On the minimum average distance of binary constant weight codes

Let β(n,M,w) denote the minimum average Hamming distance of a binary constant weight code with length n, size M and weight w. In this paper, we study the problem of determining β(n,M,w). Using the methods from coding theory and linear programming, we derive several lower bounds on the average Hamming distance of a binary constant weight code. These lower bounds enable us to determine the exact value for β(n,M,w) in several cases.     

Covering radius and dual distance

Recently a number of bounds have been obtained for the covering radius of a code with given length and cardinality. In this paper we show that—perhaps surprisingly—the covering radius of a code depends heavily on its dual distance. We consider an arbitrary finite Abelian group alphabet though in the applications the alphabet is very often the field F ₂.     

On the distance distribution of trees

Let G be a tree and k a non-negative integer. We determine best possible upper and lower bounds on the number of pairs of vertices at distance exactly k in G in terms of order alone, and in terms of order and radius or diameter.     

Bounds on permutation codes of distance four

A permutation code of length n and distance d is a set Γ of permutations from some fixed set of n symbols such that the Hamming distance between each distinct x,y∈Γ is at least d. In this note, we determine some new results on the maximum size of a permutation code with distance equal to 4, the smallest interesting value. The upper bound is improved for almost all n via an optimization problem on Young diagrams. A new recursive construction improves known lower bounds for small values of n.     

The distance of a subdivision surface to its control polyhedron

For standard subdivision algorithms and generic input data, near an extraordinary point the distance from the limit surface to the control polyhedron after m subdivision steps is shown to decay dominated by the mth power of the subsubdominant (third largest) eigenvalue. Conversely, for Loop subdivision we exhibit generic input data so that the Hausdorff distance at the mth step is greater than or equal to the mth power of the subsubdominant eigenvalue.In practice, it is important to closely predict the number of subdivision steps necessary so that the control polyhedron approximates the surface to within a fixed distance. Based on the above analysis, two such predictions are evaluated. The first is a popular heuristic that analyzes the distance only for control points and not for the facets of the control polyhedron. For a set of test polyhedra this prediction is remarkably close to the true distance. However, a concrete example shows that the prediction is not safe but can prescribe too few steps. The second approach is to first locally, per vertex neighborhood, subdivide the input net and then apply tabulated bounds on the eigenfunctions of the subdivision algorithm. This yields always safe predictions that are within one step for a set of typical test surfaces.     

Flat tori, lattices and bounds for commutative group codes

We show that commutative group spherical codes in R ⁿ , as introduced by D. Slepian, are directly related to flat tori and quotients of lattices. As consequence of this view, we derive new results on the geometry of these codes and an upper bound for their cardinality in terms of minimum distance and the maximum center density of lattices and general spherical packings in the half dimension of the code. This bound is tight in the sense it can be arbitrarily approached in any dimension. Examples of this approach and a comparison of this bound with Union and Rankin bounds for general spherical codes is also presented.     

The disjunctive domination number of a graph

Abstract

For a positive integer b, we define a set S of vertices in a graph G as a b-disjunctive dominating set if every vertex not in S is adjacent to a vertex of S or has at least b vertices in S at distance 2 from it. The b-disjunctive domination number is the minimum cardinality of such a set. This concept is motivated by the concepts of distance domination and exponential domination. In this paper, we start with some simple results, then establish bounds on the parameter especially for regular graphs and claw-free graphs. We also show that determining the parameter is NP-complete, and provide a linear-time algorithm for trees.     

Coding with injections

A permutation code of length n and minimum distance d is a set Γ of permutations from some fixed set of n symbols such that the Hamming distance between any distinct ${u,v \in \Gamma}$ is at least d. As a generalization, we introduce the problem of packing injections from an m-set, m ≤?n, sometimes called m-arrangements, relative to Hamming distance. We offer some preliminary coding-theoretic bounds, a few design-theoretic connections, and a short discussion on possible applications.     

Improving bounds on the minimum Euclidean distance for block codes by inner distance measure optimization

The minimum Euclidean distance is a fundamental quantity for block coded phase shift keying (PSK). In this paper we improve the bounds for this quantity that are explicit functions of the alphabet size q, block length n and code size |C|. For q=8, we improve previous results by introducing a general inner distance measure allowing different shapes of a neighborhood for a codeword. By optimizing the parameters of this inner distance measure, we find sharper bounds for the outer distance measure, which is Euclidean.The proof is built upon the Elias critical sphere argument, which localizes the optimization problem to one neighborhood. We remark that any code with q=8 that fulfills the bound with equality is best possible in terms of the minimum Euclidean distance, for given parameters n and |C|. This is true for many multilevel codes.     

Maximal and minimal entry in the principal eigenvector for the distance matrix of a graph

Let G=(V,E) be a simple, connected and undirected graph with vertex set V(G) and edge set E(G). Also let D(G) be the distance matrix of a graph G (Jane?i? et al., 2007) [13]. Here we obtain Nordhaus–Gaddum-type result for the spectral radius of distance matrix of a graph.A sharp upper bound on the maximal entry in the principal eigenvector of an adjacency matrix and signless Laplacian matrix of a simple, connected and undirected graph are investigated in Das (2009) [4] and Papendieck and Recht (2000) [15]. Generally, an upper bound on the maximal entry in the principal eigenvector of a symmetric nonnegative matrix with zero diagonal entries and without zero diagonal entries are investigated in Zhao and Hong (2002) [21] and Das (2009) [4], respectively. In this paper, we obtain an upper bound on minimal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs. Moreover, we present the lower and upper bounds on maximal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs.     

Average distances and distance domination numbers

Let k be a positive integer and G be a simple connected graph with order n. The average distance μ(G) of G is defined to be the average value of distances over all pairs of vertices of G. A subset D of vertices in G is said to be a k-dominating set of G if every vertex of V(G)−D is within distance k from some vertex of D. The minimum cardinality among all k-dominating sets of G is called the k-domination number γ_k(G) of G. In this paper tight upper bounds are established for μ(G), as functions of n, k and γ_k(G), which generalizes the earlier results of Dankelmann [P. Dankelmann, Average distance and domination number, Discrete Appl. Math. 80 (1997) 21-35] for k=1.     

Upper Bounds on the Dual Distance of BCH(255, k)

To obtain upper bounds on the distance of a binary linear code, many probabilistic algorithms have been proposed. The author presents a general variation to these algorithms, specific for cyclic codes, which is shown to be an improvement. As an example, the author optimizes Brouwer's algorithm to find the best upper bounds on the dual distance of BCH[255,k,d].     

Proximal Point Algorithm On Riemannian Manifolds

Abstract

In this paper we consider the minimization problem with constraints. We will show that if the set of constraints is a Riemannian manifold of nonpositive sectional curvature, and the objective function is convex in this manifold, then the proximal point method in Euclidean space is naturally extended to solve that class of problems. We will prove that the sequence generated by our method is well defined and converge to a minimizer point. In particular we show how tools of Riemannian geometry, more specifically the convex analysis in Riemannian manifolds, can be used to solve nonconvex constrained problem in Euclidean, space.     

On the Generalized Weights of a Class of Trace Codes

We build a class of codes using hermitian forms and the functional trace code. Then we give a general expression of the rth minimum distance of our code and compute general bounds for the weight hierarchy by using exponential sums. We also get the minimum distance and calculate the rth generalized Hamming weight d_r in some special cases.     

Distance between a Maximum Modulus Point and Zero Set of an Entire Function

Let f be an entire function of finite positive order. A maximum modulus point is a point w such that |f(w)|= max {|f(z)|:|z=|w|}. We obtain lower bounds for the distance between a maximum modulus point w and the zero set of f. These bounds are valid for all sufficiently large values of |w|.     

New formulations of the Hop-Constrained Minimum Spanning Tree problem via Miller-Tucker-Zemlin constraints

Given an undirected network with positive edge costs and a natural number p, the Hop-Constrained Minimum Spanning Tree problem (HMST) is the problem of finding a spanning tree with minimum total cost such that each path starting from a specified root node has no more than p hops (edges). In this paper, we develop new formulations for HMST. The formulations are based on Miller-Tucker-Zemlin (MTZ) subtour elimination constraints, MTZ-based liftings in the literature offered for HMST, and a new set of topology-enforcing constraints. We also compare the proposed models with the MTZ-based models in the literature with respect to linear programming relaxation bounds and solution times. The results indicate that the new models give considerably better bounds and solution times than their counterparts in the literature and that the new set of constraints is competitive with liftings to MTZ constraints, some of which are based on well-known, strong liftings of Desrochers and Laporte (1991).     

Different Bounds on the Different Betti Numbers of Semi-Algebraic Sets

Abstract. A classic result in real algebraic geometry due to Oleinik—Petrovskii, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers. We prove better bounds on the individual Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic hypersurfaces. As a corollary we obtain a polynomial bound on the highest Betti numbers of basic semi-algebraic sets defined by quadratic inequalities.     

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.     

Efficient Computation of Subset Influence in Regression

Abstract

The detection of influential cases is now accepted as an essential component of regression diagnostics. It is also well established that two or more cases that are individually regarded as noninfluential may act in concert to achieve a high level of joint influence. However, for the majority of data sets it is computationally infeasible to calculate the influence for all subsets of a given size. In this article we address this problem and suggest an algorithm that greatly reduces the computational effort by making use of a sequence of upper bounds on the influence value. These upper bounds are much less costly to evaluate and greatly reduce the number of subsets for which the influence value must be explicitly determined.     

Different Bounds on the Different Betti Numbers of Semi-Algebraic Sets

   Abstract. A classic result in real algebraic geometry due to Oleinik—Petrovskii, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers. We prove better bounds on the individual Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic hypersurfaces. As a corollary we obtain a polynomial bound on the highest Betti numbers of basic semi-algebraic sets defined by quadratic inequalities.