Arya Mazumdar

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.

An upper bound on the size of locally recoverable codes

In a locally recoverable or repairable code, any symbol of a codeword can be recovered by reading only a small (constant) number of other symbols. The notion of local recoverability is important in the area of distributed storage where a most frequent error-event is a single storage node failure (erasure). A common objective is to repair the node by downloading data from as few other storage node as possible. In this paper, we bound the minimum distance of a code in terms of of its length, size and locality. Unlike previous bounds, our bound follows from a significantly simple analysis and depends on the size of the alphabet being used.

Bounds on the Size of Locally Recoverable Codes

In a locally recoverable or repairable code, any symbol of a codeword can be recovered by reading only a small (constant) number of other symbols. The notion of local recoverability is important in the area of distributed storage where a most frequent error-event is a single storage node failure (erasure). A common objective is to repair the node by downloading data from as few other storage nodes as possible. In this paper, we bound the minimum distance of a code in terms of its length, size, and locality. Unlike the previous bounds, our bound follows from a significantly simple analysis and depends on the size of the alphabet being used. It turns out that the binary Simplex codes satisfy our bound with equality; hence, the Simplex codes are the first example of an optimal binary locally repairable code family. We also provide achievability results based on random coding and concatenated codes that are numerically verified to be close to our bounds.

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 Θ(n) to Θ(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.

On cooperative local repair in distributed storage

Erasure-correcting codes, that support local repair of codeword symbols, have attracted substantial attention recently for their application in distributed storage systems. In this paper we study a generalization of the usual locally recoverable codes. We consider such codes that any small set of codeword symbols is recoverable from a small number of other symbols. We call this cooperative local repair. We present bounds on the dimension of such codes as well as give explicit constructions of families of codes. Some other results regarding cooperative local repair are also presented, including an analysis for the Hadamard codes.

Coding for High-Density Recording on a 1-D Granular Magnetic Medium

In terabit-density magnetic recording, several bits of data can be replaced by the values of their neighbors in the storage medium. As a result, errors in the medium are dependent on each other and also on the data written. We consider a simple 1-D combinatorial model of this medium. In our model, we assume a setting where binary data is sequentially written on the medium and a bit can erroneously change to the immediately preceding value. We derive several properties of codes that correct this type of errors, focusing on bounds on their cardinality. We also define a probabilistic finite-state channel model of the storage medium, and derive lower and upper estimates of its capacity. A lower bound is derived by evaluating the symmetric capacity of the channel, i.e., the maximum transmission rate under the assumption of the uniform input distribution of the channel. An upper bound is found by showing that the original channel is a stochastic degradation of another, related channel model whose capacity we can compute explicitly.

On a duality between recoverable distributed storage and index coding

In this paper, we introduce a model of a single-failure locally recoverable distributed storage system. This model appears to give rise to a problem approximately dual of the well-studied index coding problem. The relation between the dimensions of an optimal index code and optimal distributed storage code of our model has been established in this paper. We also show some extensions to vector codes.

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. We present several general constructions of codes in permutations that cover a broad range of code parameters. In particular, we show that a code that corrects Hamming errors can be used to construct a code for correcting Kendall errors. For instance, from BCH codes we obtain codes correcting t Kendall errors in n memory cells that support the order of n!/ logt n! messages, for any t = 1, 2, ‥‥ We also construct families of codes that correct a number of errors that grows with n at varying rates, from Θ(n) to Θ(n2).

Cooperative local repair in distributed storage

Erasure-correcting codes, which support local repair of codeword symbols, have attracted substantial attention recently for their application in distributed storage systems. This paper investigates a generalization of the usual locally repairable codes. In particular, this paper studies a class of codes with the following property: any small set of codeword symbols can be reconstructed (repaired) from a small number of other symbols. This is referred to as cooperative local repair. The main contribution of this paper is bounds on the trade-off of the minimum distance and the dimension of such codes, as well as explicit constructions of families of codes that enable cooperative local repair. Some other results regarding cooperative local repair are also presented, including an analysis for the well-known Hadamard/Simplex codes.

On Almost Disjunct Matrices for Group Testing

In a group testing scheme, a set of tests is designed to identify a small number t of defective items among a large set (of size N) of items. In the non-adaptive scenario the set of tests has to be designed in one-shot. In this setting, designing a testing scheme is equivalent to the construction of a disjunct matrix, an M ×N matrix where the union of supports of any t columns does not contain the support of any other column. In principle, one wants to have such a matrix with minimum possible number M of rows (tests). One of the main ways of constructing disjunct matrices relies on constant weight error-correcting codes and their minimum distance. In this paper, we consider a relaxed definition of a disjunct matrix known as almost disjunct matrix. This concept is also studied under the name of weakly separated design in the literature. The relaxed definition allows one to come up with group testing schemes where a close-to-one fraction of all possible sets of defective items are identifiable. Our main contribution is twofold. First, we go beyond the minimum distance analysis and connect the average distance of a constant weight code to the parameters of an almost disjunct matrix constructed from it. Next we show as a consequence an explicit construction of almost disjunct matrices based on our average distance analysis. The parameters of our construction can be varied to cover a large range of relations for t and N. As an example of parameters, consider any absolute constant e > 0 and t proportional to N δ , δ > 0. With our method it is possible to explicitly construct a group testing scheme that identifies (1 − e) proportion of all possible defective sets of size t using only \(O\Big(t^{3/2}\sqrt{ \log(N/\epsilon)}\Big)\) tests (as opposed to O(t 2logN) required to identify all defective sets).