asymptotical stability of probabilistic boolean networks with state delays this paper devotes to establishing a bridge between asymptotical stability of a probabilistic boolean network pbn and a solution to its induced equations which are induced from the pbn s transition matrix by utilizing the semitensor product technique routinely the dynamics of a pbn with coincident state delays can be equivalently converted into that of a higher dimensional pbn without delays subsequently several novel stability criteria are derived from the standpoint of equations solution the most significant finding is that a pbn is globally asymptotically stable at a predesignated one point distribution if and only if a vector obtained by adding 1 at the bottom of this distribution is the unique nonnegative solution to pbn s induced equations moreover the influence of coincident state delays on pbn s asymptotical stability is explicitly analyzed without consideration of the convergence rate consequently such bounded state delays are verified to have no impact on pbn s stability albeit delays are time varying based on this worthwhile observation the time computational complexity of the aforementioned approach can be reduced by removing delays directly furthermore this universal procedure is summarized to reduce the time complexity of some previous results in the literature to a certain extent two examples are employed to demonstrate the feasibility and effectiveness of the obtained theoretical results

on the quadratic spans of debruijn sequences the quadratic span of a periodic binary sequence is defined to be the length of the shortest quadratic feedback shift register fsr that generates it an algorithm for computing the quadratic span of a binary sequence is described the required increase in quadratic span is determined for the special case of when a discrepancy occurs in a linear fsr that generates an initial portion of a sequence the quadratic spans of binary debruijn sequences are investigated an upper bound for the quadratic span of a debruijn sequence of span n is given this bound is attained by the class of debruijn sequences obtained from m sequences it is easy to see that a lower bound is n 1 but a lower bound of n 2 is conjectured the distributions of quadratic spans of debruijn sequences of span 3 4 5 and 6 are presented

on the density of irreducible nfsrs let n be a positive integer an nfsr of n stages is called irreducible if the family of output sequences of any nfsr of stages less than n is not included in that of the nfsr in this paper we prove that the density of the irreducible nfsrs of n stages is larger than 0 39 this implies that it is expected to find an irreducible nfsr of n stages among three randomly chosen nfsrs of n stages

stability of polynomials under coefficient perturbation let the real polynomial a 0 x n a 1 x n 1 a n 1 x a n be stable and let the real numbers b k c k geq 0 0 leq k leq n be given we present a simple determinant criterion for finding the largest t 0 geq 0 such that the polynomial alpha 0 x n alpha 1 x n 1 alpha n 1 x alpha n is stable for all alpha k in a k b k t 0 a k c k t 0 cup a k 0 leq k leq n several further observations allow us to reduce the computational cost considerably

efficient composited de bruijn sequence generators a binary de bruijn sequence with period 2 n is a sequence in which every tuple of n bits occurs exactly once de bruijn sequence generators have randomness properties that make them attractive for pseudorandom number generators and as building blocks for stream ciphers unfortunately it is very difficult to find de bruijn sequence generators with long periods e g 2 128 and most known de bruijn sequence generators are computationally quite expensive in this article we present ocdeb k n and the first hardware implementation of de bruijn sequence generators ocdeb k n efficiently computes a composited de bruijn sequence where k levels of composition are added to a de bruijn sequence of period 2 n numerically ocdeb reduces the bit operations used for computing the feedback function significantly from theta k 2 nk to theta k log k log n furthermore it enables efficient parallelization and hardware retiming comprehensive result analysis is conducted for 65 nm asic technology for example ocdeb 32 32 has an area of 643 ge with 1 45 gbps performance and with parallelization it generates up to 25 4 gbps at the cost of 4 787 ge the area of ocdeb 512 32 generating a de bruijn sequence of period 2 544 is 7 304 ge and the performance is 1 25 gbps

computing the error linear complexity spectrum of a binary sequence of period 2 sup n binary sequences with high linear complexity are of interest in cryptography the linear complexity should remain high even when a small number of changes are made to the sequence the error linear complexity spectrum of a sequence reveals how the linear complexity of the sequence varies as an increasing number of the bits of the sequence are changed we present an algorithm which computes the error linear complexity for binary sequences of period spl lscr 2 sup n using o spl lscr log spl lscr sup 2 bit operations the algorithm generalizes both the games chan 1983 and stamp martin 1993 algorithms which compute the linear complexity and the k error linear complexity of a binary sequence of period spl lscr 2 sup n respectively we also discuss an application of an extension of our algorithm to decoding a class of linear subcodes of reed muller codes

de bruijn sequences adjacency graphs and cyclotomy we study the problem of constructing de bruijn sequences by joining cycles of linear feedback shift registers lfsrs with reducible characteristic polynomials the main difficulty for joining cycles is to find the location of conjugate pairs between cycles and the distribution of conjugate pairs in cycles is defined to be adjacency graphs let l x be a characteristic polynomial and l x l 1 x l 2 x cdots l r x be a decomposition of l x into pairwise co prime factors first we show a connection between the adjacency graph of mathrm fsr l x and the association graphs of mathrm fsr l i x 1 leq i leq r by this connection the problem of determining the adjacency graph of mathrm fsr l x is decomposed to the problem of determining the association graphs of mathrm fsr l i x 1 leq i leq r which is much easier to handle then we study the association graphs of lfsrs with irreducible characteristic polynomials and give a relationship between these association graphs and the cyclotomic numbers over finite fields at last as an application of these results we explicitly determine the adjacency graphs of some lfsrs and show that our results cover the previous ones

analysis and efficient implementations of a class of composited de bruijn sequences a binary de bruijn sequence is a sequence of period 2 n 2 n in which every binary n n tuple occurs exactly once in each period a de bruijn sequence has good randomness properties such as long period ideal tuple distribution and high linear complexity and can be generated by a nonlinear feedback shift register nlfsr finding an efficient nlfsr that can generate a de bruijn sequence with a long period is a significant challenge composited construction is a technique for constructing a de bruijn sequence of period 2 n k 2 n k by an nlfsr from a de bruijn sequence of period 2 n 2 n through a composition operation repeatedly applying k k times the goal of this article is to further investigate the composited construction of de bruijn sequences with efficient hardware implementations and determine randomness properties such as linear complexity our contributions in this article are as follows first we present a generalized construction of composited de bruijn sequences that is constructed by adding a combination of conjugate pairs of different lengths in the feedback function of the composited construction which results in generating a class of de bruijn sequences of size 2 k 2 k whereas the original composited construction can generate only two sequences second we investigate the linear complexity and the correlation property of the new class of de bruijn sequences we prove theoretically that the linear complexity of this class of de bruijn sequences is optimal or close to optimal interestingly we also prove that the linear complexities of all the sequences of this class are equal which strengthens etzion s conjecture jcta 1985 ieee it 1999 about the number of de bruijn sequences with equal linear complexity this is the first known construction of de bruijn sequences of an arbitrarily long period whose linear complexities are determined theoretically finally we implement our construction in hardware to demonstrate its practicality we synthesize our implementations for a 65 nm asic and a xilinx spartan fpga and present hardware areas and performances of de bruijn sequences of periods in the range of 2 160 2 160 to 2 1056 2 1056 for instance a class of de bruijn sequences of period 2 160 2 160 resp 2 288 2 288 can be implemented with an area of 3 43 resp 6 71 kges in 65 nm asic and 83 resp 229 slices in spartan6 fpga

average and worst case number of nodes in decision diagrams of symmetric multiple valued functions we derive the average and worst case number of nodes in decision diagrams of r valued symmetric functions of n variables we show that for large n both numbers approach n sup r rl for binary decision diagrams r 2 we compute the distribution of the number of functions on n variables with a specified number of nodes subclasses of symmetric functions appear as features in this distribution for example voting functions are noted as having an average of n sup 2 6 nodes for large n compared to n sup 2 2 for general binary symmetric functions

feedback reconstruction and implementations of pseudorandom number generators from composited de bruijn sequences a binary de bruijn sequence of order n is a sequence of zeros and ones of period 2 n that contains every binary n tuple exactly once in a period of the sequence a composited construction of a de bruijn sequence is a construction of a nonlinear feedback shift register nlfsr that generates a de bruijn sequence where the composited feedback function of the nlfsr is the sum of a feedback function with k th order composition and a sum of k 1 product of sum terms the goals of this article are to perform a profound analysis of composited de bruijn sequences for use in cryptography and find an efficient implementation of the composited feedback function we first determine the lower bound of the linear complexity of a composited de bruijn sequence and then conduct a profound analysis on the composited construction by introducing the notion of the higher order d morphic preimages of a binary sequence our analysis aims at the reconstruction of a composited de bruijn sequence from a segment known as k th order d morphic order n de bruijn preimages n k dmdps of length 2 n k and k th order d morphic order n m sequence preimages n k dmmps of length 2n k for a nonlinearly and linearly generated composited de bruijn sequence respectively we also provide the success probability of finding an n k dmmp dmdp from a composited de bruijn sequence for the reconstruction furthermore we develop a new iterative technique with its parallel extension for computing the feedback function and the new technique is faster than other known techniques for producing de bruijn sequences of long period in addition we present three instances of composited de bruijn sequences of period 2 64 together with their software implementations and performances

computing generalized de bruijn sequences abstract de bruijn sequences of order n represent the set a n of all words of length n over a given alphabet a in the sense that they contain occurrences of each of these words they actually contain exactly one occurrence of each of these words recently the computational problem of representing subsets of a n by partial words which are sequences that may have holes or don t care symbols that match each letter of a was considered and shown to be in np however membership in p remained open in this paper we show that deciding if a subset s of a n is representable by a partial word can be done in polynomial time with respect to the size n s of the input we also describe a polynomial time algorithm that determines all integers h for representation by partial words with exactly h holes moreover our algorithms construct representation words our approach is graph theoretical

robust stability of polynomials with multilinearly dependent coefficient perturbations analysis of the robust stability of a polynomial with multilinearly dependent coefficient perturbations is presented in this note some sufficient conditions for forming a convex polygon with the value set of the polynomials with multilinearly dependent coefficient perturbations are obtained a zero exclusion algorithm is then given to determine the d stability of such polynomials the well known kharitonov s theorem and the edge theorem for stability analysis can be included as special eases of the authors conclusions

nonlinear shift registers that produce all vectors of weight or t it is demonstrated that all binary vectors of length and of a hamming weight that is lt can be cyclically generated by an appropriately designed nonlinear shift register

on stabilization and set stabilization of multivalued logical systems the stabilization and set stabilization of a multivalued logical system are investigated in this paper by using pinning control first new algorithms are presented to redesign the transition matrix of the multivalued logical system such that the system is stable or set stable then by using reconstruction theory of a k valued logical system the pinning nodes are selected in addition the transition matrix of the pinning control is given by solving some logical matrix equations then a pinning control design algorithm is provided finally an example is employed to illustrate the proposed control design procedure

fast time stability of temporal boolean networks in real systems most of the biological functionalities come from the fact that the connections are not active all the time based on the fact temporal boolean networks tbns are proposed in this paper and the fast time stability is analyzed via semi tensor product stp of matrices and incidence matrices first the algebraic form of a tbn is obtained based on the stp method and one necessary and sufficient condition for global fast time stability is presented moreover incidence matrices are used to obtain several sufficient conditions which reduce the computational complexity from mathcal o n2 n exponential type to mathcal o n 4 polynomial type compared with the stp method in addition the global fast time stabilization of tbns is considered and pinning controllers are designed based on the neighbors of controlled nodes rather than all the nodes finally the local fast time stability of tbns is considered based on the incidence matrices as well several examples are provided to illustrate the effectiveness of the obtained results

simple and robust binary self location patterns a simple method to generate a 2 d binary grid pattern which allows for absolute and accurate self location in a finite planar region is proposed the pattern encodes position information in a local way so that reading a small number of its black or white pixels at any place provides sufficient data from which the location can be decoded both efficiently and robustly

using the structure of subfields in the construction of goppa codes and extended goppa codes it is shown that for some location sets and some integral multiple power galois fields that some new general subclasses of goppa codes may be defined which have improved lower bounds to code dimension and minimum distance compared with ordinary goppa codes some previously published results are shown to be particular cases of these general subclasses of codes a new subclass of reversible goppa codes is also presented examples of code construction for these subclasses demonstrating the improved code parameters are presented for both non binary and binary codes

goppa and related codes invariant under a prescribed permutation we explain how to construct goppa codes expurgated or extruded goppa codes invariant under a permutation induced by an element of the projective semilinear group p spl gamma l 2 gf p sup m this follows from the knowledge of semilinear automorphism groups of generalized reed solomon codes we obtain very good codes some of them have parameters which reach the values given in the brouwer table see handbook of coding theory v s pless and w c huffman eds amsterdam the netherlands elsevier vol 1 ch 4 1998 moreover it is easy to construct quasi cyclic goppa or related codes often we obtain parameters unknown for quasi cyclic codes

output regulation of boolean control networks this note investigates the output regulation of boolean control networks bcns by using the semi tensor product of matrices and presents a number of new results firstly based on an algebraic state space representation of bcns a necessary and sufficient condition is presented for the solvability of the output regulation problem secondly an effective method is proposed for the control design of the output regulation problem by using an augmented system approach the study of an illustrative example shows that the obtained new results are effective in dealing with the output regulation of bcns

a simple proof of kharitonov s theorem an alternative proof of kharitonov s theorem on the stability of linear time invariant continuous systems under parameter variations is presented

the adjacency graphs of lfsrs with primitive like characteristic polynomials we consider the adjacency graphs of the linear feedback shift registers lfsrs with characteristic polynomials of the form l x p x where l x is a polynomial of small degree and p x is a primitive polynomial it is shown that their adjacency graphs are closely related to the association graph of l x and the cyclotomic numbers over finite fields by using this connection we give a unified method to determine their adjacency graphs as an application of the method we explicitly calculate the adjacency graphs of lfsrs with the characteristic polynomials of the form 1 x x 3 x 4 p x and construct a large class of de bruijn sequences from them

constructions for perfect maps and pseudorandom arrays a construction of perfect maps i e periodic r v binary arrays in which each n m binary matrix appears exactly once is given a similar construction leads to arrays in which only the zero n m matrix does not appear and to a construction in which only a few n m binary matrices do not appear a generalization to the nonbinary case is given the constructions involve an interesting problem in shift register theory the solution is given for almost all the case of this problem

on the computation of solutions of systems of interval polynomial equations systems of algebraic equations with interval coefficients are very common in several areas of engineering sciences the computation of the solution of such systems is a central problem when the characterization of the variables related by such systems is desired in this paper we characterize the solution of systems of algebraic equations with real interval coefficients the characterization is obtained considering the approach introduced in j comput appl math 136 2001 271

brief paper controllability of boolean control networks via the perron frobenius theory boolean control networks bcns are recently attracting considerable interest as computational models for genetic and cellular networks addressing control theoretic problems in bcns may lead to a better understanding of the intrinsic control in biological systems as well as to developing suitable protocols for manipulating biological systems using exogenous inputs we introduce two definitions for controllability of a bcn and show that a necessary and sufficient condition for each form of controllability is that a certain nonnegative matrix is irreducible or primitive respectively our analysis is based on a result that may be of independent interest namely a simple algebraic formula for the number of different control sequences that steer a bcn between given initial and final states in a given number of time steps while avoiding a set of forbidden states

an efficient algorithm for the generation of debruijn cycles an algorithm is presented for the generation of binary debruijn sequences using feedback shift registers fsrs the algorithm is based on the principle of cycle joining through which all the cycles in the cycle structure of a fsr can be joined together to one complete cycle thereby producing a debruijn sequence of period 2 sup n where n is the length of the fsr by a proper choice of the feedback function of the fsr o 2 sup 2n sup log2n debruijn sequences of period 2 sup n can be generated requiring only 3n bits of storage and at most 4n fsr shifts for the generation of the next bit in the sequence

a large class of nonlinear shift register sequences corresp the cycle structure of a binary linear shift register with connection polynomial g x 1 x 2 g x where g x is a primitive polynomial of degree m 2 over gf 2 is used to give several construction techniques for generation of shift register sequences of length l 2 m 4 it is shown that a class of nonlinear debruijn cycles where the number of elements is proportional to 2 5m can be constructed the obtained cycles can be generated by simple m stage nonlinear feedback shift registers

driven stability of nonlinear feedback shift registers with inputs driven stable nonlinear feedback shift registers nfsrs with inputs are not only able to limit error propagations in convolutional decoders but also helpful to analyze the period properties of sequences generated by a cascade connection of nfsrs in stream ciphers an nfsr is driven stable if and only if the reachable set is a subset of the basin due to lack of efficient algebraic tools the driven stability of nfsrs with inputs has been much less studied this paper continues to address this research using a boolean control network approach viewing an nfsr with input as a boolean control network we first give its boolean control network representation which is characterized with a state transition matrix some properties of the state transition matrix are then provided based on these explicit forms are given for the reachable set and the set of basin two algorithms for obtaining both the sets are provided as well compared with the exhaustive search and the existing state operator method the boolean control network approach requires lower computational complexity for those nfsrs with their stages greater than 1

the number of nonlinear shift registers that produce all vectors of weight or t it has been shown that it is possible to generate a cycle on a nonlinear shift register to contain all vectors of length n and hamming weight

controllability and observability of boolean control networks the controllability and observability of boolean control networks are investigated after a brief review on converting a logic dynamics to a discrete time linear dynamics with a transition matrix some formulas are obtained for retrieving network and its logical dynamic equations from this network transition matrix based on the discrete time dynamics the controllability via two kinds of inputs is revealed by providing the corresponding reachable sets precisely then the problem of observability is also solved by giving necessary and sufficient conditions

the properties of a class of linear fsrs and their applications to the construction of nonlinear fsrs in this paper the cycle structure and adjacency graphs of a class of linear feedback shift registers lfsrs are determined by recursively applying the d morphism to the maximum length lfsrs and representing the cycles by generating functions a new family of maximum length nonlinear feedback shift registers nfsrs are proposed based on the properties of these lfsrs the number of nfsrs in the proposed family is also considered

a survey of full length nonlinear shift register cycle algorithms shift registers have been used to generate sequences of 0 s and 1 s for over thirty years a wide variety of applications has been made of these sequences principally communications have made use of the sequences generated one particular class of shift register sequences for which applications exist is the full length nonlinear shift register sequences these sequences are periodic and of length 2 n and all 2 n different binary n tuples appear exactly one time in a periodic portion of the sequence in this paper we discuss various algorithms which have been suggested for generating these sequences

a class of multilinear uncertain polynomials to which the edge theorem is applicable this paper gives a class of multilinear uncertain polynomials to which the edge theorem is applicable in the paper a notion of symmetric uncertainty structure is introduced the main result is as follows if a multilinear uncertain polynomial has a symmetric uncertainty structure and the uncertain parameters are nonoverlapping then the corresponding value sets will be convex polygons this means that the use of the zero exclusion condition to test the robust stability of such uncertain polynomials is feasible since we need only to calculate the extreme points of the convex polygons furthermore a version of the edge theorem can be used to test the robust stability of such uncertain polynomials

the number of cross join pairs in maximum length linear sequences it has been conjectured by t chang et al 1990 that the number of cross join pairs in a maximum length linear sequence equals 2 sup n 1 1 2 sup n 1 2 6 a maximum length linear sequence an m sequence of length 2 sup n 1 is a binary sequence which satisfies a linear recurrence whose characteristic polynomial is primitive of degree n the number of primitive polynomials is given by phi 2 sup n 1 n where phi is euler s phi function a proof of the conjecture is given

linear spans of modified de bruijn sequences order n modified de bruijn sequences are created by removing a single zero from the longest run of zeros in period 2 sup n de bruijn sequences the m sequences are then the natural undisguised linear subset of modified de bruijn sequences theorems are given on the linear spans of modified de bruijn sequences and data are presented for 4

a generalized recursive construction for de bruijn sequences this paper is concerned with the construction of de bruijn sequences of span n binary sequences of period 2 n in which every binary n tuple appears as some n consecutive terms in one period of the sequence constructions in the literature are based on maximum length linear sequences algorithms which start from scratch and recursive methods which start with a single de bruijn sequence of span n and produce one of span n 1 we give a more general recursive construction which takes two de brnijn sequences of span n and produces a de bruijn sequence of span n 1 in addition for a special case of the construction the complexity shortest linear recursion that generates the sequence of the resulting sequence is determined in terms of the complexities of the ingredient sequences in particular de bruijn sequences of span n 1 with maximum complexity 2 n 1 1 are obtained from maximum complexity sequences of span n reverse complementary de bruijn sequences are also considered

a dual result to kharitonov s theorem a dual problem to kharitonov s problem involving a diamond instead of a rectangle is considered the results show that a family of polynomials with coefficients varying in the diamond is strictly hurwitz if and only if eight one dimensional exposed edges of the diamond are strictly hurwitz

an efficient implementation of the d homomorphism for generation of de bruijn sequences an efficient implementation of the d homomorphism for generating de bruijn 1946 sequences is presented the number of exclusive or operations required to generate the next bit for de bruijn sequences of order n from a de bruijn function of order k is shown to be approximately k 2 sup w n k 1 where w r is the number of one s in the binary representation of r therefore the number of required operations can be reduced to k if the de bruijn function is selected appropriately as an application the stream cipher is indicated

constructing de bruijn sequences by concatenating smaller universal cycles abstract we present sufficient conditions for when an ordering of universal cycles 1 2 m for disjoint sets s 1 s 2 s m can be concatenated together to obtain a universal cycle for s s 1 s 2 s m when s is the set of all k ary strings of length n the result of such a successful construction is a de bruijn sequence our conditions are applied to generalize two previously known de bruijn sequence constructions and then they are applied to develop three new de bruijn sequence constructions

finding matching initial states for equivalent nlfsrs in the fibonacci and the galois configurations the fibonacci and the galois configurations of nonlinear feedback shift registers nlfsrs are considered in the former the feedback is applied to the input bit of the shift register only in the latter the feedback can potentially be applied to every bit the sufficient conditions for equivalence of nlfsrs in the fibonacci and the galois configurations have been formulated previously the equivalent nlfsrs in different configurations normally have to be initialized to different states to generate the same output sequences the mapping between the initial states of two equivalent nlfsrs in the fibonacci and the galois configurations is derived in this paper

algorithms for the generation of full length shift register sequences two algorithms are presented for the generation of full length shift register cycles also referred to as de bruijn sequences the first algorithm generates 2 k cdot g n k full cycles of length 2 n using 3n k cdot g n k bits of storage where k is a free parameter in the range 1 leq k leq 2 n 4 2 and g n k is of the order of n 2 log k the second algorithm generates about 2 n 2 4 full cycles of length 2 n using about n 2 2 bits of storage in both algorithms the time required to produce the next bit from the last n bits is close to n a possible application to the construction of stream ciphers is indicated

construction of de bruijn sequences of minimal complexity it is well known that the linear complexity of a de bruijn sequence s of length 2 n is bounded below by 2 n 1 n for n geq 3 it is shown that this lower bound is attainable for all n

linear complexity of de bruijn sequences old and new results the linear complexity of a de bruijn sequence is the degree of the shortest linear recursion which generates the sequence it is well known that the complexity of a binary de bruijn sequence of length 2 sup n is bounded below by 2 sup n 1 n and above by 2 sup n 1 for n spl ges 3 we briefly survey the known knowledge in this area some new results are also presented in particular it is shown that for each interval of length 2 sup log n 1 in the above range there exist binary de bruijn sequences of length 2 sup n with linear complexity in the interval

a class of de bruijn sequences in this paper a class of linear feedback shift registers lfsrs with characteristic polynomial 1 x 3 p x is discussed where p x is a primitive polynomial of degree n 2 the cycle structure and adjacency graphs of the lfsrs are determined a new class of de bruijn sequences is constructed from these lfsrs and the number of de bruijn sequences in the class is also considered to illustrate the efficiency of constructing de bruijn sequences from these lfsrs an algorithm for producing some corresponding maximum length nonlinear feedback shift registers with time and memory complexity o n is also proposed

built in test for circuits with scan based on reseeding of multiple polynomial linear feedback shift registers we propose a new scheme for built in test bit that uses multiple polynomial linear feedback shift registers mp lfsr s the same mp lfsr that generates random patterns to cover easy to test faults is loaded with seeds to generate deterministic vectors for difficult to test faults the seeds are obtained by solving systems of linear equations involving the seed variables for the positions where the test cubes have specified values we demonstrate that mp lfsr s produce sequences with significantly reduced probability of linear dependence compared to single polynomial lfsr s we present a general method to determine the probability of encoding as a function of the number of specified bits in the test cube the length of the lfsr and the number of polynomials theoretical analysis and experiments show that the probability of encoding a test cube with s specified bits in an s stage lfsr with 16 polynomials is 1 10 sup 6 we then present the new bit scheme that allows for an efficient encoding of the entire test set here the seeds are grouped according to the polynomial they use and an implicit polynomial identification reduces the number of extra bits per seed to one bit the paper also shows methods of processing the entire test set consisting of test cubes with varied number of specified bits experimental results show the tradeoffs between test data storage and test application time while maintaining complete fault coverage

a zoo of computable binary normal sequences historically there has been a virtual absence of constructive methods to produce broad classes of certifiably random infinite sequences despite considerable interest in this endeavor previously we proved a theorem that yielded explicit algorithms to produce diverse sets of normal numbers reasonable candidates for random sequences given their limiting equidistribution of subblocks of all lengths herein we develop this algorithmic approach much further systematizing the normal number generation process in several ways we construct delineated distinct sets of normal numbers classified by the extent to which initial segments deviate from maximal irregularity with virtually any allowable specified rate of convergence to 0 of this deviation encompassing arbitrarily fast and slow rates and accommodating asymmetric behavior above or below a centered median as a corollary we provide an explicit construction of a normal number that satisfies the law of the iterated logarithm we also produce distinct families of biased normal numbers with virtually any specified rate of convergence of the bias to 0 this latter theory is in part motivated by the remarkable observation that the binary version of champernowne s number which is also normal is biased any initial segment has more 1s than 0s finally we construct an interesting normal sequence with arbitrarily fast convergence to equidistribution of singleton blocks yet arbitrarily slow convergence of pairs which has profound implications both for probability theory and for metrics to evaluate the near randomness of sequences

a scalable method for constructing galois nlfsrs with period 2 n 1 using cross join pairs a method for constructing n stage galois nlfsrs with period 2n 1 from n stage maximum length lfsrs is presented nonlinearity is introduced into state cycles by adding a nonlinear boolean function to the feedback polynomial of the lfsr each assignment of variables for which this function evaluates to 1 acts as a crossing point for the lfsr state cycle the effect of nonlinearity is cancelled and state cycles are joined back by adding a copy of the same function to a later stage of the register the presented method requires no extra time steps and it has a smaller area overhead compared to the previous approaches based on cross join pairs it is feasible for large n

on robust control invariance of boolean control networks this paper investigates the robust control invariance of boolean control networks bcns via the semi tensor product of matrices firstly based on an algebraic state space representation of bcns two necessary and sufficient conditions are presented to check whether or not a given set is a robust control invariant set under a given state feedback controller secondly by defining a series of suitable sets all possible state feedback gain matrices under which a given set is a robust control invariant set are characterized an illustrative example is presented to demonstrate the obtained new results

a recipe for randomness despite many diverse theories that address closely related themes e g probability theory algorithmic complexity cryptoanalysis and pseudorandom number generation a near void remains in constructive methods certified to yield the desired random output herein we provide explicit techniques to produce broad sets of both highly irregular finite and normal infinite sequences based on constructions and properties derived from approximate entropy apen a computable formulation of sequential irregularity furthermore for infinite sequences we considerably refine normality by providing methods for constructing diverse classes of normal numbers classified by the extent to which initial segments deviate from maximal irregularity

aperiodic and semi periodic perfect maps paterson see ibid vol 40 p 743 1993 has shown that the trivial necessary conditions are sufficient for the existence of a binary perfect map these periodic structures can be transformed very simply into corresponding aperiodic and semi periodic perfect maps however aperiodic and semi periodic perfect maps can exist for parameter sets for which the corresponding periodic perfect maps cannot in this paper it is shown by construction that binary aperiodic and semi periodic perfect maps exist for all possible parameter sets

a state of the art in structured light patterns for surface profilometry shape reconstruction using coded structured light is considered one of the most reliable techniques to recover object surfaces having a calibrated projector camera pair a light pattern is projected onto the scene and imaged by the camera correspondences between projected and recovered patterns are found and used to extract 3d surface information this paper presents an up to date review and a new classification of the existing techniques some of these techniques have been implemented and compared obtaining both qualitative and quantitative results the advantages and drawbacks of the different patterns and their potentials are discussed

a survey of combinatorial gray codes the term combinatorial gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified small way this notion generalizes the classical binary reflected gray code scheme for listing n bit binary numbers so that successive numbers differ in exactly one bit position as well as work in the 1960s and 1970s on minimal change listings for other combinatorial families including permutations and combinations the area of combinatorial gray codes was popularized by herbert wilf in his invited address at the siam conference on discrete mathematics in 1988 and his subsequent siam monograph combinatorial algorithms an update 1989 in which he posed some open problems and variations on the theme this resulted in much recent activity in the area and most of the problems posed by wilf are now solved in this paper we survey the area of combinatorial gray codes describe recent results variations and trends and highlight some open problems

disordered ground states for classical discrete state problems in one dimension it is known that one dimensional lattice problems with a discrete finite set of states per site generically have periodic ground states gss we consider slightly less generic cases in which the hamiltonian is constrained by either spin s or spatial i inversion symmetry or both we show that such constraints give rise to the possibility ofdisordered gss over a finite fraction of the coupling parameter space that is without invoking any nongeneric fine tuning of coupling constants beyond that arising from symmetry we find that such disordered gss can arise for many values of the number of statesk at each site and the ranger of the interaction the ising k 2 case is the least prone to disorder i symmetry allows for disordered gss without fine tuning only forr 5 whiles symmetry never gives rise to disordered gss

a successor rule framework for constructing k ary de bruijn sequences and universal cycles we present a simple framework for constructing k ary de bruijn sequences and more generally universal cycles via successor rules the framework is based on the often used method of joining disjoint cycles it generalizes several previously known de bruijn sequence constructions based on the pure cycling register and is applied to derive a new construction that is perhaps the simplest of all successors furthermore it generalizes an algorithm to construct binary de bruijn sequences based on any arbitrary nonsingular feedback function the framework is applied to derive and prove the correctness of successors to efficiently construct 1 universal cycles for k ary strings of length n whose weight is bounded by some w and 2 universal cycles for permutations it has also been subsequently applied to find the first universal cycle constructions for weak orders

higher order dangers and precisely constructed taxa in models of randomness the certification construction and delineation of individual infinite length random sequences have been longstanding yet incompletely resolved problems we address this topic via the study of normal numbers which often have been viewed as reasonable proxies for randomness given their limiting equidistribution of subblocks of all lengths however limitations arise within this perspective first we explicitly construct a normal number that satisfies the law of the iterated logarithm yet exhibits pairwise bias toward repeated values rendering it inappropriate for any collection of random numbers accordingly we deduce that the evaluation of higher order block dynamics even beyond limiting equidistribution and fluctuational typicality is imperative in proper evaluation of sequential randomness second we develop several criteria motivated by classical theorems for symmetric random walks which lead to algorithms for generating normal numbers that satisfy a variety of attributes for the series of initial partial sums including rates of sign changes patterns of return times to 0 and the extent of fairness of the sequence such characteristics generally are unaddressed in most evaluations of randomness more broadly we can differentiate normal numbers both on the basis of multiple distinct qualitative attributes and quantitatively via a spectrum of rates within each attribute furthermore we exhibit a toolkit of techniques to construct normal sequences that realize diverse a priori specifications including profound biases overall we elucidate the vast diversity within the category of normal sequences

invariance of the strict hurwitz property for polynomials with perturbed coefficients given a strictly hurwitz polynomial f lambda lambda n a n 1 lambda n 1 a n 2 lambda n 2 a 1 lambda a 0 it is of interest to know how much the coefficients a i can be perturbed while simultaneously preserving the strict hurwitz property for systems with n leq 4 maximal intervals of the a i are given in a recent paper by guiver and bose 1 in this note a theorem of kharitonov is exploited to obtain a general result for polynomials of any degree

clues to the hidden nature of de bruijn sequences abstract order n de bruijn sequences are the period 2 n binary sequences produced by an n stage feedback shift register theoretical results are summarized and data are presented for feedback functions generator polynomials linear spans and autocorrelation properties of modified de bruijn sequences

a generalization of kharitonov s four polynomial concept for robust stability problems with linearly dependent coefficient perturbations kharitonov s four polynomial concept is generalized to the case of linearly dependent coefficient perturbations and more general zero location regions to this end a specially constructed scalar function of a scalar variable is instrumental to the robustness analysis the present work is motivated by two fundamental limitations of kharitonov s theorem namely 1 the theorem only applies to polynomials with independent coefficient perturbations and 2 it only applies to zeros in the left hand plane

a method for constructing decodable de bruijn sequences we present two related methods of construction for de bruijn 1946 sequences both based on interleaving smaller de bruijn sequences sequences obtained using these construction methods have the advantage that they can be decoded very efficiently i e the position within the sequence of any particular window can be found very simply sequences with simple decoding algorithms are of considerable practical importance in position location applications

pseudonoise sequences based on algebraic feedback shift registers over the past half century various statistical properties of pseudorandom sequences have played important roles in a variety of applications among these properties are golomb s randomness conditions r1 balance r2 run property and r3 ideal autocorrelations as well as the closely related properties r4 shift and add and r5 de bruin uniform distribution of subblocks the purpose of this paper is to describe the relationships among these conditions and to introduce a new method for generating sequences with all these properties using algebraic feedback shift registers

a transformation from the fibonacci to the galois nlfsrs conventional nonlinear feedback shift registers nlfsrs use the fibonacci configuration in which the feedback is applied to the last bit only in this paper we show how to transform a fibonacci nlfsr into an equivalent nlfsr in the galois configuration in which the feedback can be applied to every bit such a transformation can potentially reduce the depth of the circuits implementing feedback functions thus decreasing the propagation time and increasing the throughput

robust hurwitz stability via sign definite decomposition abstract in this paper we first consider the problem of determining the robust positivity of a real function f x as the real vector x varies over a box x r l we show that it is sufficient to check a finite number of specially constructed points this is accomplished by using some results on sign definite decomposition we then apply this result to determine the robust hurwitz stability of a family of complex polynomials whose coefficients are polynomial functions of the parameters of interest we develop an eight polynomial vertex stability test that is a sufficient condition of hurwitz stability of the family this test reduces to kharitonov s well known result for the special case where the parameters are just the polynomial coefficients in this case the result is tight this test can be recursively and modularly used to construct an approximation of arbitrary accuracy to the actual stabilizing set the result is illustrated by examples

on minimum period of nonlinear feedback shift registers in grain like structure grain is one of three hardware oriented finalists of the estream project a nonlinear feedback shift register nfsr in grain like structure is a cascade connection of a linear feedback shift register lfsr into an nfsr in which the characteristic polynomial of the lfsr is primitive and the feedback function of the nfsr is nonsingular in 2011 hu and gong pointed out that the period of the sequence generated by an nfsr in grain like structure is a multiple of the period of the sequence generated by its lfsr if the initial state of the lfsr is nonzero meanwhile they proposed an open problem for fixed feedback functions of an nfsr and an lfsr determine whether the sequences generated by the nfsr in grain like structure can achieve the minimum period i e the period of the lfsr when the initial state of the lfsr is nonzero and if they can achieve provide at least one pair of the initial states of the nfsr and lfsr clearly from a security point of view it is not preferable if the sequences generated by an nfsr in grain like structure achieve the minimum period this paper converts the open problem into a problem of solving an integer equation with respect to two unknown integers that uniquely correspond to the initial states of the nfsr and lfsr by viewing the nfsr as a boolean control network based on the integer equation this paper shows that for any given initial state of an n stage nfsr and any given nonzero initial state of an m stage lfsr the probability that the sequence generated by the nfsr in grain like structure achieves the minimum period 2 m 1 is at most 2 n this implies that the probability of the cascade connection used in grain achieving the minimum period is very small

a fast algorithm for determining the complexity of a binary sequence with period 2 n corresp the complexity of a periodic sequence s is defined to be the least number of stages in a linear feedback shift register that generates s

further results on the decomposition of an nfsr into the cascade connection of an nfsr into an lfsr nonlinear feedback shift registers nfsrs are widely used in stream cipher design as building blocks in this paper we study the problem of decomposing an nfsr into the cascade connection of an nfsr into a linear feedback shift register lfsr which is a kind of concatenation of an nfsr and lfsr a necessary and sufficient condition for such decomposition is provided and other algebraic properties about such decomposition are also studied based on these theoretical results a binary decision diagram bdd based algorithm for such decomposition is proposed compared with the previous algorithm proposed by ma et al our algorithm can find more accurate candidate lfsr and the algebraic properties presented in this paper guarantee that the memory requirement during our verification is linear in the size of the bdd of the nfsrs characteristic function

fault tolerant ring embedding in de bruijn networks a method of embedding a ring in a d ary de bruijn multiprocessor network in the event of multiple node processor failures is presented in particular the algorithm guarantees that a 2 sup n n 1 node ring will be found in a binary de bruijn network with a single faulty node where 2 sup n is the total number of nodes in the network it is also shown that a d sup n 1 node ring can always be found in the presence of d 1 link failures when d is a prime power and the network contains d sup n nodes the latter is accomplished by constructing d 1 edge disjoint cycles each of length d sup n 1 a modification of the graph that allows it to admit a hamiltonian cycle in the event of d 1 edge failures is also discussed

construction of de bruijn sequences from lfsrs with reducible characteristic polynomials in this paper a family of new de bruijn sequences is proposed through the construction of maximum length nonlinear feedback shift registers nfsrs let k be a positive integer and p 0 x p 1 x ldots p k x be the primitive polynomials in mathbb f 2 x with their degrees strictly increasing and pairwise coprime we determine the cycle structure and adjacency graphs of linear feedback shift registers lfsrs with characteristic polynomial q x prod nolimits i 0 k p i x in the case that p 0 x 1 x an algorithm is proposed to produce maximum length nfsrs from these lfsrs and it is shown that the algorithm can generate o 2 2 k 1 n n stage maximum length nfsrs with memory complexity o 2 k kn and time complexity o 2 n d k kn where n and d k are the degrees of q x and p k x respectively finally we illustrate the proposed algorithm in the case of k 2 in this case we prove that for any integer n geq 8 the algorithm can produce n stage maximum length nfsrs with time complexity as low as o n rm log log n

antipersistent binary time series completely antipersistent binary time series are sequences in which every time that an n bit string appears the sequence is continued with a different bit than that after the last occurrence of this dynamics is phrased in terms of a walk on a debruijn graph and properties of transients and cycles are studied the predictability of the generated time series for an observer who sees a longer or shorter time window is investigated also for sequences that are not completely antipersistent

subclass of cyclic goppa codes the subclass of cyclic goppa codes is proposed it is proved that this subclass contains all cyclic codes of length n qm 1 with generator polynomial g x g i g j 0 i 0 1 s1 j 1 s2 n 1 gf q2m

the transformation between the galois nlfsrs and the fibonacci nlfsrs via semi tensor product of matrices in this paper a novel way is proposed to investigate the transformation between galois nonlinear feedback shift registers nlfsrs and fibonacci nlfsrs first the galois nlfsrs and fibonacci nlfsrs are regarded as two boolean networks bns and the corresponding algebraic equations are obtained based on semi tensor product stp of matrices then the definition of absolute equivalence is given to investigate the transformation between these two kinds of nlfsrs furthermore some interesting results are presented to achieve the transformation finally an example is provided to illustrate the effectiveness of obtained results

measuring the clustering effect of bwt via rle abstract the burrows wheeler transform bwt is a reversible transformation on which are based several text compressors and many other tools used in bioinformatics and computational biology the bwt is not actually a compressor but a transformation that performs a context dependent permutation of the letters of the input text that often create runs of equal letters clusters longer than the ones in the original text usually referred to as the clustering effect of bwt in particular from a combinatorial point of view great attention has been given to the case in which the bwt produces the fewest number of clusters cf 5 16 21 23 in this paper we are concerned about the cases when the clustering effect of the bwt is not achieved for this purpose we introduce a complexity measure that counts the number of equal letter runs of a word this measure highlights that there exist many words for which bwt gives an un clustering effect that is bwt produce a great number of short clusters more in general we show that the application of bwt to any word at worst doubles the number of equal letter runs moreover we prove that this bound is tight by exhibiting some families of words where such upper bound is always reached we also prove that for binary words the case in which the bwt produces the maximal number of clusters is related to the very well known artin s conjecture on primitive roots the study of some combinatorial properties underlying this transformation could be useful for improving indexing and compression strategies

computing minimum length representations of sets of words of uniform length motivated by text compression the problem of representing sets of words of uniform length by partial words i e sequences that may have some wildcard characters or holes was recently considered and shown to be in p polynomial time algorithms that construct representations were described using graph theoretical approaches as more holes are allowed representations shrink and if representation is given the set can be reconstructed we further study this problem by determining for a binary alphabet the largest possible value of the size of a set of partial words that is important in deciding the representability of a given set s of words of uniform length this largest value surprisingly is i 0 s 1 2 i where i is the number of ones in the binary representation of i a well studied digital sum and it is achieved when the cardinality of s is a power of two we show that circular representability is in p and that unlike non circular representability it is easy to decide we also consider the problem of computing minimum length representation circular total words those without holes and reduce it to a cost flow network problem

perfect maps given positive integers r s u and spl upsi an r s u spl upsi perfect map pm is defined to be a periodic r spl times s binary array in which every u spl times spl upsi binary array appears exactly once as a periodic subarray perfect maps are the natural extension of the de bruijn sequences to two dimensions in the paper the existence question for perfect maps is settled by giving constructions for perfect maps for all parameter sets subject to certain simple necessary conditions extensive use is made of previously known constructions by finding new conditions which guarantee their repeated application these conditions are expressed as bounds on the linear complexities of the periodic sequences formed from the rows and columns of perfect maps

on the distribution of de bruijn sequences of given complexity the distribution gamma c n of de bruijn sequences of order n and linear complexity c is investigated it is shown that for n geq 4 gamma 2 n 1 n equiv 0 pmod 8 and for k geq 3 gamma 2 2k 1 2k equiv 0 pmod l6 it is also shown that gamma c n equiv 0 pmod 4 for all c and n geq 3 such that cn is even

on the linear complexity of feedback registers sequences generated by arbitrary feedback registers not necessarily feedback shift registers with arbitrary feedforward functions are studied the definition of linear complexity of a sequence is generalized to the notions of strong and weak linear complexity of feedback registers a technique for finding upper bounds for the strong linear complexities of such registers is developed this technique is applied to several classes of registers it is shown that a feedback shift register in which the feedback function is of the form x sub 1 h x sub 2 x sub n can generate long periodic sequences with high linear complexities only if its linear and quadratic terms have certain specific forms

constructions of snake in the box codes for rank modulation snake in the box code is a gray code which is capable of detecting a single error gray codes are important in the context of the rank modulation scheme which was suggested recently for representing information in flash memories for a gray code in this scheme the codewords are permutations two consecutive codewords are obtained using the push to the top operation and distance measure is defined on permutations in this paper the kendall s metric is used as the distance measure we present a general method for constructing such gray codes we apply the method recursively to obtain a snake of length m2n 1 2n 1 2n 1 m2n 1 for permutations of s2n 1 from a snake of length m2n 1 for permutations of s2n 1 thus we have lim n m2n 1 s2n 1 0 4338 improving on the previous known ratio of lim n 1 n using the general method we also present a direct construction this direct construction is based on necklaces and it might yield snakes of length 2n 1 2 2n 1 for permutations of s2n 1 the direct construction was applied successfully for s7 and s9 and hence lim n m2n 1 s2n 1 0 4743

cryptographic properties of the welch gong transformation sequence generators welch gong wg transformation sequences are binary sequences of period 2 sup n 1 with two level autocorrelation these sequences were discovered by golomb gong and gaal 1998 and they verified the validity of their construction for 5 spl les n spl les 20 later no chung and yun 1998 found another way to construct the wg sequences and verified their result for 5 spl les n spl les 20 dillon 1998 first proved this result for odd n and finally dobbertin and dillon 1999 proved it for even n in this paper we investigate a two faced property of the wg transformation sequences for application in stream ciphers and pseudorandom number generators one is to present the randomness or unpredictability of the wg transformation sequences the other is to exhibit the security properties of the wg transformations regarded as boolean functions in particular we prove that the wg transformation sequences in addition to the known two level autocorrelation and three level cross correlation with m sequences have the ideal 2 tuple distribution and large linear span increasing exponentially with n moreover it can be implemented efficiently this is the first type of pseudorandom sequences with good correlation statistic properties large linear span and efficient implementation when wg transformations are regarded as boolean functions they have high nonlinearity we derive a criterion for the boolean representation of wg transformations to be r resilient and show that they are at least 1 resilient under some basis of the finite field gf 2 sup n an algorithm to find such bases is given the degree and linear span of wg transformations are presented as well