Bounds and Constructions for Two‐Dimensional Variable‐Weight Optical Orthogonal Codes

Optical orthogonal codes (1D constant‐weight OOCs or 1D CWOOCs) were first introduced by Salehi as signature sequences to facilitate multiple access in optical fibre networks. In fiber optic communications, a principal drawback of 1D CWOOCs is that large bandwidth expansion is required if a big number of codewords is needed. To overcome this problem, a two‐dimensional (2D) (constant‐weight) coding was introduced. Many optimal 2D CWOOCs were obtained recently. A 2D CWOOC can only support a single QoS (quality of service) class. A 2D variable‐weight OOC (2D VWOOC) was introduced to meet multiple QoS requirements. A 2D VWOOC is a set of 0, 1 matrices with variable weight, good auto, and cross‐correlations. Little is known on the construction of optimal 2D VWOOCs. In this paper, new upper bound on the size of a 2D VWOOC is obtained, and several new infinite classes of optimal 2D VWOOCs are obtained.     

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     

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.     

严格循环的算子代数

设X是半自共轭的Banach空间,而A是B(X)的严格循环的交换子代数,考虑算子代数A的一些性质,这些性质与算子代数的自反性有关,而且这些性质为讨论A的自反性提供了理论基础.     

Semi–cyclic Holey Group Divisible Designs and Applications to Sampling Designs and Optical Orthogonal Codes

We consider the existence problem for a semi‐cyclic holey group divisible design of type with block size 3, which is denoted by a 3‐SCHGDD of type . When t is odd and or t is doubly even and , the existence problem is completely solved; when t is singly even, many infinite families are obtained. Applications of our results to two‐dimensional balanced sampling plans and optimal two‐dimensional optical orthogonal codes are also discussed.     

Constructions for cyclic 3‐designs and improved results on cyclic Steiner quadruple systems

This paper gives some recursive constructions for cyclic 3‐designs. Using these constructions we improve Grannell and Griggs's construction for cyclic Steiner quadruple systems, and many known recursive constructions for cyclic Steiner quadruple systems are unified. Finally, some new infinite families of cyclic Steiner quadruple systems are obtained. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:178‐201, 2011     

Cyclic Designs with Block Size 4 and Related Optimal Optical Orthogonal Codes

We prove the existence of a cyclic (4p, 4, 1)-BIBD—and hence, equivalently, that of a cyclic (4, 1)-GDD of type 4 ^p —for any prime such that (p–1)/6 has a prime factor q not greater than 19. This was known only for q=2, i.e., for . In this case an explicit construction was given for . Here, such an explicit construction is also realized for .We also give a strong indication about the existence of a cyclic (4p 4, 1)-BIBD for any prime , p>7. The existence is guaranteed for p>(2q ³–3q ²+1)²+3q ² where q is the least prime factor of (p–1)/6.Finally, we prove, giving explicit constructions, the existence of a cyclic (4, 1)-GDD of type 6 ^p for any prime p>5 and the existence of a cyclic (4, 1)-GDD of type 8 ^p for any prime . The result on GDD's with group size 6 was already known but our proof is new and very easy.All the above results may be translated in terms of optimal optical orthogonal codes of weight four with =1.     

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 .     

Cyclic m‐cycle systems with m ≤ 32 or m = 2q with q a prime power

In this paper, the necessary and sufficient conditions for the existence of cyclic 2q‐cycle and m‐cycle systems of the complete graph with q a prime power and 3 ≤ m ≤ 32 are given. © 2005 Wiley Periodicals, Inc. J Combin Designs     

最优(v,{3,4},1,Q)-光正交码的构造

在光纤码分多址(OCDMA)系统中,变重量光正交码被广泛使用,以满足多种服务质量的需求.利用分圆类和斜starter给出了直接构造方法,借助有关循环差阵的递归构造方法,从而构造了两类循环填充设计.通过建立循环填充设计与变重量光正交码之间的联系,证明了当Q∈{{2/3,1/3},{3/4,1/4}}时,最优(v,{3,4},1,Q)-光正交码存在的无穷类.     

Constructions of optimal optical orthogonal codes with weight five

Several direct constructions via skew starters and Weil's theorem on character sum estimates are given in this paper for optimal (gv, 5, 1) optical orthogonal codes (OOCs) where 60 ≤ g ≤ 180 satisfying g ≡ 0 (mod 20) and v is a product of primes greater than 5. These improve the known existence results on optimal OOCs. Especially, we provide an optimal (v, 5, 1)‐OOC for any integer v ≡ 60, 420, 660, 780, 1020, 1140, 1380, 1740 (mod 1800). © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 54–69, 2005.     

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.     

码重为3和7的最优光正交码的具体构造

对光正交码(OOC)构造的关注源于它在光码分多址网络中有许多应用.截至目前,对于码重为W∈{{3,4},{3,5},{3,6},{4,5},{4,6]}的变重量光正交码的构造已经取得许多结果.然而,对于码重为W={3,7}的变重量光正交码的具体构造非常的少.给出一系列新的最优变重量光正交码(33p,{3,7},1,{4/5,1/5})-OOC的具体构造,对于任何素数p≡3(mod 4)且p≥7.     

Cyclic 4‐Colorings of Graphs on Surfaces

To attack the Four Color Problem, in 1880, Tait gave a necessary and sufficient condition for plane triangulations to have a proper 4‐vertex‐coloring: a plane triangulation G has a proper 4‐vertex‐coloring if and only if the dual of G has a proper 3‐edge‐coloring. A cyclic coloring of a map G on a surface F² is a vertex‐coloring of G such that any two vertices x and y receive different colors if x and y are incident with a common face of G. In this article, we extend the result by Tait to two directions, that is, considering maps on a nonspherical surface and cyclic 4‐colorings.     

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.     

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.     

Infinite families of 2‐designs from a class of cyclic codes

Combinatorial $t$‐designs have wide applications in coding theory, cryptography, communications, and statistics. It is well known that the supports of all codewords with a fixed weight in a code may give a $t$‐design. In this paper, we first determine the weight distributions of a class of linear codes derived from the dual of some extended cyclic codes. We then obtain infinite families of 2‐designs and explicitly compute their parameters from the supports of all the codewords with a fixed weight in the codes. By a simple counting argument, we obtain exponentially many 2‐designs.     

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 .     

A class of group divisible 3‐designs and their applications

In this article, we first show that a group divisible 3‐design with block sizes from {4, 6}, index unity and group‐type 2^m exists for every integer m≥ 4 with the exception of m = 5. Such group divisible 3‐designs play an important role in our subsequent complete solution to the existence problem for directed H‐designs DH_λ(m, r, 4, 3)s. We also consider a way to construct optimal codes capable of correcting one deletion or insertion using the directed H‐designs. In this way, the optimal single‐deletion/insertion‐correcting codes of length 4 can be constructed for all even alphabet sizes. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 136–146, 2009     

Constructions for rotational Steiner quadruple systems

A direct construction for rotational Steiner quadruple systems of order p+ 1 having a nontrivial multiplier automorphism is presented, where p≡13 (mod24) is a prime. We also give two improved product constructions. By these constructions, the known existence results of rotational Steiner quadruple systems are extended. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 353–368, 2009