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.
The goal of this paper is to construct systematic error-correcting codes for permutations and multi-permutations in the Kendall's $τ$-metric. These codes are important in new applications such as rank modulation for flash memories. The construction is based on error-correcting codes for multi-permutations and a partition of the set of permutations into error-correcting codes. For a given large enough number of information symbols $k$, and for any integer $t$, we present a construction for ${(k+r,k)}$ systematic $t$-error-correcting codes, for permutations from $S_{k+r}$, with less redundancy s…
The $c$-differential uniformity is recently proposed to reflect resistance against some variants of differential attack. Finding functions with low $c$-differential uniformity is attracting attention from many researchers. For even characteristic, it is known that permutations of low Carlitz rank have good cryptographic parameters, for example, low differential uniformity, high nonlinearity, etc. In this paper we show that permutations with low Carlitz rank have low $c$-differential uniformity. We also investigate $c$-differential uniformity of permutations with Carlitz rank 3 in detail.
Motivated by charge balancing constraints for rank modulation schemes, we introduce the notion of balanced permutations and derive the capacity of balanced permutation codes. We also describe simple interleaving methods for permutation code constructions and show that they approach capacity
The use of multiple frequency shift keying modulation with permutation codes addresses the problem of permanent narrowband noise disturbance in a power line communications system. In this paper, we extend this coded modulation scheme based on permutation codes to general codes and introduce an additional new parameter that more precisely captures a code's performance against permanent narrowband noise. As a result, we define a new class of codes, namely, equitable symbol weight codes, which are optimal with respect to this measure.
In this paper, the spectral efficiency of permutation modulation-based multiple input multiple output (MIMO) visible light communication is improved using systematically designed, multiweight codeword matrices. Soft-decision, low-complexity detection schemes are then designed for the receiver and compared with the maximum likelihood (ML) detection method. Bit error rate (BER) results show the soft-decision detection algorithm is able to decode the transmitted information without knowledge of the channel state information. This enables the mobile receiver decode information while within the fie…
We propose a novel scheme that allows MIMO system to modulate a set of permutation matrices to send more information bits, extending our initial work on the topic. This system is called Permutation Matrix Modulation (PMM). The basic idea is to employ a permutation matrix as a precoder and treat it as a modulated symbol. We continue the evolution of index modulation in MIMO by adopting all-antenna activation and obtaining a set of unique symbols from altering the positions of the antenna transmit power. We provide the analysis of the achievable rate of PMM under Gaussian Mixture Model (GMM) dis…
Synthesis of DNA molecules offers unprecedented advances in storage technology. Yet, the microscopic world in which these molecules reside induces error patterns that are fundamentally different from their digital counterparts. Hence, to maintain reliability in reading and writing, new coding schemes must be developed. In a reading technique called shotgun sequencing, a long DNA string is read in a sliding window fashion, and a profile vector is produced. It was recently suggested by Kiah et al. that such a vector can represent the permutation which is induced by its entries, and hence a rank-…
We study permutations over the set of $\ell$-grams, that are feasible in the sense that there is a sequence whose $\ell$-gram frequency has the same ranking as the permutation. Codes, which are sets of feasible permutations, protect information stored in DNA molecules using the rank-modulation scheme, and read using the shotgun sequencing technique. We construct systematic codes with an efficient encoding algorithm, and show that they are optimal in size. The length of the DNA sequences that correspond to the codewords is shown to be polynomial in the code parameters. Non-systematic with large…
Finding permutation polynomials with low differential and boomerang uniformityis an important topic in S-box designs of many block ciphers. For example, AES chooses the inverse function as its S-box, which is differentially 4-uniform and boomerang 6-uniform. Also there has been considerable research on many non-quadratic permutations which are obtained by modifying certain set of points from the inverse function. In this paper, we give a novel approach that shows that plenty of existing modifications of the inverse function are in fact affine equivalent to permutations of low Carlitz rank and …
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. The local rank-modulation, as a generalization of the rank-modulation scheme, has been recently suggested as a way of storing information in flash memory. We study constant-weight Gray codes for the local rank-modulation scheme in order to simulate conventional multi-level flash cells while retaining the benefits of rank modulation. We provide necessary conditions for the existence of cyclic and cyclic optimal Gray codes. We then specifically…
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 multi-level 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 co…
Linear constraints for a matrix polytope with no fractional vertex are investigated as intersecting research among permutation codes, rank modulations, and linear programming methods. By focusing the discussion to the block structure of matrices, new classes of such polytopes are obtained from known small polytopes. This concept, called "consolidation", is applied to find a new compact graph which is known as an approach for the graph isomorphism problem. Encoding and decoding algorithms for our new permutation codes are obtained from existing algorithms for small polytopes. The minimum distan…
We consider the problem of using polar codes with higher order modulation over AWGN channels. Unlike prior work, we focus on using modulation independent polar codes. That is, the polar codes are not re-designed based on the modulation used. Instead, we propose bit-permuted coded modulation (BPCM): a technique for using the multilevel coding (MLC) approach for an arbitrary polar code. The BPCM technique exploits a natural connection between MLC and polar codes. It involves applying bit permutations prior to mapping the polar code to a higher order modulation. The bit permutations are designed,…
We consider the problem of noisy matrix completion, in which the goal is to reconstruct a structured matrix whose entries are partially observed in noise. Standard approaches to this underdetermined inverse problem are based on assuming that the underlying matrix has low rank, or is well-approximated by a low rank matrix. In this paper, we propose a richer model based on what we term the "permutation-rank" of a matrix. We first describe how the classical non-negative rank model enforces restrictions that may be undesirable in practice, and how and these restrictions can be avoided by using the…
The rank modulation scheme has been proposed for efficient writing and storing data in non-volatile 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's $τ$-metric. 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 also prove that if such a code exists for $n$ which is not a prime then the code should have some uniform structure. We define some variations of the Kend…
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 \emph{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 o…
We study error-correcting codes for permutations under the infinity norm, motivated by a novel storage scheme for flash memories call rank modulation. In this scheme, a set of $n$ flash cells are combined to create a single virtual multi-level 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 …
Rank modulation has been recently proposed as a scheme for storing information in flash memories. While rank modulation has advantages in improving write speed and endurance, the current encoding approach is based on the "push to the top" operation that is not efficient in the general case. We propose a new encoding procedure where a cell level is raised to be higher than the minimal necessary subset - instead of all - of the other cell levels. This new procedure leads to a significantly more compressed (lower charge levels) encoding. We derive an upper bound for a family of codes that utilize…
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 $τ$-metric, which applies to charge-constrained errors, and the $\ell_\infty$-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 suc…
We construct Gray codes over permutations for the rank-modulation scheme, which are also capable of correcting errors under the infinity-metric. 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. Fina…
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 by using the "push-to-the-top" operation, and the distance measure is defined on permutations. In this paper the Kendall's $τ$-metric is used as the distance measure. We present a general method for constructing such Gray codes. We apply the method recursively to obta…
The rank modulation scheme has been proposed for efficient writing and storing data in non-volatile 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 $τ$-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 $τ$-metric, structures which have their own considerable interest. We prove t…
Local rank modulation scheme was suggested recently for representing information in flash memories in order to overcome drawbacks of rank modulation. For $0 < s\leq t\leq n$ with $s$ divides $n$, an $(s,t,n)$-LRM scheme is a local rank modulation 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 work, encoding, decoding, and asymptotic enumeration of the $(1,3,n)$-LRM scheme is studied. The te…
Local rank modulation scheme was suggested recently for representing information in flash memories in order to overcome drawbacks of rank modulation. For $s\leq t\leq n$ with $s|n$, $(s,t,n)$-LRM scheme is a local rank modulation scheme where the $n$ cells are locally viewed through a sliding window of size $t$ resulting in a sequence of small permutations which requires less comparisons and less distinct values. The distance between two windows equals to $s$. To get the simplest hardware implementation the case of sliding window of size two was presented. Gray codes and constant weight Gray c…
The rank-modulation scheme has been recently proposed for efficiently storing data in nonvolatile memories. Error-correcting codes are essential for rank modulation, however, existing results have been limited. In this work we explore a new approach, \emph{systematic error-correcting codes for rank modulation}. Systematic codes have the benefits of enabling efficient information retrieval and potentially supporting more efficient encoding and decoding procedures. We study systematic codes for rank modulation under Kendall's $τ$-metric as well as under the $\ell_\infty$-metric. In Kendall's $τ$…
Permutation codes, in the form of rank modulation, have shown promise for applications such as flash memory. One of the metrics recently suggested as appropriate for rank modulation is the Ulam metric, which measures the minimum translocation distance between permutations. Multipermutation codes have also been proposed as a generalization of permutation codes that would improve code size (and consequently the code rate). In this paper we analyze the Ulam metric in the context of multipermutations, noting some similarities and differences between the Ulam metric in the context of permutations. …
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. 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. Codes that we construct afford simple encoding and decoding algorithms of essenti…
Differential spatial modulation (DSM) was recently proposed to overcome the challenge of channel estimation in spatial modulation (SM). In this letter, we propose a gray code order of antenna index permutations for DSM. To facilitate the implementation, the well-known Trotter-Johnson ranking and unranking algorithms are adopted, which results in similar computational complexity to the existing DSM that uses the lexicographic order. The coding gain achieved by the proposed gray code order over the existing lexicographic order is also analyzed and verified via simulations, which reveals a maximu…
In the rank modulation scheme for flash memories, permutation codes have been studied. In this paper, we study perfect permutation codes in $S_n$, the set of all permutations on $n$ elements, under the Kendall τ-Metric. We answer one open problem proposed by Buzaglo and Etzion. That is, proving the nonexistence of perfect codes in $S_n$, under the Kendall τ-metric, for more values of $n$. Specifically, we present the recursive formulas for the size of a ball with radius $r$ in $S_n$ under the Kendall τ-metric. Further, We prove that there are no perfect $t$-error-correcting codes in $S_n$ unde…
Motivated by the rank modulation scheme, a recent work by Sala and Dolecek explored the study 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. In this paper, the model studied by Sala and Dolecek is extended into two constraints. A permutation $σ\in S_n$ satisfies the \emph{two-neighbor $k$-constraint} if for all $2 \leq i \leq n-1$ either $|σ(i-1)-σ(i)|\leq k$ or $|σ(i)-σ(i+1)|\leq k$, and it satisfies the \emph{asymmetri…
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 snake-in-the-box codes under Kendall's $τ$-metric, which is a Gray code capable of detecting one Kendall's $τ$-error. We answer two open problems posed by Horovitz and Etzion. Firstly, we prove the validity of a construction given by them, resulting in a snake of size $M_{2n+1}=\frac{(2n+1)!}{2}-2n+1$. Secondly, we come up with a different construction aiming at a longer snake of size $M_{2n+1}=\frac{(2n+1)…
Permutation codes are widely studied objects due to their numerous applications in various areas, such as power line communications, block ciphers, and the rank modulation scheme for flash memories. Several kinds of metrics are considered for permutation codes according to their specific applications. This paper concerns some improvements on the bounds of permutation codes under Hamming metric and Kendall's $τ$-metric respectively, using mainly a graph coloring approach. Specifically, under Hamming metric, we improve the Gilbert-Varshamov bound asymptotically by a factor $n$, when the minimum …
In the rank modulation scheme, Gray codes are very useful in the realization of flash memories. For a Gray code in this scheme, two adjacent codewords are obtained by using one "push-to-the-top" operation. Moreover, snake-in-the-box codes under the $\ell_{\infty}$-metric are Gray codes, which can be capable of detecting one $\ell_{\infty}$-error. In this paper, we give two constructions of $\ell_{\infty}$-snakes. On the one hand, inspired by Yehezkeally and Schwartz's construction, we present a new construction of the $\ell_{\infty}$-snake. The length of this $\ell_{\infty}$-snake is longer th…
The current flash memory technology focuses on the cost minimization of its static storage capacity. However, the resulting approach supports a relatively small number of program-erase cycles. This technology is effective for consumer devices (e.g., smartphones and cameras) where the number of program-erase cycles is small. However, it is not economical for enterprise storage systems that require a large number of lifetime writes. The proposed approach in this paper for alleviating this problem consists of the efficient integration of two key ideas: (i) improving reliability and endurance by r…
Permutation codes were extensively studied in order to correct different types of errors for the applications on power line communication and rank modulation for flash memory. In this paper, we introduce the neural network decoders for permutation codes to correct these errors with one-shot decoding, which treat the decoding as $n$ classification tasks for non-binary symbols for a code of length $n$. These are actually the first general decoders introduced to deal with any error type for these two applications. The performance of the decoders is evaluated by simulations with different error mo…
We introduce a modulation for unsourced massive random access whereby the transmitted symbols are rank-1 tensors constructed from Grassmannian sub-constellations. The use of a low-rank tensor structure, together with tensor decomposition in order to separate the users at the receiver, allows a convenient uncoupling between multi-user separation and single-user demapping. The proposed signaling scheme is designed for the block fading channel and multiple-antenna settings, and is shown to perform well in comparison to state-of-the-art unsourced approaches.
For odd n, the alternating group on n elements is generated by the permutations that jump an element from any odd position to position 1. We prove Hamiltonicity of the associated directed Cayley graph for all odd n not equal to 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.
We give unique recovery guarantees for matrices of bounded rank that have undergone permutations of their entries. We even do this for a more general matrix structure that we call ladder matrices. We use methods and results of commutative algebra and algebraic geometry, for which we include a preparation as needed for a general audience.
In this paper, we formally define and analyze the class of noisy permutation channels. The noisy permutation channel model constitutes a standard discrete memoryless channel (DMC) followed by an independent random permutation that reorders the output codeword of the DMC. While coding theoretic aspects of this model have been studied extensively, particularly in the context of reliable communication in network settings where packets undergo transpositions, and closely related models of DNA based storage systems have also been analyzed recently, we initiate an information theoretic study of this…
