Constructions of optimal variable‐weight optical orthogonal codes

Variable‐weight optical orthogonal code (OOC) was introduced by G‐C Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirement. In this article, new infinite classes of optimal (u, W, 1, {1/2, 1/2})‐OOCs are obtained for W={3, 4}, {3, 5} and {3, 6}. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 274–291, 2010     

On Balanced Optimal Optical Orthogonal Codes

Variable‐weight optical orthogonal codes (OOCs) were introduced by G.‐C. Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirement. In this paper, by using incomplete difference matrices and perfect relative difference families, a balanced ‐OOC is obtained for every positive integer .     

最优(24u,{3,4},1,{2/3,1/3})光正交码

Variable-weight optical orthogonal codes(OOCs) were introduced by G. C. YANG for multimedia optical CDMA systems with multiple quality of service(Qo S) requirements. In this paper, some infinite classes of optimal cyclic packing are presented. Optimal(24u, {3, 4}, 1,{2/3, 1/3})-OOCs for any positive integer u > 1 are established.     

Constructions of optimal variable-weight OOCs via quadratic residues

Variable-weight optical orthogonal code (OOC) was introduced by G. C. Yang [IEEE Trans. Commun., 1996, 44: 47–55] for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. In this paper, seven new infinite classes of optimal (v, {3, 4, 6}, 1,Q)-OOCs are constructed.     

一种OLS/OOC二维光正交码的构造与性能分析

利用不同的序列作为波长跳频序列和时间扩频序列可以构造出不同的二维光正交码在众多文献中已有所报道.在经过正交拉丁方(OLS)与跳频序列的相关性研究之后.做了以下主要工作:首先,将正交拉丁方(OLS)序列作为波长跳频序列,结合一维时间扩频序列(OOC),构造了一种OLS/OOC二维光正交码.然后,本文对构造的OLS/OOC进行了多种性能仿真和分析.相对于PC/OOC、OCFHC/OOC等二维光正交码而言,OLS/OOC的波长数并不局限于素数,更能充分利用MWOCDMA系统中的有效波长数.仿真和分析表明:码字具有很好的相关性能,码字容量直逼理论极限,为一种渐近最优二维光正交码.     

Determination of Sizes of Optimal Three‐Dimensional Optical Orthogonal Codes of Weight Three with the AM‐OPP Restriction

In this paper, we further investigate the constructions on three‐dimensional optical orthogonal codes with the at most one optical pulse per wavelength/time plane restriction (briefly AM‐OPP 3D ‐OOCs) by way of the corresponding designs. Several new auxiliary designs such as incomplete holey group divisible designs and incomplete group divisible packings are introduced and therefore new constructions are presented. As a consequence, the exact number of codewords of an optimal AM‐OPP 3D ‐OOC is finally determined for any positive integers and .     

Recursive constructions for optimal (n,4,2)‐OOCs

In [ 3 ], a general recursive construction for optical orthogonal codes is presented, that guarantees to approach the optimum asymptotically if the original families are asymptotically optimal. A challenging problem on OOCs is to obtain optimal OOCs, in particular with λ > 1. Recently we developed an algorithmic scheme based on the maximal clique problem (MCP) to search for optimal (n, 4, 2)‐OOCs for orders up to n = 44. In this paper, we concentrate on recursive constructions for optimal (n, 4, 2)‐OOCs. While “most” of the codewords can be constructed by general recursive techniques, there remains a gap in general between this and the optimal OOC. In some cases, this gap can be closed, giving recursive constructions for optimal (n, 4, 2)‐OOCs. This is predicated on reducing a series of recursive constructions for optimal (n, 4, 2)‐OOCs to a single, finite maximal clique problem. By solving these finite MCP problems, we can extend the general recursive construction for OOCs in [ 3 ] to obtain new recursive constructions that give an optimal (n · 2^x, 4, 2)‐OOC with x ≥ 3, if there exists a CSQS(n). © 2004 Wiley Periodicals, Inc.     

Bounds and Constructions on (v, 4, 3, 2) Optical Orthogonal Codes

In this paper, we are concerned about optimal (v, 4, 3, 2)‐OOCs. A tight upper bound on the exact number of codewords of optimal (v, 4, 3, 2)‐OOCs and some direct and recursive constructions of optimal (v, 4, 3, 2)‐OOCs are given. As a result, the exact number of codewords of an optimal (v, 4, 3, 2)‐OOC is determined for some infinite series.     

Optimal (4up, 5, 1) optical orthogonal codes

By a (ν, k, 1)‐OOC we mean an optical orthogonal code. In this paper, it is proved that an optimal (4p, 5, 1)‐OOC exists for prime p ≡ 1 (mod 10), and that an optimal (4up, 5, 1)‐OOC exists for u = 2, 3 and prime p ≡ 11 (mod 20). These results are obtained by applying Weil's theorem. © 2004 Wiley Periodicals, Inc.     

Optimal Two‐Dimensional Optical Orthogonal Codes with the Best Cross‐Correlation Constraint

The study of optical orthogonal codes has been motivated by an application in an optical code‐division multiple access system. From a practical point of view, compared to one‐dimensional optical orthogonal codes, two‐dimensional optical orthogonal codes tend to require smaller code length. On the other hand, in some circumstances only with good cross‐correlation one can deal with both synchronization and user identification. These motivate the study of two‐dimensional optical orthogonal codes with better cross‐correlation than auto‐correlation. This paper focuses on optimal two‐dimensional optical orthogonal codes with the auto‐correlation and the best cross‐correlation 1. By examining the structures of w‐cyclic group divisible designs and semi‐cyclic incomplete holey group divisible designs, we present new combinatorial constructions for two‐dimensional ‐optical orthogonal codes. When and , the exact number of codewords of an optimal two‐dimensional ‐optical orthogonal code is determined for any positive integers n and .     

A quasi‐symmetric 2‐(49,9,6) design

Huffman and Tonchev discovered four non‐isomorphic quasi‐symmetric 2‐(49,9,6) designs. They arise from extremal self‐dual [50,25,10] codes with a certain weight enumerator. In this note, a new quasi‐symmetric 2‐(49,9,6) design is constructed. This is established by finding a new extremal self‐dual [50,25,10] code as a neighbor of one of the four extremal codes discovered by Huffman and Tonchev. A number of new extremal self‐dual [50,25,10] codes with other weight enumerators are also found. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 173–179, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10007     

Constructions of Strictly m‐Cyclic and Semi‐Cyclic

An is a triple , where X is a set of points, is a partition of X into m disjoint sets of size n and is a set of 4‐element transverses of , such that each 3‐element transverse of is contained in exactly one of them. If the full automorphism group of an admits an automorphism α consisting of n cycles of length m (resp. m cycles of length n), then this is called m‐cyclic (resp. semi‐cyclic). Further, if all block‐orbits of an m‐cyclic (resp. semi‐cyclic) are full, then it is called strictly cyclic. In this paper, we construct some infinite classes of strictly m‐cyclic and semi‐cyclic , and use them to give new infinite classes of perfect two‐dimensional optical orthogonal codes with maximum collision parameter and AM‐OPPTS/AM‐OPPW property.     

Spreads, arcs, and multiple wavelength codes

We present several new families of multiple wavelength (2-dimensional) optical orthogonal codes (2D-OOCs) with ideal auto-correlation λ_a=0 (codes with at most one pulse per wavelength). We also provide a construction which yields multiple weight codes. All of our constructions produce codes that are either optimal with respect to the Johnson bound (J-optimal), or are asymptotically optimal and maximal. The constructions are based on certain pointsets in finite projective spaces of dimension k over GF(q) denoted PG(k,q).     

2-dimensional optical orthogonal codes from singer groups

We present several new families of (Λ×T,w,λ) (2D) wavelength/time optical orthogonal codes (2D-OOCs) with λ=1,2. All families presented are either optimal with respect to the Johnson bound (J-optimal) or are asymptotically optimal. The codes presented have more flexible dimensions and weight than the J-optimal families appearing in the literature. The constructions are based on certain pointsets in finite projective spaces of dimension k over GF(q) denoted PG(k,q). This finite geometries framework gives structure to the codes providing insight. We establish that all 2D-OOCs constructed are in fact maximal (in that no new codeword may be added to the original whereby code cardinality is increased).     

Bayesian optimal life‐testing plan under the balanced two sample type‐II progressive censoring scheme

Joint progressive censoring schemes are quite useful to conduct comparative life‐testing experiment of different competing products. Recently, Mondal and Kundu (“A New Two Sample Type‐II Progressive Censoring Scheme,” Commun Stat‐Theory Methods; 2018) introduced a joint progressive censoring scheme on two samples known as the balanced joint progressive censoring (BJPC) scheme. Optimal planning of such progressive censoring scheme is an important issue to the experimenter. This article considers optimal life‐testing plan under the BJPC scheme using the Bayesian precision and D‐optimality criteria, assuming that the lifetimes follow Weibull distribution. In order to obtain the optimal BJPC life‐testing plans, one needs to carry out an exhaustive search within the set of all admissible plans under the BJPC scheme. However, for large sample size, determination of the optimal life‐testing plan is difficult by exhaustive search technique. A metaheuristic algorithm based on the variable neighborhood search method is employed for computation of the optimal life‐testing plan. Optimal plans are provided under different scenarios. The optimal plans depend upon the values of the hyperparameters of the prior distribution. The effect of different prior information on optimal scheme is studied.     

Bounds on the Dimensions of 2‐Spontaneous Emission Error Designs

Quantum jump codes are quantum error‐correcting codes which correct errors caused by quantum jumps. A t‐spontaneous emission error design (t‐SEED) was introduced by Beth et al. in 2003 [T. Beth, C. Charnes, M. Grassl, G. Alber, A. Delgado, and M. Mussinger, A new class of designs which protect against quantum jumps, Des Codes Cryptogr 29 (2003), 51–70.] to construct quantum jump codes. The number of designs (dimension) in a t‐SEED corresponds to the number of orthogonal basis states in a quantum jump code. A nondegenerate t‐SEED is optimal if it has the largest possible dimension. In this paper, we investigate the bounds on the dimensions of 2‐SEEDs systematically. The exact dimensions of optimal 2‐ SEEDs are almost determined, with five possible exceptions in doubt. General upper bounds on dimensions of 2‐ SEEDs are demonstrated, the corresponding leave graphs are described, and several exceptional cases are studied in details. Meanwhile, we employ 2‐homogenous groups to obtain new lower bounds on the dimensions of 2‐ SEEDs for prime power orders v and general block sizes k.     

Self‐Dual Codes and the Nonexistence of a Quasi‐Symmetric 2‐(37,9,8) Design with Intersection Numbers 1 and 3

We prove that a certain binary linear code associated with the incidence matrix of a quasi‐symmetric 2‐(37, 9, 8) design with intersection numbers 1 and 3 must be contained in an extremal doubly even self‐dual code of length 40. Using the classification of extremal doubly even self‐dual codes of length 40, we show that a quasi‐symmetric 2‐(37, 9, 8) design with intersection numbers 1 and 3 does not exist.     

An improved product construction of rotational Steiner quadruple systems

An improved product construction is presented for rotational Steiner quadruple systems. Direct constructions are also provided for small orders. It is known that the existence of a rotational Steiner quadruple system of order υ+1 implies the existence of an optimal optical orthogonal code of length υ with weight four and index two. New infinite families of orders are also obtained for both rotational Steiner quadruple systems and optimal optical orthogonal codes. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 433–443, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10025     

Analytic solutions of a (2 + 1)‐dimensional variable‐coefficient Broer–Kaup system

Based on a Riccati equation and one of its new generalized solitary solutions constructed by the Exp‐function method, new analytic solutions with free parameters and arbitrary functions of a (2 + 1)‐dimensional variable‐coefficient Broer–Kaup system are obtained. These free parameters and arbitrary functions reveal that the (2 + 1)‐dimensional variable‐coefficient Broer–Kaup system has rich spatial structures. As an illustrative example, two new spatial structures are shown by setting the arbitrary functions as different Jacobi elliptic functions. Compared with tanh‐function method and its extensions, the method proposed in this paper is more powerful and it can be applied to other nonlinear evolution equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Unified multipliers‐free theory of dual‐primal domain decomposition methods

The concept of dual‐primal methods can be formulated in a manner that incorporates, as a subclass, the non preconditioned case. Using such a generalized concept, in this article without recourse to “Lagrange multipliers,” we introduce an all‐inclusive unified theory of nonoverlapping domain decomposition methods (DDMs). One‐level methods, such as Schur‐complement and one‐level FETI, as well as two‐level methods, such as Neumann‐Neumann and preconditioned FETI, are incorporated in a unified manner. Different choices of the dual subspaces yield the different dual‐primal preconditioners reported in the literature. In this unified theory, the procedures are carried out directly on the matrices, independently of the differential equations that originated them. This feature reduces considerably the code‐development effort required for their implementation and permit, for example, transforming 2D codes into 3D codes easily. Another source of this simplification is the introduction of two projection‐matrices, generalizations of the average and jump of a function, which possess superior computational properties. In particular, on the basis of numerical results reported there, we claim that our jump matrix is the optimal choice of the B operator of the FETI methods. A new formula for the Steklov‐Poincaré operator, at the discrete level, is also introduced. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009