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

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 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

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

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

hierarchical testing designs for pattern recognition we explore the theoretical foundations of a twenty questions approach to pattern recognition the object of the analysis is the computational process itself rather than probability distributions bayesian inference or decision boundaries statistical learning our formulation is motivated by applications to scene interpretation in which there are a great many possible explanations for the data one background is statistically dominant and it is imperative to restrict intensive computation to genuinely ambiguous regions the focus here is then on pattern filtering given a large set mathcal y of possible patterns or explanations narrow down the true one y to a small random subset widehat y subset mathcal y of detected patterns to be subjected to further more intense processing to this end we consider a family of hypothesis tests for y a versus the nonspecific alternatives y ac each test has null type i error and the candidate sets a subset mathcal y are arranged in a hierarchy of nested partitions these tests are then characterized by scope a power or type ii error and algorithmic cost we consider sequential testing strategies in which decisions are made iteratively based on past outcomes about which test to perform next and when to stop testing the set y is then taken to be the set of patterns that have not been ruled out by the tests performed the total cost of a strategy is the sum of the testing cost and the postprocessing cost proportional to y and the corresponding optimization problem is analyzed as might be expected under mild assumptions good designs for sequential testing strategies exhibit a steady progression from broad scope coupled with low power to high power coupled with dedication to specific explanations in the assumptions ensuring this property a key role is played by the ratio cost power these ideas are illustrated in the context of detecting rectangles amidst clutter

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

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

asymptotic profile in selection mutation equations gauss versus cauchy distributions abstract in this paper we study the asymptotic large time behaviour of a selection mutation competition model for a population structured with respect to a phenotypic trait when the rate of mutation is very small we assume that the reproduction is asexual and that the mutations can be described by a linear integral operator we are interested in the interplay between the time variable t and the rate e of mutations we show that depending on 0 the limit e 0 with t e can lead to population number densities which are either gaussian like when is small or cauchy like when is large

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

context computation and optimal roc performance in hierarchical models it is widely recognized that human vision relies on contextual information typically arising from each of many levels of analysis local gradient information otherwise ambiguous is seen as part of a smooth contour or sharp angle in the context of an object s boundary or corner a stroke or degraded letter unreadable by itself contributes to the perception of a familiar word in the context of the surrounding strokes and letters the iconic dalmatian dog stays invisible until a multitude of clues about body parts and posture and figure and ground are coherently integrated context is always based on knowledge about the composition of parts that make up a whole as in the arrangement of strokes that make up a letter the arrangement of body parts that make up an animal or the poses and postures of individuals that make up a mob from this point of view the hierarchy of contextual information available to an observer derives from the compositional nature of the world being observed we will formulate this combinatorial viewpoint in terms of probability distributions and examine the computational implications whereas optimal recognition performance in this formulation is np complete we will give mathematical and experimental evidence that a properly orchestrated computational algorithm can achieve nearly optimal recognition within a feasible number of operations we will interpret the notions of bottom up and top down processing as steps in the staging of one such orchestration

explicit solutions for replicator mutator equations extinction versus acceleration we consider a class of nonlocal reaction diffusion problems referred to as replicator mutator equations in evolutionary genetics by using explicit changes of unknown function we show that they are equivalent to the heat equation and therefore compute their solution explicitly based on this we then prove that in the case of beneficial mutations in asexual populations solutions dramatically depend on the tails of the initial data they can be global become extinct in finite time or even be defined for no positive time in the first case we prove that solutions are accelerating and in many cases converge for large times to some universal gaussian profile this sheds light on the biological relevance of such models

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

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

coarse to fine face detection we study visual selection detect and roughly localize all instances of a generic object class such as a face in a greyscale scene measuring performance in terms of computation and false alarms our approach is sequential testing which is coarse to fine in both in the exploration of poses and the representation of objects all the tests are binary and indicate the presence or absence of loose spatial arrangements of oriented edge fragments starting from training examples we recursively find larger and larger arrangements which are decomposable which implies the probability of an arrangement appearing on an object decays slowly with its size detection means finding a sufficient number of arrangements of each size along a decreasing sequence of pose cells at the beginning the tests are simple and universal accommodating many poses simultaneously but the false alarm rate is relatively high eventually the tests are more discriminating but also more complex and dedicated to specific poses as a result the spatial distribution of processing is highly skewed and detection is rapid but at the expense of isolated false alarms which presumably could be eliminated with localized more intensive processing

binary linear quasi perfect codes are normal whether quasi perfect codes are normal is addressed let c be a code of length n dimension k covering radius r and minimal distance d it is proved that c is normal if d or 2r 1 hence all quasi perfect codes are normal consequently any n k r binary linear code with minimal distance d or 2r 1 is normal

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

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

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

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

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

travelling wave solutions for a non local evolutionary epidemic system abstract in this work we study the travelling wave solutions for a spatially distributed system of equations modelling the evolutionary epidemiology of plant pathogen interaction here the mutation process is described using a non local convolution operator in the phenotype space using dynamical system ideas coupled with refined estimates on the asymptotic behaviour of the profiles we prove that the wave solutions have a rather simple structure this analysis allows us to reduce the infinite dimensional travelling wave profile system of equations to a four dimensional ode system the latter is used to prove the existence of travelling wave solutions for any wave speed larger than a minimal wave speed c provided some parameters condition expressed using the principle eigenvalue of some integral operator it is also used to prove that any travelling wave solution connects two determined stationary states

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

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

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

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

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

orphans of the first order reed muller codes if c is a code an orphan is a coset that is not a descendant orphans arise naturally in the investigation of the covering radius case c has only even weight vectors and minimum distance of at least four cosets that are orphans are characterized and then the existence is proved of a family of orphans of first order reed muller codes r 1 m for m

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

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

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

the covering radius of extremal self dual code d11 and its application d11 constructed by conway and sloane is an extremal self dual 56 28 12 code with weight enumerator 1 8190y sup 12 622314y sup 16 11699688y sup 20 we find that the covering radius of d11 is 10 this shows that the inequality r c sub 56 spl les 10 cannot be improved any more where r c sub 56 is the covering radius of any extremal doubly even 56 28 12 code c sub 56 and we use it to get ten new 58 29 10 codes

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