Constructions of optimal difference systems of sets

Difference systems of sets (DSSs) are combinatorial configurations which were introduced in 1971 by Levenstein for the construction of codes for synchronization. In this paper, we present two kinds of constructions of difference systems of sets by using disjoint difference families and a special type of difference sets, respectively. As a consequence, new infinite classes of optimal DSSs are obtained.     

A generalization of combinatorial designs and related codes

In this paper we study a certain generalization of combinatorial designs related to almost difference sets, namely the t-adesign, which was coined by Ding (Codes from difference sets, 2015). It is clear that 2-adesigns are partially balanced incomplete block designs which naturally arise in many combinatorial and statistical problems. We discuss some of their basic properties and give several constructions of 2-adesigns (some of which correspond to new almost difference sets and some to new almost difference families), as well as two constructions of 3-adesigns. We discuss basic properties of the incidence matrices and make an initial investigation into the codes which they generate. We find that many of the codes have good parameters in the sense they are optimal or have relatively high minimum distance.     

Weighted external difference families and R-optimal AMD codes

In this paper, we provide a mathematical framework for characterizing AMD codes that are R-optimal. We introduce a new combinatorial object, the reciprocally-weighted external difference family (RWEDF), which corresponds precisely to an R-optimal weak AMD code. This definition subsumes known examples of existing optimal codes, and also encompasses combinatorial objects not covered by previous definitions in the literature. By developing structural group-theoretic characterizations, we exhibit infinite families of new RWEDFs, and new construction methods for known objects such as near-complete EDFs. Examples of RWEDFs in non-abelian groups are also discussed.     

New constructions for perfect hash families and related structures using combinatorial designs and codes

In this paper, we consider explicit constructions of perfect hash families using combinatorial methods. We provide several direct constructions from combinatorial structures related to orthogonal arrays. We also simplify and generalize a recursive construction due to Atici, Magliversas, Stinson and Wei [3]. Using similar methods, we also obtain efficient constructions for separating hash families which result in improved existence results for structures such as separating systems, key distribution patterns, group testing algorithms, cover‐free families and secure frameproof codes. © 2000 John Wiley & Sons, Inc. J Combin Designs 8:189–200, 2000     

A new combinatorial approach to the construction of constant composition codes

Constant composition codes(CCCs)are a new generalization of binary constant weight codes and have attracted recent interest due to their numerous applications. In this paper, a new combinatorial approach to the construction of CCCs is proposed, and used to establish new optimal CCCs.     

On constructions for optimal two-dimensional optical orthogonal codes

Two-dimensional optical orthogonal codes (2-D OOCs) are of current practical interest in fiber-optic code-division multiple-access networks as they enable optical communication at lower chip rate to overcome the drawbacks of nonlinear effects in large spreading sequences of one-dimensional codes. A 2-D OOC is said to be optimal if its cardinality is the largest possible. In this paper, we develop some constructions for optimal 2-D OOCs using combinatorial design theory. As an application, these constructions are used to construct an infinite family of new optimal 2-D OOCs with auto-correlation 1 and cross-correlation 1.     

Constructions of partitioned difference families

Partitioned difference families (PDFs) arise from constructions of optimum constant composition codes. In this paper, a number of infinite classes of PDFs are constructed, based on known cyclotomic difference families in GF(q). A general approach which obtains PDFs from difference packings and coverings in abelian group is also presented.     

Constructions of optimal GDRP’s of type

Generalized doubly resolvable packings (GDRPs) represent a combinatorial characterization of constant composition codes (CCCs). In this paper, we develop a number of general constructions of GDRPs of type λ¹μ^m−1. As a consequence, a new series of optimal CCCs is obtained.     

A class of optimal constant composition codes from GDRPs

As a common generalization of constant weight binary codes and permutation codes, constant composition codes (CCCs) have attracted recent interest due to their numerous applications. In this paper, a class of new CCCs are constructed using design-theoretic techniques. The obtained codes are optimal in the sense of their sizes. This result is established, for the most part, by means of a result on generalized doubly resolvable packings which is of combinatorial interest in its own right.      

Frame difference families and resolvable balanced incomplete block designs

Frame difference families, which can be obtained via a careful use of cyclotomic conditions attached to strong difference families, play an important role in direct constructions for resolvable balanced incomplete block designs. We establish asymptotic existences for several classes of frame difference families. As corollaries new infinite families of 1-rotational \((pq+1,p+1,1)\)-RBIBDs over \({\mathbb {F}}_{p}^+ \times {\mathbb {F}}_{q}^+\) are derived, and the existence of \((125q+1,6,1)\)-RBIBDs is discussed. We construct (v, 8, 1)-RBIBDs for \(v\in \{624,\) \(1576,2976,5720,5776,10200,14176,24480\}\), whose existence were previously in doubt. As applications, we establish asymptotic existences for an infinite family of optimal constant composition codes and an infinite family of strictly optimal frequency hopping sequences.     

A Construction of Optimal Constant Composition Codes

In this paper, a construction of optimal constant composition codes is developed, and used to derive some series of new optimal constant composition codes meeting the upper bound given by [13].     

Optimal and perfect difference systems of sets

Difference systems of sets (DSS) were introduced in 1971 by Levenstein for the construction of codes for synchronization, and are closely related to cyclic difference families. In this paper, algebraic constructions of difference systems of sets using functions with optimum nonlinearity are presented. All the difference systems of sets constructed in this paper are perfect and optimal. One conjecture on difference systems of sets is also presented.     

Recursive constructions of secure codes and hash families using difference function families

To protect copyrighted digital data against piracy, codes with different secure properties such as frameproof codes, secure frameproof codes, codes with identifiable parent property (IPP codes), traceability codes (TA codes) are introduced. In this paper, we study these codes together with related combinatorial objects called separating and perfect hash families. We introduce for the first time the notion of difference function families and use these difference function families to give generalized recursive techniques that can be used for any kind of secure codes and hash families. We show that some previous recursive techniques are special cases of these new techniques.     

两类组合设计及其在认证码中的应用

W．Ogata等定义了两种新的组合设计：外差族（EDF）与外平衡不完全区组设计（E-BIBD）．本文首先用有限域中的分圆类给出EDF的一个构造；接着用EBIBD构造出具有完善保密性的最优分裂A-码，然后证明了由满足一定条件的两个EBIBD通过上述方法构造出的两个认证码是同构的.     

Cyclotomic construction of strong external difference families in finite fields

Strong external difference families (SEDFs) and their generalizations GSEDFs and BGSEDFs in a finite abelian group G are combinatorial designs introduced by Paterson and Stinson (Discret Math 339: 2891–2906, 2016) and have applications in communication theory to construct optimal strong algebraic manipulation detection codes. In this paper we firstly present some general constructions of these combinatorial designs by using difference sets and partial difference sets in G. Then, as applications of the general constructions, we construct series of SEDF, GSEDF and BGSEDF in finite fields by using cyclotomic classes. Particularly, we present an \((n,m,k,\lambda )=(243,11,22,20)\)-SEDF in \((\mathbb {F}_q,+)\ (q=3^5=243)\) by using the cyclotomic classes of order 11 in \(\mathbb {F}_q\) which answers an open problem raised in Paterson and Stinson (2016).     

Equidistant frequency permutation arrays and related constant composition codes

In this paper we study the special class of equidistant constant composition codes of type CCC(n, d, μ ^m ) (where n = m μ), which correspond to equidistant frequency permutation arrays; we also consider related codes with composition “close to” μ ^m . We establish various properties of these objects and give constructions for optimal families of codes.     

Constructing difference families through an optimization approach: Six new BIBDs

In this paper we formulate the construction of difference families as a combinatorial optimization problem. A tabu search algorithm is used to find an optimal solution to the optimization problem for various instances of difference families. In particular, we construct six new difference families which lead to an equal number of new balanced incomplete block designs with parameters: (49, 98, 18, 9, 3), (61, 122, 20, 10, 3), (46, 92, 20, 10, 4), (45, 90, 22, 11, 5), (85, 255, 24, 8, 2) and (34, 85, 30, 12, 10). © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 261–273, 2000     

一类新的广义零差分平衡函数及其应用

2008-2009年,丁存生在构造最佳常组合码与优化及完善差分系统中首次引入了零差分平衡(简称ZDB)函数的概念,据此学者们构造出了最佳组成权重码和最优跳频序列.作者将零差分平衡函数的定义推广到一般的广义零差分平衡函数,并利用2分圆陪集构造了一类广义零差分平衡函数,由此构造出一类新的常组合码和差分系统.     

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.     

Constructions of External Difference Families and Disjoint Difference Families

External difference families (EDFs) are a type of new combinatorial designs originated from cryptography. In this paper, some earlier ideas of recursive and cyclotomic constructions of combinatorial designs are extended, and a number of classes of EDFs and disjoint difference families are presented. A link between a subclass of EDFs and a special type of (almost) difference sets is set up.