APPLICATION OF MINIMUM PROJECTION UNIFORMITY CRITERION IN COMPLEMENTARY DESIGNS

In this article, we consider the characterization problem in design theory. The objective is to characterize minimum projection uniformity for two-level designs in terms of their complementary designs. Here, the complementary design means a design in which all the Hamming distances of any two runs are the same, which generalizes the concept of a pair of complementary designs in the literature. Based on relationships of the uniformity pattern between a pair of complementary designs, we propose a minimum projection uniformity （MPU） rule to assess and compare two-level factorials.     

Some properties of double designs in terms of lee discrepancy

Doubling is a simple but powerful method of constructing two-level fractional factorial designs with high resolution. This article studies uniformity in terms of Lee discrepancy of double designs. We give some linkages between the uniformity of double design and the aberration case of the original one under different criteria. Furthermore, some analytic linkages between the generalized wordlength pattern of double design and that of the original one are firstly provided here, which extend the existing findings. The lower bound of Lee discrepancy for double designs is also given.     

Uniform supersaturated design and its construction

Supersaturated designs are factorial designs in which the number of main effects is greater than the number of experimental runs. In this paper, a discrete discrepancy is proposed as a measure of uniformity for supersaturated designs, and a lower bound of this discrepancy is obtained as a benchmark of design uniformity. A construction method for uniform supersaturated designs via resolvable balanced incomplete block designs is also presented along with the investigation of properties of the resulting designs. The construction method shows a strong link between these two different kinds of designs     

A new class of Latin hypercube designs with high-dimensional hidden projective uniformity

Latin hypercube design is a good choice for computer experiments. In this paper, we construct a new class of Latin hypercube designs with some high-dimensional hidden projective uniformity. The construction is based on a new class of orthogonal arrays of strength two which contain higher strength orthogonal arrays after their levels are collapsed. As a result, the obtained Latin hypercube designs achieve higher-dimensional uniformity when projected onto the columns corresponding to higher strength orthogonal arrays, as well as twodimensional projective uniformity. Simulation study shows that the constructed Latin hypercube designs are significantly superior to the currently used designs in terms of the times of correctly identifying the significant effects.     

Uniformity pattern and related criteria for two-level factorials

In this paper,the study of projection properties of two-level factorials in view ofgeometry is reported.The concept of uniformity pattern is defined.Based on this new con-cept,criteria of uniformity resolution and minimum projection uniformity are proposed forcomparing two-level factorials.Relationship between minimum projection uniformity andother criteria such as minimum aberration,generalized minimum aberration and orthogo-nality is made explict.This close relationship raises the hope of improving the connectionbetween uniform design theory and factorial design theory.Our results provide a justifi-cation of orthogonality,minimum aberration,and generalized minimum aberration from anatural geometrical interpretation.     

正交设计的最新发展和应用(Ⅱ)—均匀正交设计

方开泰等．正交设计的最新发展和应用（Ⅱ）—均匀正交设计．令Ｌ（ｎ;ｑｓ）为一切正交表Ｌｎ（ｑｓ）之集合,Ｍ为试验点分布于试验区域的均匀性测度。给定（ｎ,ｑ,ｓ）,在Ｌ（ｎ;ｑｓ）上具有最好均匀性（在测度ＭＦ）的设计称为无匀正交设计,并表为ＵＬｎ（ｑｓ）。本讲座以ＵＬ９（３４）为例说明均匀正交设计在估计和混杂方面的优良性质。在附录中列出了七个均匀正交表,它们都是最近获得的     

正交设计的最新发展和应用(Ⅱ)-均匀正交设计

方开泰等．正交设计的最新发展和应用（Ⅱ）—均匀正交设计．令Ｌ（ｎ;ｑｓ）为一切正交表Ｌｎ（ｑｓ）之集合,Ｍ为试验点分布于试验区域的均匀性测度。给定（ｎ,ｑ,ｓ）,在Ｌ（ｎ;ｑｓ）上具有最好均匀性（在测度ＭＦ）的设计称为无匀正交设计,并表为ＵＬｎ（ｑｓ）。本讲座以ＵＬ９（３４）为例说明均匀正交设计在估计和混杂方面的优良性质。在附录中列出了七个均匀正交表,它们都是最近获得的     

A note on lower bound of centered L2-discrepancy on combined designs

This note provides a theoretical justification of optimal foldover plans in terms of uniformity. A new lower bound of the centered L ₂-discrepancy values of combined designs is obtained, which can be used as a benchmark for searching optimal foldover plans. Our numerical results show that this lower bound is sharper than existing results when more factors reverse the signs in the initial design.     

Uniformity in double designs

In constructing two-level fractional factorial designs, the so-called doubling method has been employed. In this paper, we study the problem of uniformity in double designs. The centered L2-discrepancy is employed as a measure of uniformity. We derive results connecting the centered L2-discrepancy value of D（X） and generalized wordlength pattern of X, which show the uniformity relationship between D（X） and X. In addition, we also obtain lower bounds of centered L2-discrepancy value of D（X）, which can be used to assess uniformity of D（X）.     

On the boomerang uniformity of permutations of low Carlitz rank

Finding permutation polynomials with low differential and boomerang uniformity is an important topic in S-box designs of many block ciphers. For example, AES chooses the inverse function as its S-box, which is differentially 4-uniform and boomerang 6-uniform. Also there has been considerable research on many non-quadratic permutations which are modifications of the inverse function. In this paper, we give a novel approach which shows that plenty of existing modifications of the inverse function are in fact affine equivalent to permutations of low Carlitz rank, and those modifications cannot be APN. We also present the complete list of permutations of Carlitz rank 3 having the boomerang uniformity six, and give the complete classification of the differential uniformities of permutations of Carlitz rank 3. As an application, we provide all the involutions of Carlitz rank 3 having the boomerang uniformity six.     

An algorithmic approach to finding factorial designs with generalized minimum aberration

Factorial designs are arguably the most widely used designs in scientific investigations. Generalized minimum aberration (GMA) and uniformity are two important criteria for evaluating both regular and non-regular designs. The generation of GMA designs is a non-trivial problem due to the sequential optimization nature of the criterion. Based on an analytical expression between the generalized wordlength pattern and a uniformity measure, this paper converts the generation of GMA designs to a constrained optimization problem, and provides effective algorithms for solving this particular problem. Moreover, many new designs with GMA or near-GMA are reported, which are also (nearly) optimal under the uniformity measure.     

混水平均匀设计的构造

我们用离散偏差来度量部分因子设计的均匀性,本文的目的在于寻找一些构造混水平均匀设计的方法,这些方法比文献中已有的方法更简单且计算成本更低.我们得到了离散偏差的一个下界,如果一个U 型设计的离散偏差值达到这个下界,那么该设计是—个均匀设计.我们建立了均匀设计与组合设计理论中一致可分解设计之间的联系.通过一致可分解设计,我们提出了一些构造均匀设计的新方法,同时也给出了许多均匀设计存在的无穷类.     

Minimum secondary aberration fractional factorial split-plot designs in terms of consulting designs

It is very powerful for constructing nearly saturated factorial designs to characterize fractional factorial (FF) designs through their consulting designs when the consulting designs are small. Mukerjee and Fang employed the projective geometry theory to find the secondary wordlength pattern of a regular symmetrical fractional factorial split-plot (FFSP) design in terms of its complementary subset, but not in a unified form. In this paper, based on the connection between factorial design theory and coding theory, we obtain some general and unified combinatorial identities that relate the secondary wordlength pattern of a regular symmetrical or mixed-level FFSP design to that of its consulting design. According to these identities, we further establish some general and unified rules for identifying minimum secondary aberration, symmetrical or mixed-level, FFSP designs through their consulting designs.     

Indexing text with approximate q-grams

We present a new index for approximate string matching. The index collects text q-samples, that is, disjoint text substrings of length q, at fixed intervals and stores their positions. At search time, part of the text is filtered out by noticing that any occurrence of the pattern must be reflected in the presence of some text q-samples that match approximately inside the pattern. Hence the index points out the text areas that could contain occurrences and must be verified. The index parameters permit load balancing between filtering and verification work, and provide a compromise between the space requirement of the index and the error level for which the filtration is still efficient. We show experimentally that the index is competitive against others that take more space, being in fact the fastest choice for intermediate error levels, an area where no current index is useful.     

Some results on resolvable incomplete block designs

This paper is concerned with the uniformity of a certain kind of resolvable incomplete block (RIB for simplicity) design which is called the PRIB design here. A sufficient and necessary condition is obtained, under which a PRIB design is the most uniform in the sense of a discrete discrepancy measure, and the uniform PRIB design is shown to be connected. A construction method for such designs via a kind of U-type designs is proposed, and an existence result of these designs is given. This method sets up an important bridge between PRIB designs and U-type designs.     

The design of composite permutations with applications to DES-like S-boxes

This paper presents an iterative construction method for building composite permutations. Its efficiency is based on the concepts of pre-computation and equivalence classes. Equivalence class representatives of permutations on four bits are pre-computed. These class representatives can serve as input to the construction method, however, the results are also of independent interest for applications in cryptography. A well-known example of a cryptosystem using composite permutations for its Substitution boxes (S-boxes) is the Data Encryption Standard (DES). Throughout the paper, DES-like S-boxes are defined as mappings satisfying all design criteria as disclosed by one of the designers of DES. All permutations on four bits with DES-like properties are identified. Starting with pre-computed representatives of classes with such permutations, two iterations of a specialized version of the algorithm are applied to obtain bounds on the minimum differential uniformity and minimum non-linear uniformity of DES-like S-boxes. It is established that the two values cannot be less than eight, and that DES-like S-boxes for which the values are both equal to 12 do exist. In addition, if the non-linear uniformity of each of the four permutations in a DES-like S-box is at most six, as in all DES S-boxes, then its non-linear uniformity cannot be less than ten and its minimum differential uniformity equals 12.     

Complementary design theory for sliced equidistance designs

The complementary design theory is powerful for searching for an optimal design when its complementary design is smaller. This paper introduces a new class of sliced equidistance designs and develops the corresponding complementary design theory under the generalized minimum aberration criterion. Two rules are established to search for a generalized minimum aberration design through its complementary design in a sliced equidistance design. As a result, the developed theory covers the related results for the whole designs being saturated designs as special cases. Some examples are presented to illustrate its usefulness.     

Some Results on Minimum Aberration Mixed-level （2^r）2^n Factorial Designs

Mukerjee and Wu(2001) employed projective geometry theory to find the wordlength pattern of a regular mixed factorial design in terms of its complementary set, but only for the numbers of words of length 3 or 4.In this paper,by introducing a concept of consulting design and based on the connection between factorial design theory and cod- ing theory,we obtain some combinatorial identities that relate the wordlength pattern of a regular mixed-level (2~r)2~n factorial design to that of its consulting design.Consequently,a general rule for identifying minimum aberration (2~r)2~n factorial designs through their con- sulting designs is established.It is an improvement and generalization of the related result in Mukerjee and Wu(2001).     

任意凸多面体上布点均匀性的度量

均匀性度量是构作均匀设计的基础,本文从距离概念出发,通过对称的方法,得到一种新的距离函数-势函数,并将势函数作为衡量任意凸多面体上布点均匀性好坏的准则.数值例子和多变量Kendall 协和系数检验表明,当试验区域限制在单位立方体上时,势函数与目前常用的两种偏差-中心化L_2-偏差和可卷L_2.偏差在度量布点均匀性方面结论一致.     

Connection Among Some Optimal Criteria for Symmetrical Fractional Factorial Designs

A fundamental and practical question for fractional factorial designs is the issue of optimal factor assignment. Recently, some new criteria, such as generalized minimum aberration, WV-criterion, NB-criterion and uniformity criterion are proposed for comparing and selecting fractions. In this paper, we indicate that these criteria agree quite well for symmetrical fraction factorial designs.