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

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

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

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

holomorphic discrete series and harmonic series unitary irreducible representations of non compact lie groups sp 2n r u p q and so 2n generalised characters of the infinite dimensional holomorphic discrete series unitary irreducible representations of the non compact groups u p q sp 2n r and so 2n are explicitly expressed in terms of characters of finite dimensional unitary group representations these formulae are remarkably succinct despite involving certain infinite series of schur functions similar formulae are derived for harmonic series unitary representation of both u p q and sp 2n r consideration of the branching rules from u p q to u q p and from sp 2n r to u n enables holomorphic representations to be identified as a subset of the harmonic representations the branching rules are established in full generality and are then used in the evaluation of tensor products of both holomorphic and harmonic representations in the case of the former a known result is recast in terms of closed formulae involving schur functions and for the latter various generalisations of these formulae are given a conjecture is also made regarding what might be the simplest possible formulae covering all holomorphic and harmonic representations of sp 2n r and u p q illustrative examples are presented

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

division algebras pseudo orthogonal groups and spinors the groups so nu 1 so nu so nu 1 so nu 1 1 and so nu 2 2 nu 1 2 4 8 and their spin representations are described in terms of the division algebras r c h and o

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

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

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

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

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

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

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

the remarkable algebra so 2n its representations its clifford algebra and potential applications properties of the real lie algebra so 2n and its finite dimensional representations are described the structure of the clifford algebras associated with the two fundamental spinor representations are determined and it is found in particular that for so 4n there is a remarkable asymmetry one representation is real the other complex though equivalent to its complex conjugate and the associated clifford algebras are inequivalent generalised dirac matrices are defined acting within the direct sum of these two spinor representations of so 2n these matrices generate a different clifford algebra not determined uniquely by those representations and their commutators do not span a representation of so 2n the isomorphisms of so 6 with su 3 1 and of so 8 with so 6 2 are described together with the corresponding mappings between finite dimensional irreducible representations a possible application to unified field theory is discussed

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

triality principle and g2 group in spinor language the presented paper is aimed at providing a systematic study of a relation between octonions and spinors corresponding to s7 7 sphere starting from a natural point of view enabling us to endow spinor space s 8 0 with octonion algebra structures as a result we arrive at formulations of triality principle in its finite form in terms of vector and fundamental representations of spin 8 group both for spinors and vectors the group of automorphisms of octonion algebra as well as its lie algebra gains clear interpretation in the context the method proposed is purely algebraic and could be applied as well to c 4 4 clifford algebra corresponding to 4s7 indefinite 7 sphere geometry

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

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

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

on the pless construction and ml decoding of the 48 24 12 quadratic residue code we present a method for maximum likelihood decoding of the 48 24 12 quadratic residue code this method is based on projecting the code onto a subcode with an acyclic tanner graph and representing the set of coset leaders by a trellis diagram this results in a two level coset decoding which can be considered a systematic generalization of the wagner rule we show that unlike the 24 12 8 golay code the 48 24 12 code does not have a pless construction which has been an open question in the literature it is determined that the highest minimum distance of a 48 24 binary code having a pless 1986 construction is 10 and up to equivalence there are three such codes

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