permutation sampled uniformly at random, we study the asymptotics of the writhe, and obtain a non‐Gaussian limit distribution. This work is motivated by the study of random knots. A model for random framed knots is described, which refines the Petaluma model, studied with Hass, Linial, and Nowik (Discrete Comput Geom, 2016). The distribution of the framing in this model is equivalent to the writhe of random permutations. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 121–142, 2017     

On the self‐dual 𝔽5‐codes constructed from Hadamard matrices of order 24

There are exactly 60 inequivalent Hadamard matrices of order 24. In this note, we give a classification of the self‐dual ??₅‐codes of length 48 constructed from the Hadamard matrices of order 24. © 2004 Wiley Periodicals, Inc.     

2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) designs with automorphisms of odd prime order,and their related Hadamard matrices and codes

We present the full classification of Hadamard 2-(31,15,7), Hadamard 2-(35, 17,8) and Menon 2-(36,15,6) designs with automorphisms of odd prime order. We also give partial classifications of such designs with automorphisms of order 2. These classifications lead to related Hadamard matrices and self-dual codes. We found 76166 Hadamard matrices of order 32 and 38332 Hadamard matrices of order 36, arising from the classified designs. Remarkably, all constructed Hadamard matrices of order 36 are Hadamard equivalent to a regular Hadamard matrix. From our constructed designs, we obtained 37352 doubly-even [72,36,12] codes, which are the best known self-dual codes of this length until now.      

Twisted product of cocycles and factorization of semi‐regular relative difference sets

In this article, we introduce what we call twisted Kronecker products of cocycles of finite groups and show that the twisted Kronecker product of two cocycles is a Hadamard cocycle if and only if the two cocycles themselves are Hadamard cocycles. This enables us to generalize some known results concerning products and factorizations of central semi‐regular relative difference sets. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 431–441, 2008     

一类具有双恢复集的局部恢复码的构造

假设C是有限域Fq上的[n,k]线性码,如果码字的每个坐标是其它至多r个坐标的函数,称C是(n,k,r)局部恢复码,这里r是较小的数.在分布式存储系统中,具有多个恢复集的局部恢复码使得数据在系统中更具实际意义,因为它可以避免热数据的频繁访问.引入代数函数域、特别是Hermite函数域去构造局部恢复码,这类局部恢复码具有双恢复集,并且码长可以突破字符集的大小的限制.结果表明,此构造方法得出的最小距离下界明显地改进了Alexander Barg的最小距离的下界.