construction of nonlinear resilient boolean functions using small affine functions in this correspondence we use affine functions on a small number of variables to construct resilient functions on a large number of variables we show that by properly combining these functions it is possible to achieve high nonlinearity and high algebraic degree an important contribution of the correspondence is to show that for each order of resiliency m it is possible to find infinitely many odd and even positive integers n such that it is possible to construct maximum degree n variable m resilient functions having nonlinearity strictly greater than 2 sup n 1 2 sup spl lfloor n 2 spl rfloor we also present construction of some important functions on a small number of variables

periodic orbit maslov indices for systems with time reversal symmetry we show that in time reversal invariant systems a pair of periodic orbits related by time reversal symmetry have the same maslov index previously this result had been implicitly assumed in the semiclassical derivation of the gaussian orthogonal ensemble spectral form factor

minimum aberration construction results for nonregular two level fractional factorial designs summary nonregular two level fractional factorial designs are designs which cannot be specified in terms of a set of defining contrasts the aliasing properties of nonregular designs can be compared by using a generalisation of the minimum aberration criterion called minimum g2 aberration until now the only nontrivial designs that are known to have minimum g2 aberration are designs for n runs and m n 5 factors in this paper a number of construction results are presented which allow minimum g2 aberration designs to be found for many of the cases with n 16 24 32 48 64 and 96 runs and m n 2 2 factors

secondary constructions of highly nonlinear boolean functions and disjoint spectra plateaued functions in this paper we modify a generalized indirect sum construction to construct functions with high nonlinearity by utilizing the modified construction highly nonlinear functions in n m variables can be obtained from known bent functions in n variables and highly nonlinear functions in m variables it is possible to obtain new n 15 variable functions with nonlinearity 2 n 1 5 1 2 n 1 5 1 2 20x2 n 2 and new 12 variable 2 resilient functions with nonlinearity 2000 and algebraic degree 8 which achieve optimal algebraic immunity moreover the modified construction can also be used as an iterative construction of a quadruple of disjoint spectra plateaued functions in addition we present sufficient conditions for a quadruple of disjoint spectra plateaued functions to have no nonzero linear structure

on design orthogonality maximin distance and projection uniformity for computer experiments space filling designs are widely used in both computer and physical experiments column orthogonality maximin distance and projection uniformity are three basic and popular space filling criteria

on the covering radius of r 1 m in r 3 m we prove that the covering radius of r 1 11 in r 3 11 is 992 and that the covering radius of r 1 13 in r 3 13 is 4032 both not exceeding the quadratic bound

blocking schemes for definitive screening designs in earlier work jones and nachtsheim proposed a new class of screening designs called definitive screening designs as originally presented these designs are three level designs for quantitative factors that provide estimates of main effects that are unbiased by any second order effect and require only one more than twice as many runs as there are factors definitive screening designs avoid direct confounding of any pair of second order effects and for designs that have more than five factors project to efficient response surface designs for any two or three factors recently jones and nachtsheim expanded the applicability of these designs by showing how to include any number of two level categorical factors however methods for blocking definitive screening designs have not been addressed in this article we develop orthogonal blocking schemes for definitive screening designs we separately consider the cases where all of the factors are quantitative and where there is a mix of quantitative and tw

a generalized discrepancy and quadrature error bound an error bound for multidimensional quadrature is derived that includes the koksma hlawka inequality as a special case this error bound takes the form of a product of two terms one term which depends only on the integrand is defined as a generalized variation the other term which depends only on the quadrature rule is defined as a generalized discrepancy the generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the l p star discrepancy and p that arises in the study of lattice rules

a bayesian variable selection approach for analyzing designed experiments with complex aliasing experiments using designs with complex aliasing patterns are often performed for example twolevel nongeometric plackett burman designs multilevel and mixed level fractional factorial designs two level fractional factorial designs with hard to control factors and supersaturated designs hamada and wu proposed an iterative guided stepwise regression strategy for analyzing the data from such designs that allows entertainment of interactions their strategy provides a restricted search in a rather large model space however this article provides an efficient methodology based on a bayesian variable selection algorithm for searching the model space more thoroughly we show how the use of hierarchical priors provides a flexible and powerful way to focus the search on a reasonable class of models the proposed methodology is demonstrated with four examples three of which come from actual industrial experiments

on the weight distributions of optimal cosets of the first order reed muller codes we study the weight distributions of cosets of the first order reed muller code r 1 m for odd m whose minimum weight is greater than or equal to the so called quadratic bound some general restrictions on the weight distribution of a coset of r 1 m are obtained by partitioning its words according to their weight divisibility most notably we show that there are exactly five weight distributions for optimal cosets of r 1 7 in r 5 7 and that these distributions are related to the degree of the function generating the coset moreover for any odd m spl ges 9 we exhibit optimal cubic cosets of r 1 m whose weights take on exactly five values

selecting 2m p designs using a minimum aberration criterion when some two factor interactions are important we consider the problem of selecting appropriate 2m p designs when some two factor interactions are important current methods in the literature select designs that permit estimation of the postulated model consisting of the main effects and important two factor interactions under the assumption that all of the other effects are negligible when the effects not in the postulated model are not negligible they will bias the estimates of the effects in the model to minimize the contamination of these nonnegligible effects on the model we propose and study a minimum aberration criterion we then discuss the application of this new aberration criterion to compromise plans finally we examine how to search for the best designs according to the criterion and present some results for designs of 16 and 32 runs

strong orthogonal arrays and associated latin hypercubes for computer experiments this paper introduces constructs and studies a new class of arrays called strong orthogonal arrays as suitable designs for computer experiments a strong orthogonal array of strength t enjoys better space filling properties than a comparable orthogonal array in all dimensions lower than t while retaining the space filling properties of the latter in t dimensions latin hypercubes based on strong orthogonal arrays of strength t are more space filling than comparable orthogonal array based latin hypercubes in all g dimensions for any 2 g t 1 copyright 2013 oxford university press

blocked nonregular two level factorial designs this article discusses the optimal blocking criteria for nonregular two level designs we extend the optimal blocking criteria of cheng and wu to nonregular designs by adapting the g and g2 minimum aberration criteria discussed by tang and deng to define word length pattern for nonregular designs we extend the notion of word to nonregular designs through a polynomial representation of factorial designs we define treatment resolution and block resolution for evaluating the degrees of aliasing and confounding we propose four new criteria which we use to search for optimal blocking schemes of 12 run 16 run and 20 run two level orthogonal arrays

indicator function and its application in two level factorial designs a two level factorial design can be uniquely represented by a polynomial indicator function therefore properties of factorial designs can be studied through their indicator functions this paper shows that the indicator function is an effective tool in studying two level factorial designs the indicator function is used to generalize the aberration criterion of a regular two level fractional factorial design to all two level factorial designs an important identity of generalized aberration is proved the connection between a uniformity measure and aberration is also extended to all two level factorial designs

multi layer designs for computer experiments space filling designs such as latin hypercube designs lhds are widely used in computer experiments however finding an optimal lhd with good space filling properties is computationally cumbersome on the other hand the well established factorial designs in physical experiments are unsuitable for computer experiments owing to the redundancy of design points when projected onto a subset of factor space in this work we present a new class of space filling designs developed by splitting two level factorial designs into multiple layers the method takes advantage of many available results in factorial design theory and therefore the proposed multi layer designs mlds are easy to generate moreover our numerical study shows that mlds can have better space filling properties than optimal lhds

patterson wiedemann type functions on 21 variables with nonlinearity greater than bent concatenation bound nonlinearity is one of the most challenging combinatorial property in the domain of boolean function research obtaining nonlinearity greater than the bent concatenation bound for odd number of variables continues to be one of the most sought after combinatorial research problems the pioneering result in this direction has been discovered by patterson and wiedemann in 1983 ieee it which considered boolean functions on 5 times 3 15 variables that are invariant under the actions of the cyclic group gf 2 5 ast cdot gf 2 3 ast as well as the group of frobenius automorphisms some of these boolean functions possess nonlinearity greater than the bent concatenation bound the next possible option for exploring such functions is on 7 times 3 21 variables however obtaining such functions remained elusive for more than three decades even after substantial efforts as evident in the literature in this paper we exploit combinatorial arguments together with heuristic search to demonstrate such functions for the first time

on the covering radius of first order generalized reed muller codes we generalize to any finite fq fields a theorem about covering radius of codes of strength 2 proved by helleseth and coworkers then using this result and partial covering radius we give bounds for the covering radius of first order generalized reed muller codes finally using magma we get some improvements for f3

optimal foldover plans for two level fractional factorial designs a commonly used follow up experiment strategy involves the use of a foldover design by reversing the signs of one or more columns of the initial design defining a foldover plan as the collection of columns whose signs are to be reversed in the foldover design this article answers the following question given a 2k p design with k factors and p generators what is its optimal foldover plan we obtain optimal foldover plans for 16 and 32 runs and tabulate the results for practical use most of these plans differ from traditional foldover plans that involve reversing the signs of one or all columns there are several equivalent ways to generate a particular foldover design we demonstrate that any foldover plan of a 2k p fractional factorial design is equivalent to a core foldover plan consisting only of the p out of k factors furthermore we prove that there are exactly 2k p foldover plans that are equivalent to any core foldover plan of a 2k p design and demonstrate how these foldover plans can be const

geometric isomorphism and minimum aberration for factorial designs with quantitative factors factorial designs have broad applications in agricultural engineering and scientific studies in constructing and studying properties of factorial designs traditional design theory treats all factors as nominal however this is not appropriate for experiments that involve quantitative factors for designs with quantitative factors level permutation of one or more factors in a design matrix could result in different geometric structures and thus different design properties in this paper indicator functions are introduced to represent factorial designs a polynomial form of indicator functions is used to characterize the geometric structure of those designs geometric isomorphism is defined for classifying designs with quantitative factors based on indicator functions a new aberration criteria is proposed and some minimum aberration designs are presented

the bakerian lecture 1987 quantum chaology bounded or driven classical systems often exhibit chaos exponential instability that persists but their quantum counterparts do not nevertheless there are new regimes of quantum behaviour that emerge in the semiclassical limit and depend on whether the classical orbits are regular or chaotic and this motivates the following definition definition quantum chaology is the study of semiclassical but nonclassical behaviour characteristic of systems whose classical motion exhibits chaos this is illustrated by the statistics of energy levels on scales comparable with the mean level spacing of order h n for n freedoms these fall into universality classes for classically chaotic systems the statistics are those of random matrices real symmetric or complex hermitian depending on the presence or absence of time reversal symmetry for classically regular ones the statistics are poisson on larger scales of order h i e classically small but semiclassically large universality breaks down these phenomena are being explained by representing spectra in terms of classical closed orbits universal spectral behaviour has its origin in very long orbits non universal behaviour depends only on short ones

the covering radius of the 2 15 16 reed muller code is at least 16276 two interesting cosets of the first order reed muller code of block length 2 15 are described they provide counterexamples to a conjecture on the covering radius

space filling fractional factorial designs fractional factorial designs are widely used in various scientific investigations and industrial applications level permutation of factors could alter their geometrical structures and statistical properties this article studies space filling properties of fractional factorial designs under two commonly used space filling measures discrepancy and maximin distance when all possible level permutations are considered the average discrepancy is expressed as a linear combination of generalized word length pattern for fractional factorial designs with any number of levels and any discrepancy defined by a reproducing kernel generalized minimum aberration designs are shown to have good space filling properties on average in terms of both discrepancy and distance several novel relationships between distance distribution and generalized word length pattern are derived it is also shown that level permutations can improve space filling properties for many existing saturated designs a two step construction pro

a connection between uniformity and aberration in regular fractions of two level factorials summary we show a link between the two apparently unrelated areas of uniformity and minimum aberration with reference to regular fractions of two level factorials we derive an expression for the centred l2 discrepancy measure for uniformity in terms of the word length pattern this result indicates in particular excellent behaviour of minimum aberration designs with regard to uniformity and provides further justification for the popular criterion of minimum aberration

on the covering radius of codes the covering radius r of a code is the maximal distance of any vector from the code this work gives a number of new results concerning t n k the minimal covering radius of any binary code of length n and dimension k for example t n 4 and t n 5 are determined exactly and reasonably tight bounds on t n k are obtained for any k when n is large these results are found by using several new constructions for codes with small covering radius one construction the amalgamated direct sum involves a quantity called the norm of a code codes with norm leq 2 r 1 are called normal and may be combined efficiently the paper concludes with a table giving bounds on t n k for n leq 64

theory of j characteristics for fractional factorial designs and projection justification of minimum g2 aberration deng tang 1999 introduced the generalised resolution and minimum g aberration criteria for assessing nonregular fractional factorials in tang deng 1999 a relaxed variant of minimum g aberration called minimum g 2 aberration is proposed and studied these criteria are defined using a set of j values called j characteristics in this paper we show that a factorial design is uniquely determined by its j characteristics just as a regular factorial design is uniquely determined by its defining relation the theorem is given through an explicit formula that relates the set of design points to that of j characteristics through this formula projection justification of minimum g 2 aberration is established

better foldover fractions for resolution iii 2k p designs this article describes a family of resolution iiidesigns for which the usual advice regarding foldover reversing all factors is ill advised the smallest such designs have 16 runs for designs in this family alternative foldover fractions not only increase the resolution to iv but also separate some of the aliased two factor interactions although resolution iv designs obtained by reversing all factors provide fewer than half the degrees of freedom for estimating two factor interactions the recommended foldovers for designs in this class permit estimation of many more two factor interactions this result implies that designs in this class make attractive initial designs even if they are not minimum aberration

on the construction of cryptographically significant boolean functions using objects in projective geometry spaces recently several construction methods of highly nonlinear boolean functions with relatively good algebraic properties were proposed these approaches manage in optimizing most of the relevant cryptographic criteria but not all of them at the same time usually either the nonlinearity bounds are rather loose though the actual nonlinearity is relatively high or the functions do not provide a good resistance to fast algebraic cryptanalysis in this paper we develop a theoretical framework for using objects in suitable projective geometry spaces for construction of highly nonlinear boolean functions this allows us to establish tight bounds on the nonlinearity using simple counting arguments thus avoiding rather complicated estimates of certain trace sums our method generates a class of almost fully optimized functions that is the functions apart from very high nonlinearity also have the maximum algebraic degree and optimal algebraic immunity compared to the classes of functions proposed by carlet and feng wang and zeng our functions achieve a slightly better nonlinearity which is traded off against a little worse resistance against fast algebraic attacks on the other hand compared to the functions by tang and tu and deng our nonlinearity is somewhat lower but the algebraic properties are slightly better

using the folded over 12 run plackett burman design to consider interactions this article demonstrates that the folded over 12 run plackett burman design is useful for considering up to 12 factors in 24 runs even if one anticipates that some two factor interactions may be significant the properties of this design are investigated and a sequential procedure for analyzing the data from such a design is proposed the performance of the procedure is investigated through the analysis of real and constructed examples and through a small simulation study applications to other folded over plackett burman designs are also briefly discussed

two level orthogonal screening designs with 24 28 32 and 36 runs abstractthe potential of two level orthogonal designs to fit models with main effects and two factor interaction effects is commonly assessed through the correlation between contrast vectors involving these effects we study the complete catalog of nonisomorphic orthogonal two level 24 run designs involving 3 23 factors and we identify the best few designs in terms of these correlations by modifying an existing enumeration algorithm we identify the best few 28 run designs involving 3 14 factors and the best few 36 run designs in 3 18 factors as well based on a complete catalog of 7570 designs with 28 runs and 27 factors we also seek good 28 run designs with more than 14 factors finally starting from a unique 31 factor design in 32 runs that minimizes the maximum correlation among the contrast vectors for main effects and two factor interactions we obtain 32 run designs that have low values for this correlation to demonstrate the added value of our work we provide a detailed comparison of our desi

modifications of patterson wiedemann functions for cryptographic applications three basic properties of boolean functions to be useful for cryptographic purposes are balancedness high algebraic degree and high nonlinearity in addition strict avalanche criteria and propagation characteristics are required for design of s boxes we introduce methods to modify the patterson wiedemann 19983 1990 and bent functions to achieve the above cryptographic properties in the process we are able to answer some open questions about boolean functions

space filling properties of good lattice point sets we study space filling properties of good lattice point sets and obtain some general theoretical results we show that linear level permutation does not decrease the minimum distance for good lattice point sets and we identify several classes of such sets with large minimum distance based on good lattice point sets some maximin distance designs are also constructed

projections of definitive screening designs by dropping columns selection and evaluation abstract definitive screening designs permit the study of many quantitative factors in a few runs more than twice the number of factors in practical applications researchers often require a desig

on the existence of saturated and nearly saturated asymmetrical orthogonal arrays we develop a combinatorial condition necessary for the existence of a saturated asymmetrical orthogonal array of strength 2 this condition limits the choice of integral solutions to the system of equations in the bose bush approach and can thus strengthen considerably the bose bush approach as applied to a symmetrical part of such an array as a consequence several nonexistence results follow for saturated and nearly saturated orthogonal arrays of strength 2 one of these leads to a partial settlement of an issue left open in a paper by wu zhang and wang nonexistence of a class of saturated asymmetrical orthogonal arrays of strength 4 is briefly discussed

search for boolean functions with excellent profiles in the rotation symmetric class for the first time boolean functions on 9 variables having nonlinearity 241 are discovered that remained as an open question in literature for almost three decades such functions are found by heuristic search in the space of rotation symmetric boolean functions rsbfs this shows that there exist boolean functions on n odd variables having nonlinearity 2n 1 2n 1 2 if and only if n 7 using similar search technique balanced boolean functions on 9 10 and 11 variables are attained having autocorrelation spectra with maximum absolute value 2lceiln 2rceil on odd number of variables earlier such functions were known for 15 21 variables there was no evidence of such functions at all on even number of variables in certain cases our functions can be affinely transformed to obtain first order resiliency or first order propagation characteristics moreover 10 variable functions having first order resiliency and nonlinearity 492 are presented that had been posed as an open question at crypto 2000 the functions reported in this paper are discovered using a suitably modified steepest descent based iterative heuristic search in the rsbf class along with proper affine transformations it seems elusive to get a construction technique to match such functions

lower bounds and stochastic optimization algorithms for uniform designs with three or four levels new lower bounds for three and four level designs under the centered l 2 discrepancy are provided we describe necessary conditions for the existence of a uniform design meeting these lower bounds we consider several modifications of two stochastic optimization algorithms for the problem of finding uniform or close to uniform designs under the centered l 2 discrepancy besides the threshold accepting algorithm we introduce an algorithm named balance pursuit heuristic this algorithm uses some combinatorial properties of inner structures required for a uniform design using the best specifications of these algorithms we obtain many designs whose discrepancy is lower than those obtained in previous works as well as many new low discrepancy designs with fairly large scale moreover some of these designs meet the lower bound i e are uniform designs

systematic constructions of rotation symmetric bent functions 2 rotation symmetric bent functions and bent idempotent functions rotation symmetric bent functions and their generation two rotation symmetric bent functions are two classes of cryptographically significant boolean functions however few constructions have been presented in the literature which either have restriction on integer n or have algebraic degree no more than 4 in this paper for any even integer n ge 4 three classes of bent functions are presented respectively most notably the proposed n variable rotation symmetric bent functions and two rotation symmetric bent functions can have any possible algebraic degree ranging from 2 to n 2 besides we obtain bent idempotent functions with the maximal algebraic degree n 2

an approach to the construction of asymmetrical orthogonal arrays abstract use of asymmetrical orthogonal arrays oa s for planning industrial experiments with mixed levels has become increasingly popular in addition to applications to experimental investigations in many disciplines they can also be used in the balanced repeated replications method of inference from general stratified survey samples in this article we propose a general approach to the construction of asymmetrical oa s with economic run size and flexibility in the choice of factor levels as applications of this approach we construct several general classes of arrays that include numerous existing classes of arrays as special cases a large number of known asymmetrical oa s can be reproduced by our approach in a unified and simple manner many new arrays are obtained a list of asymmetrical oa s with moderate run size is given our approach consists of three steps we first construct an oa as the generalized kronecker sum of an oa and some difference matrices we then add to the constructed oa co

9 variable boolean functions with nonlinearity 242 in the generalized rotation symmetric class we give a new lower bound to the covering radius of the first order reed muller code rm 1 n where n 9 11 13 equivalently we present the n variable boolean functions for n 9 11 13 with maximum nonlinearity found till now in 2006 9 variable boolean functions having nonlinearity 241 which is strictly greater than the bent concatenation bound of 240 have been discovered in the class of rotation symmetric boolean functions rsbfs by kavut maitra and yucel to improve this nonlinearity result we have firstly defined some subsets of the n variable boolean functions as the generalized classes of k rsbfs and k dsbfs k dihedral symmetric boolean functions where k is a positive integer dividing n secondly utilizing a steepest descent like iterative heuristic search algorithm we have found 9 variable boolean functions with nonlinearity 242 within the classes of both 3 rsbfs and 3 dsbfs thirdly motivated by the fact that rsbfs are invariant under a special permutation of the input vector we have classified all possible permutations up to the linear equivalence of boolean functions that are invariant under those permutations

uniform designs limit aliasing summary when fitting a linear regression model to data aliasing can adversely affect the estimates of the model coefficients and the decision of whether or not a term is significant optimal experimental designs give efficient estimators assuming that the true form of the model is known while robust experimental designs guard against inaccurate estimates caused by model misspecification although it is rare for a single design to be both maximally efficient and robust it is shown here that uniform designs limit the effects of aliasing to yield reasonable efficiency and robustness together aberration and resolution measure how well fractional factorial designs guard against the effects of aliasing here it is shown that the definitions of aberration and resolution may be generalised to other types of design using the discrepancy

design selection and classification for hadamard matrices using generalized minimum aberration criteria deng and tang 1999 and tang and deng 1999 proposed and justified two criteria of generalized minimum aberration for general two level fractional factorial designs the criteria are defined using a set of values called j characteristics in this article we examine the practical use of the criteria in design selection specifically we consider the problem of classifying and ranking designs that are based on hadamard matrices a theoretical result on j characteristics is developed to facilitate the computation the issue of design selection is further studied by linking generalized aberration with the criteria of efficiency and estimation capacity our studies reveal that generalized aberration performs quite well under these familiar criteria

optimal projective three level designs for factor screening and interaction detection orthogonal arrays oas are widely used in industrial experiments for factor screening suppose that only a few of the factors in the experiments turn out to be important an oa can be used not only for screening factors but also for detecting interactions among a subset of active factors in this article a set of optimality criteria is proposed to assess the performance of designs for factor screening projection and interaction detection and a three step approach is proposed to search for optimal designs combinatorial and algorithmic construction methods are proposed for generating new designs permutations of levels are used for improving the eligibility and estimation efficiency of the projected designs the techniques are then applied to search for best three level designs with 18 and 27 runs many new efficient and practically useful nonregular designs are found and their properties are discussed

on the lower bounds of the second order nonlinearities of some boolean functions the rth order nonlinearity of a boolean function is an important cryptographic criterion in analyzing the security of stream as well as block ciphers it is also important in coding theory as it is related to the covering radius of the reed muller code r r n in this paper we deduce the lower bounds of the second order nonlinearities of the following two types of boolean functions 1 f l x tr 1 n lx d with d 2 2 r 2 r 1 and l f 2 n where n 6r 2 f x y tr 1 t xy 2 i 1 where x y f 2 t n 2t n 6 and i is an integer such that 1

a bayesian approach to the design and analysis of fractionated experiments specifying a prior distribution for the large number of parameters in the statistical model is a critical step in a bayesian approach to the design and analysis of experiments this article shows that the prior distribution can be induced from a functional prior on the underlying transfer function the functional prior requires specification of only a few hyperparameters and thus can be easily implemented in practice the usefulness of the approach is demonstrated through the analysis of some experiments the article also proposes a new class of design criteria and establishes their connections with the minimum aberration criterion

permuting regular fractional factorial designs for screening quantitative factors fractional factorial designs are widely used in screening experiments they are often chosen by the minimum aberration criterion which regards factor levels as symbols for designs with quantitative factors however permuting the levels for one or more factors could alter their geometrical structures and statistical properties we provide a justification of the minimum aberration criterion for quantitative factors and study level permutations for regular fractional factorial designs in order to improve their efficiency for screening quantitative factors we show how regular designs can be linearly permuted to reduce contamination of nonnegligible interactions on the estimation of linear effects without increasing the run size we further show that such linear permutations are unique under the minimum aberration criterion and the best level permutations can be determined without an exhaustive search we establish additional theoretical results for three level designs and obtain the best level permutations for regular designs with 27 and 81 runs we illustrate the practical benefits of level permutation with an antiviral drug combination experiment

designs of variable resolution prior information or background knowledge may suggest that interactions arise only within certain factors when such knowledge is available we propose using a new class of designs designs of variable resolution several constructions are presented statistical justifications for using such designs from minimum g 2 aberration and design efficiency perspectives are provided copyright 2012 oxford university press

nonregular designs with desirable projection properties in many industrial applications the experimenter is interested in estimating some of the main effects and two factor interactions in this article we rank two level orthogonal designs based on the number of estimable models containing a subset of main effects and their associated two factor interactions by ranking designs in this way the experimenter can directly assess the usefulness of the experimental plan for the purpose in mind we apply the new ranking criterion to the class of all non isomorphic two level orthogonal designs with 16 and 20 runs and introduce a computationally efficient algorithm based on two theoretical results which will aid in finding designs with larger run sizes catalogs of useful designs with 16 20 24 and 28 runs are presented

generalized minimum aberration for asymmetrical fractional factorial designs by studying treatment contrasts and anova models we propose a generalized minimum aberration criterion for comparing asymmetrical fractional factorial designs the criterion is independent of the choice of treatment contrasts and thus model free it works for symmetrical and asymmetrical designs regular and nonregular designs in particular it reduces to the minimum aberration criterion for regular designs and the minimum g 2 aberration criterion for two level nonregular designs in addition by exploring the connection between factorial design theory and coding theory we develop a complementary design theory for general symmetrical designs which covers many existing results as special cases

a general criterion for factorial designs under model uncertainty motivated by two industrial experiments in which rather extreme prior knowledge was used to choose the design we show that the qb criterion which aims to improve the estimation in as many models as possible by incorporating experimenters prior knowledge along with an approximation to the as criterion is more general and has a better statistical interpretation than many standard criteria the generalization and application of the criterion to different types of designs are presented the relationships between qb and other criteria for different situations are explored it is shown that the e s2 criterion is a special case of qb and several aberration type criteria are limiting cases of our criterion so that qb provides a bridge between alphabetic optimality and aberration the two case studies illustrate the potential benefits of the qb criterion r programs for calculating qb are available online as supplemental materials

cryptographically significant boolean functions with five valued walsh spectra we describe methods to construct balanced resp 1 resilient functions of odd number of variables n achieving the bent concatenation nonlinearity and having algebraic degree n1 resp n 2 the technique is to algebraically modify the concatenation of two properly chosen n 1 variable maiorana mcfarland bent functions the constructed functions can be used with certain recursive operators to provide higher order resilient functions with maximum possible degree and high nonlinearity such functions are well suited for stream cipher applications interestingly all the constructed functions have a five valued walsh spectr

on cryptographic properties of the cosets of r 1 m we introduce a new approach for the study of weight distributions of cosets of the reed muller code of order 1 our approach is based on the method introduced by kasami 1968 using pless 1963 identities by interpreting some equations we obtain a necessary condition for a coset to have a high minimum weight most notably we are able to distinguish such cosets which have three weights only we then apply our results to the problem of the nonlinearity of boolean functions we particularly study the links between this criterion and the propagation characteristics of a function

incorporating prior information in optimal design for model selection an important use of experimental designs is in screening in which experimenters seek to identify significant effects both main effects and potentially interactions from a large set of candidate effects this article goes further than identification of effects introducing a design criterion that seeks to maximize the ability to discriminate between models motivated by the work of meyer steinberg and box the bayesian criterion is based on the hellinger distance between predictive distributions under competing models a bound for the criterion is obtained greatly improving interpretability the set of all possible models to compare is huge and not all models are equally plausible this challenge is addressed through prior distributions on the space of models that indicate preference for intuitively appealing models such as those with few effects more low order than high order effects and inheritance structure between active main effects and interactions techniques for evaluating the criterion a

a trigonometric approach to quaternary code designs with application to one eighth and one sixteenth fractions the study of good nonregular fractional factorial designs has received significant attention over the last two decades recent research indicates that designs constructed from quaternary codes qc are very promising in this regard the present paper shows how a trigonometric approach can facilitate a systematic understanding of such qc designs and lead to new theoretical results covering hitherto unexplored situations we focus attention on one eighth and one sixteenth fractions of two level factorials and show that optimal qc designs often have larger generalized resolution and projectivity than comparable regular designs moreover some of these designs are found to have maximum projectivity among all designs

two level designs to estimate all main effects and two factor interactions we study the design of two level experiments with n runs and n factors large enough to estimate the interaction model which contains all the main effects and all the two factor interactions yet an effect hierarchy assumption suggests that main effect estimation should be given more prominence than the estimation of two factor interactions orthogonal arrays oas favor main effect estimation however complete enumeration becomes infeasible for cases relevant for practitioners we develop a partial enumeration procedure for these cases and we establish upper bounds on the d efficiency for the interaction model based on arrays that have not been generated by the partial enumeration we also propose an optimal design procedure that favors main effect estimation designs created with this procedure have smaller d efficiencies for the interaction model than d optimal designs but standard errors for the main effects in this model are improved generated oas for 7 10 factors and 32 72 runs are smaller or have a higher d efficiency than the smallest oas from the literature designs obtained with the new optimal design procedure or strength 3 oas which have main effects that are not correlated with two factor interactions are recommended if main effects unbiased by possible two factor interactions are of primary interest d optimal designs are recommended if interactions are of primary interest supplementary materials for this article are available online 2017 american statistical association and the american society for quality

on some cosets of the first order reed muller code with high minimum weight we study a family of particular cosets of the first order reed muller code r 1 m those generated by special codewords the idempotents thus we obtain new maximal weight distributions of cosets of r 1 7 and 84 distinct almost maximal weight distributions of cosets of r 1 9 that is with minimum weight 240 this leads to crypotographic applications in the context of stream ciphers

on the norm and covering radius of the first order reed muller codes let spl rho 1 m and n 1 m be the covering radius and norm of the first order reed muller code r 1 m respectively it is known that spl rho 1 2k 1 spl les lower bound 2 sup 2k 2 sup 2k 1 2 and n 1 2k 1 spl les 2 lower bound 2 sup 2k 2 sup 2k 1 2 k 0 we prove that spl rho 1 2k 1 spl les 2 lower bound 2 sup 2k 1 2 sup 2k 3 2 and n 1 2k 1 spl les 4 lower bound 2 sup 2k 1 2 sup 2k 3 2 k 0 we also discuss the connections of the two new bounds with other coding theoretic problems

covering radius survey and recent results all known results on covering radius are presented as well as some new results there are a number of upper and lower bounds including asymptotic results a few exact determinations of covering radius some extensive relations with other aspects of coding theory through the reed muller codes and new results on the least covering radius of any linear n k code there is also a recent result on the complexity of computing the covering radius

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

the newton radius of codes for a binary linear code c of minimum distance d if t d 1 2 then there are errors of weight t which are not uniquely correctable however in many cases there are also errors of weight t which are uniquely correctable the newton radius of a code is defined to be the largest weight of a uniquely correctable error bounds and exact values of the newton radius are given for several classes of codes

construction of two level nonregular designs of strength three with large run sizes abstracttwo level orthogonal arrays of strength 3 permit the study of the main effects and the two factor interactions of the experimental factors these arrays are classified into regular and nonr

quasi orthogonal sequences for code division multiple access systems the notion of quasi orthogonal sequence qos as a means of increasing the number of channels in synchronous code division multiple access cdma systems that employ walsh sequences for spreading information signals and separating channels is introduced it is shown that a qos sequence may be regarded as a class of bent almost bent functions possessing in addition a certain window property such sequences while increasing the system capacity minimize interference to the existing set of walsh sequences the window property gives the system the ability to handle variable data rates a general procedure of constructing qoss from well known families of binary sequences with good correlation including the kasami and gold 1967 sequence families as well as from the binary kerdock code is provided examples of qoss are presented for small lengths some examples of quaternary qoss drawn from family a are also included

projective three level main effects designs robust to model uncertainty this paper is concerned with designing experiments under the assumption that an analysis strategy is used that considers interactions in addition to main effects a criterion which averages an approximation to as efficiency over lower dimensional projections of the design is introduced to compare designs a columnwise design procedure is used to construct three level designs for six factors in 18 runs the projection efficiencies of these designs are explored and compared with designs obtained from the l 18 orthogonal array results are also given for designs with 14 and 17 runs

projection properties of plackett and burman designs the projection properties of the 2 r q p fractional factorials are well known and have been used effectively in a number of published examples of experimental investigations the plackett and burman designs also have interesting projective properties knowledge of which allows the experimenter to follow up an initial plackett and burman design with runs that increase the initial resolution for the factors that appear to matter and thus permit efficient separation of effects of interest projections of designs into 2 5 dimensions are discussed and the 12 run case is given in detail a numerical example illustrates the practical uses of these projections

some results on the covering radii of reed muller codes let r r m be the rth order reed muller code of length 2 sup m and let rho r m be its covering radius r

generalised minimum aberration construction results for symmetrical orthogonal arrays generalised minimum aberration is a recently established design criterion for the whole class of orthogonal arrays and fractional factorial designs the criterion is as its name suggests a generalisation of minimum aberration for regular designs and of minimum g 2 aberration for two level designs the aim of the criterion is to find designs which minimise in a certain sense the aliasing between main effects and interactions in this paper theoretical results are developed for finding symmetrical orthogonal arrays with generalised minimum aberration for more than two factor levels

application of threshold accepting to the evaluation of the discrepancy of a set of points efficient routines for multidimensional numerical integration are provided by quasi monte carlo methods these methods are based on evaluating the integrand at a set of representative points of the integration area a set may be called representative if it shows a low discrepancy however in dimensions higher than two and for a large number of points the evaluation of discrepancy becomes infeasible the use of the efficient multiple purpose heuristic threshold accepting offers the possibility to obtain at least good approximations to the discrepancy of a given set of points this paper presents an implementation of the threshold accepting heuristic an assessment of its performance for some small examples and results for larger sets of points with unknown discrepancy

maximum nonlinearity of symmetric boolean functions on odd number of variables in this correspondence we establish that for odd n the maximum nonlinearity achievable by an n variable symmetric boolean function is 2 sup n 1 2 sup n 1 sup 2 and characterize the set of functions which achieve this value of nonlinearity in particular we show that for each odd n spl ges 3 there are exactly four possible symmetric boolean functions achieving the nonlinearity 2 sup n 1 2 sup n 1 2

on the degree nonlinearity algebraic thickness and nonnormality of boolean functions with developments on symmetric functions the two main criteria evaluating from cryptographic viewpoint the complexity of boolean functions are the nonlinearity and the algebraic degree two other criteria can also be considered the algebraic thickness and the nonnormality simple proofs are given that asymptotically almost all boolean functions have high algebraic thicknesses and are deeply nonnormal as well as they have high algebraic degrees and high nonlinearities we also study in detail the relationship between nonnormality and nonlinearity we derive simple proofs of known results on symmetric boolean functions and we prove several new and more general results on a class containing all symmetric functions

periodic orbits and a correlation function for the semiclassical density of states through the rule that there is one quantum state per planck cell in phase space classical mechanics supplies an average semiclassical density of states for a system with bound motion can it supply more refined averages such as the correlation function for the density of states i shall argue that for the two extreme cases of integrable classical motion and ergodic classical motion information can indeed be obtained on not quite this but a closely related correlation function the results for the two types of motion are markedly different

generalized maiorana mcfarland construction of resilient boolean functions with high nonlinearity and good algebraic properties a new framework concerning the construction of small order resilient boolean functions whose nonlinearity is strictly greater than 2 n 1 2 n 2 is given first a generalized maiorana mcfarland construction technique is described which extends the current approaches by combining the usage of affine and nonlinear functions in a controllable manner it is shown that for any given m this technique can be used to construct a large class of n variable n both even and odd m resilient degree optimized boolean functions with currently best known nonlinearity this class may also provide functions with excellent algebraic properties measured through the resistance to fast algebraic attacks if the number of n 2 variable affine subfunctions used in the construction is relatively low due to a potentially low hardware implementation cost along with overall good cryptographic properties this class of functions is an attractive candidate for the use in certain stream cipher schemes

optimal fractions of two level factorials under a baseline parameterization two level fractional factorial designs are considered under a baseline parameterization the criterion of minimum aberration is formulated in this context and optimal designs under this criterion are investigated the underlying theory and the concept of isomorphism turn out to be significantly different from their counterparts under orthogonal parameterization and this is reflected in the optimal designs obtained copyright 2012 oxford university press

optimal foldover plans for two level nonregular orthogonal designs this article considers optimal foldover plans for nonregular designs by using the indicator function we define words with fractional lengths the extended word length pattern is then used to select among nonregular foldover designs some general properties of foldover designs are obtained using the indicator function we prove that the full foldover plan that reverses the signs of all factors is optimal for all 12 run and 20 run orthogonal designs the optimal foldover plans for all 16 run regular and nonregular orthogonal designs are constructed and tabulated for practical use optimal foldover plans for higher order orthogonal designs can be constructed in a similar manner

semiclassical form factor for spectral and matrix element fluctuations of multidimensional chaotic systems we present a semiclassical calculation of the generalized form factor kab tau which characterizes the fluctuations of matrix elements of the operators a and b in the eigenbasis of the hamiltonian of a chaotic system our approach is based on some recently developed techniques for the spectral form factor of systems with hyperbolic and ergodic underlying classical dynamics and f 2 degrees of freedom that allow us to go beyond the diagonal approximation first we extend these techniques to systems with f 2 then we use these results to calculate kab tau we show that the dependence on the rescaled time tau time in units of the heisenberg time is universal for both the spectral and the generalized form factor furthermore we derive a relation between kab tau and the classical time correlation function of the weyl symbols of a and b

generalized resolution for orthogonal arrays the generalized word length pattern of an orthogonal array allows a ranking of orthogonal arrays in terms of the generalized minimum aberration criterion xu and wu 2001 we provide a statistical interpretation for the number of shortest words of an orthogonal array in terms of sums of r values based on orthogonal coding or sums of squared canonical correlations based on arbitrary coding directly related to these results we derive two versions of generalized resolution for qualitative factors both of which are generalizations of the generalized resolution by deng and tang 1999 and tang and deng 1999 we provide a sufficient condition for one of these to attain its upper bound and we provide explicit upper bounds for two classes of symmetric designs factor wise generalized resolution values provide useful additional detail