A construction of resolvable nested 3‐designs

A method of constructing resolvable nested 3‐designs from an affine resolvable 3‐design is proposed with one example. © 2004 Wiley Periodicals, Inc.     

A recursive construction for Room designs

It is shown that if there is a Room design of sidev ₁ and a Room design of sidev ₂ containing a subdesign of sidev ₃, then there exists a design of side v₁ (v₂ — v₃)+v₃, provided thats = v ₂ — v₃ 6. Further, ifs 0, each of the 3 initial designs is isomorphic to a subdesign of the resultant design. It is also shown that there exist Room designs of sidev for all Fermat primesv > 65537.     

A recursive construction of asymmetric 1-factorizations

W. D. Wallis has recently shown that forn 4, the complete graph on 2n points (denotedK _2n ) has two non-isomorphic 1-factorizations. In this paper we prove that forn 5,K _2n has a 1-factorization with no symmetry at all and that as n increases without bound, the number of pairwise non-isomorphic asymmetric 1-factorizations ofK _2n also increases without bound.The work of B. A. Anderson was partially supported by an Arizona State University Summer Faculty Fellowship.     

A general recursive construction for quadruple systems

A Steiner-quadruple system of order υ is an ordered pair (X, Q), where X is a set of cardinality υ, and Q is a set of 4-subsets of X, called blocks, with the property that every 3-subset of X is contained in a unique block. In this paper we show that if there exists a quadruple system of order V with a subsystem of order υ, then there exists a quadruple system of order 3V ? 2υ with subsystems of orders V and υ. Hanani has given a proof of this result for υ = 1, and in a previous paper, the author has proved the case when V ≡ 2υ(mod 6). The construction given here proves all remaining cases, and has many applications to other existence problems for 3-designs.     

A recursive construction of nonbinary de Bruijn sequences

This paper presents a method to find new de Bruijn sequences based on ones of lesser order. This is done by mapping a de Bruijn cycle to several vertex disjoint cycles in a de Bruijn digraph of higher order and then connecting these cycles into one full cycle. We present precise formulae for the locations where those cycles can be rejoined into one full cycle. We obtain an exponentially large class of distinct de Bruijn cycles. This method generalizes the Lempel construction of binary de Bruijn sequences as well as its efficient implementation by Annextein.     

A recursive method for computing interpolants

In this paper, we describe a recursive method for computing interpolants defined in a space spanned by a finite number of continuous functions in R^d${\mathbb{R}}^d$. We apply this method to construct several interpolants such as spline interpolants, tensor product interpolants and multivariate polynomial interpolants. We also give a simple algorithm for solving a multivariate polynomial interpolation problem and constructing the minimal interpolation space for a given finite set of interpolation points.     

A recursive construction of 1-rotational Steiner 2-designs

A Steiner 2-design S(2,k, v) is said to be 1-rotational if it admits an automorphism whose cycle structure is a (v ? 1)-cycle and a fixed point. In this paper, a recursive construction of 1-rotational Steiner 2-designs is given.     

A recursive construction for universal cycles of 2-subspaces

We prove that universal cycles of 2-dimensional subspaces of vector spaces over any finite field F exist, i.e., if V is a finite-dimensional vector space over F, there is a cycle of vectors v₁,v₂,…,v_n such that each 2-dimensional subspace of V occurs exactly once as the span of consecutive vectors.     

Random recursive construction of Salem sets

We introduce a random recursive method for constructing random Salem sets inR ^d. The method is inspired by Salem's construction [13] of certain signular monotonic functions. This work contains parts of the author's forthcoming doctoral thesis [2] which were presented at theConference on Harmonic Analysis from the Pichorides Viewpoint in Anogia Academic Village on Crete in July 1995.     

A recursive construction of t‐wise uniform permutations

We present a recursive construction of a (2t + 1)‐wise uniform set of permutations on 2n objects using a combinatorial design, a t‐wise uniform set of permutations on n objects and a (2t + 1)‐wise uniform set of permutations on n objects. Using the complete design in this procedure gives a t‐wise uniform set of permutations on n objects whose size is at most t²ⁿ, the first non‐trivial construction of an infinite family of t‐wise uniform sets for . If a non‐trivial design with suitable parameters is found, it will imply a corresponding improvement in the construction. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 531–540, 2015     

A degree by degree recursive construction of Hermite spline interpolants

Based on the classical Hermite spline interpolant _H2n−1$H_{2n-1}$, which is the piecewise interpolation polynomial of class Cⁿ⁻¹$C^{n-1}$ and degree 2n−1$2n-1$, a piecewise interpolation polynomial _H2n$H_{2n}$ of degree 2n$2n$ is given. The formulas for computing _H2n$H_{2n}$ by _H2n−1$H_{2n-1}$ and computing _H2n+1$H_{2n+1}$ by _H2n$H_{2n}$ are shown. Thus a simple recursive method for the construction of the piecewise interpolation polynomial set {_Hj}$\left\{H_j\right\}$ is presented. The piecewise interpolation polynomial _H2n$H_{2n}$ satisfies the same interpolation conditions as the interpolant _H2n−1$H_{2n-1}$, and is an optimal approximation of the interpolant _H2n+1$H_{2n+1}$. Some interesting properties are also proved.     

A characterization of almost resolvable spaces

A space is said to be resolvable if it has two disjoint dense subsets. It is shown thatX is a Baire space with no resolvable open subsets iff every real function defined onX has a dense set of points of continuity. Thus almost resolvable spaces, as defined by Bolstein, are shown to be characterized as the union of a first category set and a closed resolvable set.     

A class of explicitly resolvable evolution equations

We consider a class of evolution equations with “scalar nonlinearities” and the associated steady equations. An explicit representation of solutions is obtained in terms of the solution of a scalar nonlinear functional differential equation. Convergence to an equilibrium solution is discussed.     

On the direct construction of recursive MDS matrices

MDS matrices allow to build optimal linear diffusion layers in the design of block ciphers and hash functions. There has been a lot of study in designing efficient MDS matrices suitable for software and/or hardware implementations. In particular recursive MDS matrices are considered for resource constrained environments. Such matrices can be expressed as a power of simple companion matrices, i.e., an MDS matrix \(M = C_g^k\) for some companion matrix corresponding to a monic polynomial \(g(X) \in \mathbb {F}_q[X]\) of degree k. In this paper, we first show that for a monic polynomial g(X) of degree \(k\ge 2\), the matrix \(M = C_g^k\) is MDS if and only if g(X) has no nonzero multiple of degree \(\le 2k-1\) and weight \(\le k\). This characterization answers the issues raised by Augot et al. in FSE-2014 paper to some extent. We then revisit the algorithm given by Augot et al. to find all recursive MDS matrices that can be obtained from a class of BCH codes (which are also MDS) and propose an improved algorithm. We identify exactly what candidates in this class of BCH codes yield recursive MDS matrices. So the computation can be confined to only those potential candidate polynomials, and thus greatly reducing the complexity. As a consequence we are able to provide formulae for the number of such recursive MDS matrices, whereas in FSE-2014 paper, the same numbers are provided by exhaustively searching for some small parameter choices. We also present a few ideas making the search faster for finding efficient recursive MDS matrices in this class. Using our approach, it is possible to exhaustively search this class for larger parameter choices which was not possible earlier. We also present our search results for the case \(k=8\) and \(q=2^{16}\).     

On the construction of a nonlinear recursive predictor

In this paper, we present a novel approach for constructing a nonlinear recursive predictor. Given a limited time series data set, our goal is to develop a predictor that is capable of providing reliable long-term forecasting. The approach is based on the use of an artificial neural network and we propose a combination of network architecture, training algorithm, and special procedures for scaling and initializing the weight coefficients. For time series arising from nonlinear dynamical systems, the power of the proposed predictor has been successfully demonstrated by testing on data sets obtained from numerical simulations and actual experiments.     

Maximal resolvable packings and minimal resolvable coverings of triples by quadruples

Determination of maximal resolvable packing number and minimal resolvable covering number is a fundamental problem in designs theory. In this article, we investigate the existence of maximal resolvable packings of triples by quadruples of order v (MRPQS(v)) and minimal resolvable coverings of triples by quadruples of order v (MRCQS(v)). We show that an MRPQS(v) (MRCQS(v)) with the number of blocks meeting the upper (lower) bound exists if and only if v≡0 (mod 4). As a byproduct, we also show that a uniformly resolvable Steiner system URS(3, {4, 6}, {r₄, r₆}, v) with r₆≤1 exists if and only if v≡0 (mod 4). All of these results are obtained by the approach of establishing a new existence result on RH(6²ⁿ) for all n≥2. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 209–223, 2010     

Equitable resolvable coverings

In an earlier article, Willem H. Haemers has determined the minimum number of parallel classes in a resolvable 2‐(qk,k,1) covering for all k ≥ 2 and q = 2 or 3. Here, we complete the case q = 4, by construction of the desired coverings using the method of simulated annealing. Secondly, we look at equitable resolvable 2‐(qk,k,1) coverings. These are resolvable coverings which have the additional property that every pair of points is covered at most twice. We show that these coverings satisfy k < 2q ? , and we give several examples. In one of these examples, k > q. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 113–123, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10024