Minimum degree of minimal defect -extendable bipartite graphs

A near perfect matching is a matching saturating all but one vertex in a graph. If G is a connected graph and any n independent edges in G are contained in a near perfect matching, then G is said to be defect n-extendable. If for any edge e in a defect n-extendable graph G, G−e is not defect n-extendable, then G is minimal defect n-extendable. The minimum degree and the connectivity of a graph G are denoted by δ(G) and κ(G) respectively. In this paper, we study the minimum degree of minimal defect n-extendable bipartite graphs. We prove that a minimal defect 1-extendable bipartite graph G has δ(G)=1. Consider a minimal defect n-extendable bipartite graph G with n≥2, we show that if κ(G)=1, then δ(G)≤n+1 and if κ(G)≥2, then 2≤δ(G)=κ(G)≤n+1. In addition, graphs are also constructed showing that, in all cases but one, there exist graphs with minimum degree that satisfies the established bounds.     

RELATIVE WIDTH OF SMOOTH CLASSES OF MULTIVARIATE PERIODIC FUNCTIONS WITH RESTRICTIONS ON ITERATED LAPLACE DERIVATIVES IN THE L2-METRIC

For two subsets W and V of a Banach space X, let Kn(W, V, X) denote the relative Kolmogorov n-width of W relative to V defined by Kn (W, V, X) := inf sup Ln f∈W g∈V∩Ln inf ‖f-g‖x,where the infimum is taken over all n-dimensional linear subspaces Ln of X. Let W2(△r) denote the class of 2w-periodic functions f with d-variables satisfying ∫[-π,π]d |△rf(x)|2dx ≤ 1,while △r is the r-iterate of Laplace operator △. This article discusses the relative Kolmogorov n-width of W2(△r) relative to W2(△r) in Lq([-r, πr]d) (1 ≤ q ≤∞), and obtain its weak asymptotic result.     

Asymmetric Binary Covering Codes

An asymmetric binary covering code of length n and radius R is a subset of the n-cube Q_n such that every vector xQ_n can be obtained from some vector c by changing at most R 1's of c to 0's, where R is as small as possible. K⁺(n,R) is defined as the smallest size of such a code. We show K⁺(n,R)Θ(2ⁿ/n^R) for constant R, using an asymmetric sphere-covering bound and probabilistic methods. We show K⁺(n,n− )= +1 for constant coradius iff n ( +1)/2. These two results are extended to near-constant R and , respectively. Various bounds on K⁺ are given in terms of the total number of 0's or 1's in a minimal code. The dimension of a minimal asymmetric linear binary code ([n,R]⁺-code) is determined to be min{0,n−R}. We conclude by discussing open problems and techniques to compute explicit values for K⁺, giving a table of best-known bounds.     

Cyclic Difference Packing and Covering Arrays

Let n and k be positive integers. Let C_q be a cyclic group of order q. A cyclic difference packing (covering) array, or a CDPA(k, n; q) (CDCA(k, n; q)), is a k × n array (a_ij) with entries a_ij (0 ≤ i ≤ k−1, 0 ≤ j ≤ n−1) from C_q such that, for any two rows t and h (0 ≤ t < h ≤ k−1), every element of C_q occurs in the difference list at most (at least) once. When q is even, then n ≤ q−1 if a CDPA(k, n; q) with k ≥ 3 exists, and n ≥ q+1 if a CDCA(k, n; q) with k ≥ 3 exists. It is proved that a CDCA(4, q+1; q) exists for any even positive integers, and so does a CDPA(4, q−1; q) or a CDPA(4, q−2; q). The result is established, for the most part, by means of a result on cyclic difference matrices with one hole, which is of interest in its own right.     

New Linear Codes with Covering Radius 2 and Odd Basis

On the way of generalizing recent results by Cock and the second author, it is shown that when the basis q is odd, BCH codes can be lengthened to obtain new codes with covering radius R=2. These constructions (together with a lengthening construction by the first author) give new infinite families of linear covering codes with codimension r=2k+1 (the case q=3, r=4k+1 was considered earlier). New code families with r=4k are also obtained. An updated table of upper bounds on the length function for linear codes with 24, R=2, and q=3,5 is given.     

An updated table of binary/ternary mixed covering codes

An updated table of K_2,3(b,t;R)—the minimum cardinality of a code with b binary coordinates, t ternary coordinates, and covering radius R—is presented for b + t ≤ 13, R ≤ 3. The results include new explanations of short binary and ternary covering codes, several new constructions and codes, and a general lower bound for R = 1. © 2004 Wiley Periodicals, Inc.     

New Results on Codes with Covering Radius 1 and Minimum Distance 2

The minimal cardinality of a q-ary code of length n and covering radius at most R is denoted by K_q(n, R); if we have the additional requirement that the minimum distance be at least d, it is denoted by K_q(n, R, d). Obviously, K_q(n, R, d) K_q(n, R). In this paper, we study instances for which K_q(n,1,2) > K_q(n, 1) and, in particular, determine K₄(4,1,2)=28 > 24=K₄(4,1).Supported in part by the Academy of Finland under grant 100500.     

Optimal binary covering codes of length 2j

The minimum size of a binary covering code of length n and covering radius r is denoted by K(n,r), and codes of this length are called optimal. For j > 0 and n = 2^j, it is known that K(n,1) = 2 · K(n?1,1) = 2^{n ? j}. Say that two binary words of length n form a duo if the Hamming distance between them is 1 or 2. In this paper, it is shown that each optimal binary covering code of length n = 2^j, j > 0, and covering radius 1 is the union of duos in just one way, and that the closed neighborhoods of the duos form a tiling of the set of binary words of length n. Methods of constructing such optimal codes from optimal covering codes of length n ? 1 (that is, perfect single‐error‐correcting codes) are discussed. The paper ends with the construction of an optimal covering code of length 16 that does not contain an extension of any optimal covering code of length 15. © 2005 Wiley Periodicals, Inc. J Combin Designs     

Nonexistence of [n, 5, d]q Codes Attaining the Griesmer Bound for q4 ?2q2 ?2q+1 ≤ d≤ q4?2q2 ?q

We prove that there does not exist a [q⁴+q³−q²−3q−1, 5, q⁴−2q²−2q+1]_q code over the finite field for q≥ 5. Using this, we prove that there does not exist a [g_q(5, d), 5, d]_q code with q⁴ −2q² −2q +1 ≤ d ≤ q⁴ −2q² −q for q≥ 5, where g_q(k,d) denotes the Griesmer bound.MSC 2000: 94B65, 94B05, 51E20, 05B25     

Some Upper Bounds on the Covering Radii of Linear Codes Over Fq and Their Applications

We show that the covering radius R of an [n,k,d] code over F_q is bounded above by R n-n _q(k, d/q). We strengthen this bound when R d and find conditions under which equality holds.As applications of this and other bounds, we show that all binary linear codes of lengths up to 15, or codimension up to 9, are normal. We also establish the normality of most codes of length 16 and many of codimension 10. These results have applications in the construction of codes that attain t[n,k,/it>], the smallest covering radius of any binary linear [n,k].We also prove some new results on the amalgamated direct sum (ADS) construction of Graham and Sloane. We find new conditions assuring normality of the ADS; covering radius 1 less than previously guaranteed for ADS of codes with even norms; good covering codes as ADS without the hypothesis of normality, from concepts p- stable and s- stable; codes with best known covering radii as ADS of two, often cyclic, codes (thus retaining structure so as to be suitable for practical applications).     

Bounds for short covering codes and reactive tabu search

Given a prime power q, c_q(n,R) denotes the minimum cardinality of a subset H in such that every word in this space differs in at most R coordinates from a multiple of a vector in H. In this work, two new classes of short coverings are established. As an application, a new optimal record-breaking result on the classical covering code is obtained by using short covering. We also reformulate the numbers c_q(n,R) in terms of dominating set on graphs. Departing from this reformulation, the reactive tabu search (a variation of tabu search heuristics) is developed to obtain new upper bounds on c_q(n,R). The algorithm is described and conclusions on the results are drawn; they identify the advantages of using the reactive mechanism for this problem. Tables of lower and upper bounds on c_q(n,R), q=3,4, n≤7, and R≤3, are also presented.     

Linear codes with covering radius 3

The shortest possible length of a q-ary linear code of covering radius R and codimension r is called the length function and is denoted by ℓ _q (r, R). Constructions of codes with covering radius 3 are here developed, which improve best known upper bounds on ℓ _q (r, 3). General constructions are given and upper bounds on ℓ _q (r, 3) for q = 3, 4, 5, 7 and r ≤ 24 are tabulated.     

Average Width and Optimal Interpolation of the Sobolev-Wiener Class Wpq(R) in the Metric Lq(R)

In this paper, we determine the exact value of average n − K width _n(W^r_pq(R), L_q(R)) of Sobolev-Wiener class W^r_pq(R) in the metric L_q(R) for 1 > q ≥ p > ∞ and get the value of _n(W^r_p(R), L_qp(R)) for the dual case. We also solve the optimal interpolation problems of W^r_pq(R) in the metric L_q(R) and W^r_p(R) in the metric L_qp(R) for 1 < q ≤ p < ∞.     

Classification of binary covering codes

The minimum size of a binary covering code of length n and covering radius r is denoted by K (n, r) and corresponding codes are called optimal. In this article a classification up to equivalence of all optimal covering codes having either length at most 8 or cardinality at most 4 is completed. Moreover, we prove that K (9, 2) = 16. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 391–401, 2000     

On the Rate of Approximation of Closed Jordan Curves by Lemniscates

As proved by Hilbert, it is, in principle, possible to construct an arbitrarily close approximation in the Hausdorff metric to an arbitrary closed Jordan curve Γ in the complex plane {z} by lemniscates generated by polynomials P(z). In the present paper, we obtain quantitative upper bounds for the least deviations H _n (Γ) (in this metric) from the curve Γ of the lemniscates generated by polynomials of a given degree n in terms of the moduli of continuity of the conformal mapping of the exterior of Γ onto the exterior of the unit circle, of the mapping inverse to it, and of the Green function with a pole at infinity for the exterior of Γ. For the case in which the curve Γ is analytic, we prove that H _n (Γ) = O(q _n ), 0 ≤ q = q(Γ) < 1, n → ∞.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 861–876.Original Russian Text Copyright ©2005 by O. N. Kosukhin.     

Greedily Finding a Dense Subgraph

Given an n-vertex graph with nonnegative edge weights and a positive integer k ≤ n, our goal is to find a k-vertex subgraph with the maximum weight. We study the following greedy algorithm for this problem: repeatedly remove a vertex with the minimum weighted-degree in the currently remaining graph, until exactly k vertices are left. We derive tight bounds on the worst case approximation ratio R of this greedy algorithm: (1/2 + n/2k)² − O(n ^− 1/3) ≤ R ≤ (1/2 + n/2k)² + O(1/n) for k in the range n/3 ≤ k ≤ n and 2(n/k − 1) − O(1/k) ≤ R ≤ 2(n/k − 1) + O(n/k²) for k < n/3. For k = n/2, for example, these bounds are 9/4 ± O(1/n), improving on naive lower and upper bounds of 2 and 4, respectively. The upper bound for general k compares well with currently the best (and much more complicated) approximation algorithm based on semidefinite programming.     

Constructing Covering Codes with Given Automorphisms

Let K(n,r) denote the minimum cardinality of a binary covering code of length n and covering radius r. Constructions of covering codes give upper bounds on K(n,r). It is here shown how computer searches for covering codes can be sped up by requiring that the code has a given (not necessarily full) automorphism group. Tabu search is used to find orbits of words that lead to a covering. In particular, a code D found which proves K(13,1) 704, a new record. A direct construction of D given, and its full automorphism group is shown to be the general linear group GL(3,3). It is proved that D is a perfect dominating set (each word not in D is covered by exactly one word in D) and is a counterexample to the recent Uniformity Conjecture of Weichsel.     

New upper bounds on the minimum size of covering designs

Let D = {B₁, B₂,…, B_b} be a finite family of k-subsets (called blocks ) of a v-set X(v) = {1, 2,…, v} (with elements called points ). Then D is a (v, k, t) covering design or covering if every t-subset of X(v) is contained in at least one block of D. The number of blocks, b, is the size of the covering, and the minimum size of the covering is called the covering number , denoted C(v, k, t). This article is concerned with new constructions of coverings. The constructions improve many upper bounds on the covering number C(v, k, t) © 1998 John Wiley & Sons, Inc. J Combin Designs 6:21–41, 1998     

On the number of queries necessary to identify a permutation

Let p and q be two permutations over {1, 2,…, n}. We denote by m(p, q) the number of integers i, 1 ≤ i ≤ n, such that p(i) = q(i). For each fixed permutation p, a query is a permutation q of the same size and the answer a(q) to this query is m(p, q). We investigate the problem of finding the minimum number of queries required to identify an unknown permutation p. A polynomial-time algorithm that identifies a permutation of size n by O(n · log₂n) queries is presented. The lower bound of this problem is also considered. It is proved that the problem of determining the size of the search space created by a given set of queries and answers is #P-complete. Since this counting problem is essential for the analysis of the lower bound, a complete analysis of the lower bound appears infeasible. We conjecture, based on some preliminary analysis, that the lower bound is Ω(n · log₂n).     

Decomposable graphs and definitions with no quantifier alternation

Let D(G) be the minimum quantifier depth of a first order sentence Φ that defines a graph G up to isomorphism. Let D₀(G) be the version of D(G) where we do not allow quantifier alternations in Φ. Define q₀(n) to be the minimum of D₀(G) over all graphs G of order n.We prove that for all n we have

log^*n−log^*log^*n−2≤q₀(n)≤log^*n+22,