permutation arrays under the chebyshev distance an n d permutation array pa is a subset of sn with the property that the distance under some metric between any two permutations in the array is at least d they became popular recently for communication over power lines motivated by an application to flash memories in this paper the metric used is the chebyshev metric a number of different constructions are given as well as bounds on the size of such pa

the alexander orbach conjecture holds in high dimensions we examine the incipient infinite cluster iic of critical percolation in regimes where mean field behavior has been established namely when the dimension d is large enough or when d 6 and the lattice is sufficiently spread out we find that random walk on the iic exhibits anomalous diffusion with the spectral dimension d s frac 4 3 that is p t x x t 2 3 o 1 this establishes a conjecture of alexander and orbach j phys lett paris 43 625 631 1982 en route we calculate the one arm exponent with respect to the intrinsic distance

inapproximability results for the weight problems of subgroup permutation codes a subgroup permutation code is a set of permutations on n symbols with the property that its elements are closed under the operation of composition in this paper we give inapproximability results for the minimum and maximum weight problems of subgroup permutation codes under several well known metrics based on previous works we prove that under hamming lee cayley kendall s tau ulam s and lp distance metrics 1 there is no polynomial time 2log1 en approximation algorithm for the minimum weight problem for any constant e 0 unless np dtime 2polylog n quasi polynomial time and 2 there is no polynomial time r approximation algorithm for the minimum weight problem for any constant r 1 unless p np under l metric we prove that it is np hard to approximate the minimum weight problem within factor 2 e for any constant e 0 we also prove that for any constant e 0 it is np hard to approximate the maximum weight within p 3 2 e under lp distance metric and within 3 2 e under hamming lee cayley kendall s tau and ulam s distance metrics

the survival probability for critical spread out oriented percolation above 4 1 dimensions i induction we consider critical spread out oriented percolation above 4 1 dimensions our main result is that the extinction probability at time n i e the probability for the origin to be connected to the hyperplane at time n but not to the hyperplane at time n 1 decays like 1 bn 2 as n to infty where b is a finite positive constant this in turn implies that the survival probability at time n i e the probability that the origin is connected to the hyperplane at time n decays like 1 bn as n to infty the latter has been shown in an earlier paper to have consequences for the geometry of large critical clusters and for the incipient infinite cluster the present paper is part i in a series of two papers in part ii we derive a lace expansion for the survival probability adapted so as to deal with point to plane connections this lace expansion leads to a nonlinear recursion relation for the survival probability in part i we use this recursion relation to deduce the asymptotics via induction

lp decodable multipermutation codes in this paper we introduce a new way of constructing and decoding multipermutation codes multipermutations are the permutations of a multiset that generally consist of duplicate entries we first introduce a class of binary matrices called multipermutation matrices each of which corresponds to a unique and distinct multipermutation by enforcing a set of linear constraints on these matrices we define a new class of codes that we term linear program lp decodable multipermutation codes in order to decode these codes using an lp thereby enabling soft decoding we characterize the convex hull of multipermutation matrices this characterization allows us to relax the coding constraints to a polytope and to derive two lp decoding problems these two problems are respectively formulated by relaxing the maximum likelihood decoding problem and the minimum chebyshev distance decoding problem because these codes are non linear we also study efficient encoding and decoding algorithms we first describe an algorithm that maps consecutive integers one by one to an ordered list of multipermutations based on this algorithm we develop an encoding algorithm for a code proposed by shieh and tsai a code that falls into our class of lp decodable multipermutation codes regarding decoding algorithms we propose an efficient distributed decoding algorithm based on the alternating direction method of multipliers finally we observe from the simulation results that the soft decoding techniques we introduce can significantly outperform hard decoding techniques that are based on quantized channel outputs

snake in the box codes for rank modulation motivated by the rank modulation scheme with applications to flash memory we consider gray codes capable of detecting a single error also known as snake in the box codes we study two error metrics kendall s metric which applies to charge constrained errors and the l metric which is useful in the case of limited magnitude errors in both cases we construct snake in the box codes with rate asymptotically tending to 1 we also provide efficient successor calculation functions as well as ranking and unranking functions finally we also study bounds on the parameters of such codes

poset resolutions and lattice linear monomial ideals abstract we introduce the class of lattice linear monomial ideals and use the lcm lattice to give an explicit construction for their minimal free resolution the class of lattice linear ideals includes among others the class of monomial ideals with a linear free resolution and the class of scarf monomial ideals our main tool is a new construction by tchernev that produces from a map of posets p n n a sequence of multigraded modules and maps

generalized gray codes for local rank modulation we consider the local rank modulation scheme in which a sliding window going over a sequence of real valued variables induces a sequence of permutations local rank modulation is a generalization of the rank modulation scheme which has been recently suggested as a way of storing information in flash memory we study gray codes for the local rank modulation scheme in order to simulate conventional multilevel flash cells while retaining the benefits of rank modulation unlike the limited scope of previous works we consider code constructions for the entire range of parameters including the code length sliding window size and overlap between adjacent windows we show that the presented codes have asymptotically optimal rate we also provide efficient encoding decoding and next state algorithms

permutation codes for the gaussian channel the initial vector problem is considered for group codes for the gaussian channel and in particular for codes generated by permutation groups variant i and variant ii permutation codes are defined and optimal initial vectors for these codes are constructed a numerical algorithm for constructing optimal group codes is presented together with examples of this algorithm applied to several permutation groups

ulam ball size analysis for permutation and multipermutation codes correcting translocation errors permutation and multipermutation codes in the ulam metric have been suggested for use in non volatile memory storage systems such as flash memory devices in this paper we introduce a new method to calculate permutation ball sizes in the ulam metric using young tableaux and prove the non existence of non trivial perfect permutation codes in the ulam metric we then extend the study to multipermutations providing upper and lower bounds on multipermutation ulam ball sizes and resulting upper and lower bounds on the maximal size of multipermutation codes in the ulam metric

perfect binary codes constructions properties and enumeration properties of nonlinear perfect binary codes are investigated and several new constructions of perfect codes are derived from these properties an upper bound on the cardinality of the intersection of two perfect codes of length n is presented and perfect codes whose intersection attains the upper bound are constructed for all n as an immediate consequence of the proof of the upper bound the authors obtain a simple closed form expression for the weight distribution of a perfect code furthermore they prove that the characters of a perfect code satisfy certain constraints and provide a sufficient condition for a binary code to be perfect the latter result is employed to derive a generalization of the construction of phelps 1983 which is shown to give rise to some perfect codes that are nonequivalent to the perfect codes obtained from the known constructions moreover for any m spl ges 4 the authors construct full rank perfect binary codes of length 2 sup m 1 these codes are obviously nonequivalent to any of the previously known perfect codes furthermore the latter construction exhibits the existence of full rank perfect tilings finally they construct a set of 2 2 sup cn nonequivalent perfect codes of length n for sufficiently large n and a constant c 0 5 spl epsiv precise enumeration of the number of codes in this set provides a slight improvement over the results reported by phelps

decoding frequency permutation arrays under chebyshev distance a frequency permutation array fpa of length n m and distance d is a set of permutations on a multiset over m symbols where each symbol appears exactly times and the distance between any two elements in the array is at least d fpa generalizes the notion of permutation array in this paper under the chebyshev distance we first prove lower and upper bounds on the size of fpa then we give several constructions of fpas and some of them come with efficient encoding and decoding capabilities moreover we show one of our designs is locally decodable i e we can decode a message bit by reading at most 1 symbols which has an interesting application to private information retrieval

a product construction for perfect codes over arbitrary alphabets corresp a general product construction for perfect single error correcting codes over an arbitrary alphabet is presented given perfect single error correcting codes of lengths n m and q 1 over an alphabet of order q one can construct perfect single error correcting codes of length q 1 nm n m over the same alphabet moreover if there exists a perfect single error correcting code of length q 1 over an alphabet of order q then there exist perfect single error correcting codes of length n n q t 1 q 1 and t 0 an integer finally connections between projective planes of order q and perfect codes of length q 1 over an alphabet of order q are discussed

systematic error correcting codes for permutations and multi permutations multi permutations and in particular permutations appear in various applications in an information theory new applications such as rank modulation for flash memories have suggested the need to consider error correcting codes for multi permutations in this paper we study systematic error correcting codes for multi permutations in general and for permutations in particular for a given number of information symbols k and for any integer t we present a construction of k r k systematic t error correcting codes for permutations of length k r where the number of redundancy symbols r is relatively small in particular for a given t and for sufficiently large k we obtain r t 1 while a lower bound on the number of redundancy symbols is shown to be t the same construction is also applied to obtain related systematic error correcting codes for any types of multi permutations

lp decodable permutation codes based on linearly constrained permutation matrices a set of linearly constrained permutation matrices are proposed for constructing a class of permutation codes the main feature of this class of permutation codes called linear programming lp decodable permutation codes is this lp decodability it is demonstrated that the lp decoding performance of the proposed class of permutation codes is characterized by the vertices of the code polytope of the code two types of linear constraints are discussed one is structured constraints and the other is random constraints the structured constraints allow an efficient encoding algorithm on the other hand the random constraints enable us to use probabilistic methods for analyzing several code properties such as the average cardinality and the average weight distribution

infinity norm permutation covering codes from cyclic groups we study covering codes of permutations with the ell infty metric we provide a general code construction which combines short building block codes into a single long code we focus on cyclic transitive groups as building blocks determining their exact covering radius and showing a linear time algorithm for finding a covering codeword when used in the general construction we show that the resulting covering code asymptotically out performs the best known code while maintaining linear time decoding we also bound the covering radius of relabeled cyclic transitive groups under conjugation showing that the covering radius is quite robust while relabeling cannot reduce the covering radius by much the downside is that we prove the covering radius cannot be increased by more than 1 when using relabeling

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

lee metric bch codes and their application to constrained and partial response channels shows that each code in a certain class of bch codes over gf p specified by a code length n spl les p sup m 1 and a runlength r spl les p 1 2 of consecutive roots in gf p sup m has minimum lee distance spl ges 2r for the very high rate range these codes approach the sphere packing bound on the minimum lee distance furthermore for a given r the length range of these codes is twice as large as that attainable by berlekamp s 1984 extended negacyclic codes the authors present an efficient decoding procedure based on euclid s algorithm for correcting up to r 1 errors and detecting r errors that is up to the number of lee errors guaranteed by the designed minimum lee distance 2r bounds on the minimum lee distance for r spl ges p 1 2 are provided for the reed solomon case i e when the bch code roots are in gf p the authors present two applications first lee metric bch codes can be used for protecting against bitshift errors and synchronization errors caused by insertion and or deletion of zeros in d k constrained channels second the code construction with its decoding algorithm can be formulated over the integer ring providing an algebraic approach to correcting errors in partial response channels where matched spectral null codes are used

systematic error correcting codes for rank modulation the rank modulation scheme has been recently proposed for efficiently storing data in nonvolatile memories in this paper we explore n k d systematic error correcting codes for rank modulation such codes have length n k information symbols and minimum distance d systematic codes have the benefits of enabling efficient information retrieval in conjunction with memory scrubbing schemes we study systematic codes for rank modulation under kendall s t metric as well as under the l metric in kendall s t metric we present k 2 k 3 systematic codes for correcting a single error which have optimal rates unless systematic perfect codes exist we also study the design of multierror correcting codes and provide a construction of k t 1 k 2t 1 systematic codes for large enough k we use nonconstructive arguments to show that for rank modulation systematic codes achieve the same capacity as general error correcting codes finally in the l metric we construct two n k d systematic multierror correcting codes the first for the case of d 0 1 and the second for d n in the latter case the codes have the same asymptotic rate as the best codes currently known in this metric

scaling limit for the ant in a simple high dimensional labyrinth we prove that after suitable rescaling the simple random walk on the trace of a large critical branching random walk in mathbb z d converges to the brownian motion on the integrated super brownian excursion when d 14

random walk on the incipient infinite cluster for oriented percolation in high dimensions we consider simple random walk on the incipient infinite cluster for the spread out model of oriented percolation on mathbb z d times mathbb z in dimensions d 6 we obtain bounds on exit times transition probabilities and the range of the random walk which establish that the spectral dimension of the incipient infinite cluster is frac 4 3 and thereby prove a version of the alexander orbach conjecture in this setting the proof divides into two parts one part establishes general estimates for simple random walk on an arbitrary infinite random graph given suitable bounds on volume and effective resistance for the random graph a second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d 6 by extending results about critical oriented percolation obtained previously via the lace expansion

spectral enumerators for certain additive error correcting codes over integer alphabets the paper contains a study of certain block codes with integer coordinate symbols devised for the correction of some types of additive errors the codes are defined by a check equation over a finite abelian group thus they appear as an extension of the varshamov and constantin rao codes the main results of the paper are concerned with the error correcting capability including a decoding method for generalized varshamov codes and principally with enumeration problems a general formula for spectral enumerators is obtained and applied to the varshamov and constantin rao codes

local rank modulation for flash memories local rank modulation lrm scheme was suggested for representing the information in flash memories in order to overcome the drawbacks of rank modulation for 0 with s dividing n an s t n lrm scheme is a lrm scheme where the n cells are locally viewed cyclically through a sliding window of size t resulting in a sequence of small permutations which requires less comparisons and less distinct values the gap between two such windows equals to s in this paper encoding decoding and asymptotic enumeration of the 1 t n lrm scheme are studied

snake in the box codes for rank modulation under kendall s tau metric for a gray code in the scheme of rank modulation for flash memories the codewords are permutations and two consecutive codewords are obtained using a push to the top operation we consider the snake in the box code under kendall s tau metric which is a gray code capable of detecting one kendall s tau error we answer two open problems posed by horovitz and etzion first we prove the validity of a construction given by them resulting in a snake of size m 2n 1 2n 1 2 2n 1 second we come up with a different construction aiming at a larger snake of size m 2n 1 2n 1 2 2n 3 the construction is applied successfully to s 7

codes in permutations and error correction for rank modulation codes for rank modulation have been recently proposed as a means of protecting flash memory devices from errors we study basic coding theoretic problems for such codes representing them as subsets of the set of permutations of n elements equipped with the kendall tau distance we derive several lower and upper bounds on the size of codes these bounds enable us to establish the exact scaling of the size of optimal codes for large values of n we also show the existence of codes whose size is within a constant factor of the sphere packing bound for any fixed number of errors

on the labeling problem of permutation group codes under the infinity metric we consider codes over permutations under the infinity norm given such a code we show that a simple relabeling operation which produces an isomorphic code may drastically change the minimal distance of the code thus we may choose a code structure for efficient encoding procedures and then optimize the code s minimal distance via relabeling to establish that the relabeling problem is hard and is of interest we formally define it and show that all codes may be relabeled to get a minimal distance at most 2 on the other hand the decision problem of whether a code may be relabeled to distance 2 or more is shown to be np complete and calculating the best achievable minimal distance after relabeling is proved to be hard to approximate up to a factor of 2 we then consider general bounds on the relabeling problem we specifically construct the optimal relabeling for transitive cyclic groups we conclude with the main result a general probabilistic bound which we then use to show both the agl p group and the dihedral group on p elements may be relabeled to a minimal distance of p o p in p

bounds for permutation rate distortion we study the rate distortion relationship in the set of permutations endowed with the kendall tau metric and the chebyshev metric the ell infty metric this paper is motivated by the application of permutation rate distortion to the average case and worst case distortion analysis of algorithms for ranking with incomplete information and approximate sorting algorithms for the kendall tau metric we provide bounds for various distortion regimes while for the chebyshev metric we present bounds that are valid for all distortions and are especially accurate for small distortions in addition for the chebyshev metric we provide a construction for covering codes

on the capacity of constrained permutation codes for rank modulation motivated by the rank modulation scheme a recent study by sala and dolecek explored the idea of constraint codes for permutations the constraint studied by them is inherited by the inter cell interference phenomenon in flash memories where high level cells can inadvertently increase the level of low level cells a permutation sigma in s n satisfies the single neighbor k constraint if sigma i 1 sigma i leq k for all 1 leq i leq n 1 in this paper this model is extended into two constraints a permutation sigma in s n satisfies the two neighbor k constraint if for all 2 leq i leq n 1 sigma i sigma i 1 leq k or sigma i 1 sigma i leq k and it satisfies the asymmetric two neighbor k constraint if for all 2 leq i leq n 1 sigma i 1 sigma i or sigma i 1 sigma i we show that the capacity of the first constraint is 1 epsilon 2 in case that k theta n epsilon and the capacity of the second constraint is 1 regardless for any positive k we also extend our results and study the capacity of these two constraints combined with error correcting codes in the kendall tau metric

snake in the box codes for rank modulation under kendall s tau metric in s_ 2n 2 snake in the box codes under kendall s tau metric are studied in the rank modulation scheme for flash memories where codewords are a subset of permutations in s n with minimal kendall s tau distance two and two cyclically consecutive codewords are connected via a push to the top operation studies so far restrict the push to the top operations only on odd indices resulting in a snake consisting of permutations with the same parity and thus the minimal distance constraint is easily satisfied asymptotically optimal snake codes have been constructed this way in s 2n 1 as for s 2n 2 this framework keeps the last element fixed and thus a snake in s 2n 2 is equivalent to a snake in s 2n 1 which is rather trivial if one wants to do better then it is inevitable to have some push to the top operations on even indices resulting in a combination of odd and even permutations in the snake which increases the difficulty to guarantee the minimal kendall s tau distance constraint thus horovitz and etzion pose the open problem to prove or disprove that the size of the largest snake in s 2n 2 is not larger than the size of the largest snake in s 2n 1 a first step toward this problem is a negative answer by wang and fu who construct a snake in s 2n 2 with exactly one more permutation than an optimal snake in s 2n 1 in this paper we give an explicit construction of a snake in s 2n 2 with size asymptotically approaching 1 4 s 2n 2

efficiently computing the permanent and hafnian of some banded toeplitz matrices we present a new efficient method for computing the permanent and hafnian of certain banded toeplitz matrices the method covers non trivial cases for which previous known methods do not apply the main idea is to use the elements of the first row and column which determine the entire toeplitz matrix to construct a digraph in which certain paths correspond to permutations that the permanent and hafnian count since counting paths can be done efficiently the permanent and hafnian for those matrices is easily obtainable

compression in the space of permutations we investigate lossy compression source coding of data in the form of permutations this problem has direct applications in the storage of ordinal data or rankings and in the analysis of sorting algorithms we analyze the rate distortion characteristic for the permutation space under the uniform distribution and the minimum achievable rate of compression that allows a bounded distortion after recovery our analysis is with respect to different practical and useful distortion measures including kendall tau distance spearman s footrule chebyshev distance and inversion ell 1 distance we establish equivalence of source code designs under certain distortions and show simple explicit code designs that incur low encoding decoding complexities and are asymptotically optimal finally we show that for the mallows model a popular nonuniform ranking model on the permutation space both the entropy and the maximum distortion at zero rate are much lower than the uniform counterparts which motivates the future design of efficient compression schemes for this model

longest increasing subsequences from patience sorting to the baik deift johansson theorem we describe a simple one person card game patience sorting its analysis leads to a broad circle of ideas linking young tableaux with the longest increasing subsequence of a random permutation via the schensted correspondence a recent highlight of this area is the work of baik deift johansson which yields limiting probability laws via hard analysis of toeplitz determinants

constructions of rank modulation codes rank modulation is a way of encoding information to correct errors in flash memory devices as well as impulse noise in transmission lines modeling rank modulation involves construction of packings of the space of permutations equipped with the kendall tau distance as our main set of results we present several general constructions of codes in permutations that cover a broad range of code parameters in particular we show a number of ways in which conventional error correcting codes can be modified to correct errors in the kendall space our constructions are nonasymptotic and afford simple encoding and decoding algorithms of essentially the same complexity as required to correct errors in the hamming metric as an example from binary bose chaudhuri hocquenghem codes we obtain codes correcting t kendall errors in n memory cells that support the order of n log2n t messages for any constant t 1 2 we give many examples of rank modulation codes with specific parameters turning to asymptotic analysis we construct families of rank modulation codes that correct a number of errors that grows with n at varying rates from n to n2 one of our constructions gives rise to a family of rank modulation codes for which the tradeoff between the number of messages and the number of correctable kendall errors approaches the optimal scaling rate

limited magnitude error correcting gray codes for rank modulation we construct error correcting codes over permutations under the infinity metric which are also gray codes in the context of rank modulation i e are generated as simple circuits in the rotator graph these errors model limited magnitude or spike errors for which only single error detecting gray codes are currently known surprisingly the error correcting codes we construct achieve a better asymptotic rate than that of presently known constructions not having the gray property and exceed the gilbert varshamov bound additionally we present efficient ranking and unranking procedures as well as a decoding procedure that runs in linear time finally we also apply our methods to solve an outstanding issue with error detecting rank modulation gray codes also known in this context as snake in the box codes under a different metric the kendall tau metric in the group of permutations over an even number of elements s 2n where we provide asymptotically optimal codes

correcting charge constrained errors in the rank modulation scheme we investigate error correcting codes for a the rank modulation scheme with an application to flash memory devices in this scheme a set of n cells stores information in the permutation induced by the different charge levels of the individual cells the resulting scheme eliminates the need for discrete cell levels overcomes overshoot errors when programming cells a serious problem that reduces the writing speed and mitigates the problem of asymmetric errors in this paper we study the properties of error correcting codes for charge constrained errors in the rank modulation scheme in this error model the number of errors corresponds to the minimal number of adjacent transpositions required to change a given stored permutation to another erroneous one a distance measure known as kendall s distance we show bounds on the size of such codes and use metric embedding techniques to give constructions which translate a wealth of knowledge of codes in the lee metric to codes over permutations in kendall s metric specifically the one error correcting codes we construct are at least half the ball packing upper bound

perfect snake in the box codes for rank modulation for odd n the alternating group on n elements is generated by the permutations that jump an element from an odd position to position 1 we prove hamiltonicity of the associated directed cayley graph for all odd n neq 5 a result of rankin implies that the graph is not hamiltonian for n 5 this solves a problem arising in rank modulation schemes for flash memory our result disproves a conjecture of horovitz and etzion and proves another conjecture of yehezkeally and schwartz

computing the ball size of frequency permutations under chebyshev distance abstract let s n be the set of all permutations over the multiset 1 1 ts m m where n m a frequency permutation array fpa of minimum distance d is a subset of s n in which every two elements have distance at least d fpas have many applications related to error correcting codes in coding theory the gilbert varshamov bound and the sphere packing bound are derived from the size of balls of certain radii we propose two efficient algorithms that compute the ball size of frequency permutations under chebyshev distance here it is equivalent to computing the permanent of a special type of matrix which generalizes the toepliz matrix in some sense both methods extend previous known results the first one runs in o 2 d d 2 376 log n time and o 2 d d 2 space the second one runs in o 2 d d d n time and o 2 d d space for small constants and d both are efficient in time and use constant storage space

quasi cross lattice tilings with applications to flash memory we consider lattice tilings of n by a shape we call a k k n quasi cross such lattices form perfect error correcting codes which correct a single limited magnitude error with prescribed maximal magnitudes k and k of positive error and negative error respectively the ratio of which k k is called the balance ratio these codes can be used to correct both disturb and retention errors in flash memories which are characterized by having limited magnitudes and different signs for any rational 0 1 we construct an infinite family of k k n quasi cross lattice tilings with balance ratio k k we also provide a specific construction for an infinite family of 2 1 n quasi cross lattice tilings the constructions are related to group splitting and modular b1 sequences in addition we study bounds on the parameters of lattice tilings by quasi crosses and express them in terms of the arm lengths of the quasi crosses and the dimension we also prove constraints on group splitting a specific case of which shows that the parameters of the lattice tiling by 2 1 n quasi crosses are the only ones possible for these quasi crosses

constant weight gray codes for local rank modulation we consider the local rank modulation lrm scheme in which a sliding window going over a sequence of real valued variables induces a sequence of permutations lrm is a generalization of the rank modulation scheme which has been recently suggested as a way of storing information in flash memory we study constant weight gray codes for the lrm scheme in order to simulate conventional multilevel flash cells while retaining the benefits of rank modulation we present a practical construction of codes with asymptotically optimal rate and weight asymptotically half the length thus having an asymptotically optimal charge difference between adjacent cells next we turn to examine the existence of optimal codes by specifically studying codes of weight 2 and 3 in the former case we upper bound the code efficiency proving that there are no such asymptotically optimal cyclic codes in contrast for the latter case we construct codes which are asymptotically optimal we conclude by providing necessary conditions for the existence of cyclic and cyclic optimal gray codes

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

distance increasing mappings from binary vectors to permutations that increase hamming distances by at least two in this correspondence for any k ges 2 we first propose two constructions of n k distance increasing mappings dims from the set of binary vectors of length n to the set of permutations of the same length that strictly increase the hamming distance by at least k except when it is obviously not possible next we prove that for any k ges 2 there is a smallest positive integer nk such that an n k dim can be constructed for any n ges nk an explicit upper bound on nk is also given

construction of the incipient infinite cluster for spread out oriented percolation above 4 1 dimensions we construct the incipient infinite cluster measure iic for sufficiently spread out oriented percolation on d for d 1 4 1 we consider two different constructions for the first construction we define n e by taking the probability of the intersection of an event e with the event that the origin is connected to x n d summing this probability over x d and normalising the sum to get a probability measure we let n and prove existence of a limiting measure the iic for the second construction we condition the connected cluster of the origin in critical oriented percolation to survive to time n and let n under the assumption that the critical survival probability is asymptotic to a multiple of n 1 we prove existence of a limiting measure with in addition we study the asymptotic behaviour of the size of the level set of the cluster of the origin and the dimension of the cluster of the origin under our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super brownian motion for d 1 4 1

constructions of permutation arrays a permutation array pa of length n and minimum distance d is a set of permutations of n elements such that any two permutations coincide in at most n d positions some constructions of pas are given

bounds on the size of permutation codes with the kendall tau metric the rank modulation scheme has been proposed for efficient writing and storing data in nonvolatile memory storage error correction in the rank modulation scheme is done by considering permutation codes in this paper we consider codes in the set of all permutations on n elements s n using the kendall tau metric the main goal of this paper is to derive new bounds on the size of such codes for this purpose we also consider perfect codes diameter perfect codes and the size of optimal anticodes in the kendall tau metric structures which have their own considerable interest we prove that there are no perfect single error correcting codes in s n where n 4 is a prime or 4 leq n leq 10 we present lower bounds on the size of optimal anticodes with odd diameter as a consequence we obtain a new upper bound on the size of codes in s n with even minimum kendall tau distance we present larger single error correcting codes than the known ones in s 5 and s 7

perfect constant weight codes in his pioneering work from 1973 delsarte conjectured that there are no nontrivial perfect codes in the johnson scheme many attempts were made during the years which followed to prove delsarte s conjecture but only partial results have been obtained we survey all these attempts and prove some new results having the same flavor we also present a new method taking a different approach which we hope can lead to the settling of this conjecture we show how this new method rules out sets of parameters as well as specific given parameters

electrical resistance of the low dimensional critical branching random walk we show that the electrical resistance between the origin and generation n of the incipient infinite oriented branching random walk in dimensions d 0 this answers a question of barlow et al commun math phys 278 385 431 2008

an axiomatic approach to constructing distances for rank comparison and aggregation we propose a new family of distance measures on rankings derived through an axiomatic approach that consider the nonuniform relevance of the top and bottom of ordered lists and similarities between candidates the proposed distance functions include specialized weighted versions of the kendall distance and the cayley distance and are suitable for comparing rankings in a number of applications including information retrieval and rank aggregation in addition to proposing the distance measures and providing the theoretical underpinnings for their applications we also analyze algorithmic and computational aspects of weighted distance based rank aggregation we present an aggregation method based on approximating weighted distance measures by a generalized version of spearman s footrule distance as well as a markov chain method inspired by pagerank where transition probabilities of the markov chain reflect the chosen weighted distances

correcting limited magnitude errors in the rank modulation scheme we study error correcting codes for permutations under the infinity norm motivated by a novel storage scheme for flash memories called rank modulation in this scheme a set of n flash cells are combined to create a single virtual multilevel cell information is stored in the permutation induced by the cell charge levels spike errors which are characterized by a limited magnitude change in cell charge levels correspond to a low distance change under the infinity norm we define codes protecting against spike errors called limited magnitude rank modulation codes lmrm codes and present several constructions for these codes some resulting in optimal codes these codes admit simple recursive and sometimes direct encoding and decoding procedures we also provide lower and upper bounds on the maximal size of lmrm codes both in the general case and in the case where the codes form a subgroup of the symmetric group in the asymptotic analysis the codes we construct outperform the gilbert varshamov like bound estimate

codes correcting single errors of limited magnitude an error model with symmetric errors of limited magnitude is considered limited magnitude means that the size of any error is limited by a number smaller usually much smaller than the alphabet size several constructions of codes correcting a single error are given in some cases the codes are perfect or quasi perfect

improved lower bounds on the size of balls over permutations with the infinity metric we study the size or volume of balls in the metric space of permutations s n under the infinity metric we focus on the regime of balls with radius r rho cdot n 1 rho in 0 1 i e a radius that is a constant fraction of the maximum possible distance we provide new lower bounds on the size of such balls these new lower bounds reduce the asymptotic gap to the known upper bounds to at most 0 029 bits per symbol additionally they imply an improved ball packing bound for error correcting codes and an improved upper bound on the size of optimal covering codes

error correction in flash memories via codes in the ulam metric we consider rank modulation codes for flash memories that allow for handling arbitrary charge drop errors unlike classical rank modulation codes used for correcting errors that manifest themselves as swaps of two adjacently ranked elements the proposed translocation rank codes account for more general forms of errors that arise in storage systems translocations represent a natural extension of the notion of adjacent transpositions and as such may be analyzed using related concepts in combinatorics and rank modulation coding our results include derivation of the asymptotic capacity of translocation rank codes construction techniques for asymptotically good codes as well as simple decoding methods for one class of constructed codes as part of our exposition we also highlight the close connections between the new code family and permutations with short common subsequences deletion and insertion error correcting codes for permutations and permutation codes in the hamming distance

permutation arrays for powerline communication and mutually orthogonal latin squares we develop a connection between permutation arrays that are used in powerline communication and well studied combinatorial objects mutually orthogonal latin squares mols from this connection many new results on permutation arrays can be obtained

optimal permutation recovery in permuted monotone matrix model motivated by recent research on quantifying bacterial growth dynamics based on genome assemblies we consider a permuted monotone matrix model y z where the rows represent different samples th

non intersection of transient branching random walks let g be a cayley graph of a nonamenable group with spectral radius rho 1 it is known that branching random walk on g with offspring distribution mu is transient i e visits the origin at most finitely often almost surely if and only if the expected number of offspring overline mu satisfies overline mu le rho 1 benjamini and muller groups geom dyn 6 231 247 2012 conjectured that throughout the transient supercritical phase 1 overline mu le rho 1 and in particular at the recurrence threshold overline mu rho 1 the trace of the branching random walk is tree like in the sense that it is infinitely ended almost surely on the event that the walk survives forever this is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely we prove this conjecture along with several other related conjectures made by the same authors a central contribution of this work is the introduction of the notion of local unimodularity which we expect to have several further applications in the future

conceptual proofs of l log l criteria for mean behavior of branching processes the kesten stigum theorem is a fundamental criterion for the rate of growth of a supercritical branching process showing that an l log l condition is decisive in critical and subcritical cases results of kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations we give conceptual proofs of these theorems based on comparisons of galton watson measure to another measure on the space of trees this approach also explains yaglom s exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution

on the complexity of unsigned translocation distance translocation is one of the basic operations for genome rearrangement translocation distance is the minimum number of translocations required to transform one genome into the other in this paper we show that computing the translocation distance for unsigned genomes is np hard moreover we show that approximating the translocation distance for unsigned genomes within ratio 1 00017 is np hard

product constructions for perfect lee codes a well known conjecture of golomb and welch is that the only nontrivial perfect codes in the lee and manhattan metrics have length two or minimum distance three this problem and related topics were subject for extensive research in the last 40 years in this paper two product constructions for perfect lee codes and diameter perfect lee codes are presented these constructions yield a large number of nonlinear perfect codes and nonlinear diameter perfect codes in the lee and manhattan metrics a short survey and other related problems on perfect codes in the lee and manhattan metrics are also discussed

codes correcting erasures and deletions for rank modulation error correcting codes for permutations have received considerable attention in the past few years especially in applications of the rank modulation scheme for flash memories while codes over several metrics have been studied such as the kendall tau ulam and hamming distances no recent research has been carried out for erasures and deletions over permutations in rank modulation flash memory cells represent a permutation which is induced by their relative charge levels we explore problems that arise when some of the cells are either erased or deleted in each case we study how these erasures and deletions affect the information carried by the remaining cells in particular we study models that are symbol invariant where unaffected elements do not change their corresponding values from those in the original permutation or permutation invariant where the remaining symbols are modified to form a new permutation with fewer elements our main approach in tackling these problems is to build upon the existing works of error correcting codes and leverage them in order to construct codes in each model of deletions and erasures the codes we develop are in certain cases asymptotically optimal while in other cases such as for codes in the ulam distance improve upon the state of the art results