arXiv subjects

Alexander Vardy

Publications and source records attributed to Alexander Vardy.

At least 55 records · Page 3Linked to original sources

arXiv2012

Coding for the Lee and Manhattan Metrics with Weighing Matrices

This paper has two goals. The first one is to discuss good codes for packing problems in the Lee and Manhattan metrics. The second one is to consider weighing matrices for some of these coding problems. Weighing matrices were considered as building blocks for codes in the Hamming metric in various constructions. In this paper we will consider mainly two types of weighing matrices, namely conference matrices and Hadamard matrices, to construct codes in the Lee (and Manhattan) metric. We will show that these matrices have some desirable properties when considered as generator matrices for codes in these metrics. Two related packing problems will be considered. The first is to find good codes for error-correction (i.e. dense packings of Lee spheres). The second is to transform the space in a way that volumes are preserved and each Lee sphere (or conscribed cross-polytope), in the space, will be transformed to a shape inscribed in a small cube.

cs.IT↗

arXiv2012

q-Analogs of Steiner Systems

A Steiner structure $\dS = \dS_q[t,k,n]$ is a set of $k$-dimensional subspaces of $\F_q^n$ such that each $t$-dimensional subspace of $\F_q^n$ is contained in exactly one subspace of $\dS$. Steiner structures are the $q$-analogs of Steiner systems; they are presently known to exist only for $t = 1$, $t=k$, and\linebreak for $k = n$. The existence of nontrivial $q$-analogs of Steiner systems has occupied mathematicians for over three decades. In fact, it was conjectured that they do not exist. In this paper, we show that nontrivial Steiner structures do exist. First, we describe a general method which may be used to produce Steiner structures. The method uses two mappings in a finite field: the Frobenius map and the cyclic shift map. These maps are applied to codes in the Grassmannian, in order to form an automorphism group of the Steiner structure. Using this method, assisted by an exact-cover computer search, we explicitly generate a Steiner structure $\dS_2[2,3,13]$. We conjecture that many other Steiner structures, with different parameters, exist.

math.CO↗

arXiv2012

Rewriting Codes for Flash Memories

Flash memory is a non-volatile computer memory comprising blocks of cells, wherein each cell can take on q different values or levels. While increasing the cell level is easy, reducing the level of a cell can be accomplished only by erasing an entire block. Since block erasures are highly undesirable, coding schemes - known as floating codes (or flash codes) and buffer codes - have been designed in order to maximize the number of times that information stored in a flash memory can be written (and re-written) prior to incurring a block erasure. An (n,k,t)q flash code C is a coding scheme for storing k information bits in $n$ cells in such a way that any sequence of up to t writes can be accommodated without a block erasure. The total number of available level transitions in n cells is n(q-1), and the write deficiency of C, defined as δ(C) = n(q-1)-t, is a measure of how close the code comes to perfectly utilizing all these transitions. In this paper, we show a construction of flash codes with write deficiency O(qk\log k) if q \geq \log_2k, and at most O(k\log^2 k) otherwise. An (n,r,\ell,t)q buffer code is a coding scheme for storing a buffer of r \ell-ary symbols such that for any sequence of t symbols it is possible to successfully decode the last r symbols that were written. We improve upon a previous upper bound on the maximum number of writes t in the case where there is a single cell to store the buffer. Then, we show how to improve a construction by Jiang et al. that uses multiple cells, where n\geq 2r.

cs.IT↗

arXiv2012

Automorphisms of Codes in the Grassmann Scheme

Two mappings in a finite field, the Frobenius mapping and the cyclic shift mapping, are applied on lines in PG($n,p$) or codes in the Grassmannian, to form automorphisms groups in the Grassmanian and in its codes. These automorphisms are examined on two classical coding problems in the Grassmannian. The first is the existence of a parallelism with lines in the related projective geometry and the second is the existence of a Steiner structure. A computer search was applied to find parallelisms and codes. A new parallelism of lines in PG(5,3) was formed. A parallelism with these parameters was not known before. A large code which is only slightly short of a Steiner structure was formed.

math.CO↗

arXiv2012

List Decoding of Polar Codes

We describe a successive-cancellation \emph{list} decoder for polar codes, which is a generalization of the classic successive-cancellation decoder of Arıkan. In the proposed list decoder, up to $L$ decoding paths are considered concurrently at each decoding stage. Then, a single codeword is selected from the list as output. If the most likely codeword is selected, simulation results show that the resulting performance is very close to that of a maximum-likelihood decoder, even for moderate values of $L$. Alternatively, if a "genie" is allowed to pick the codeword from the list, the results are comparable to the current state of the art LDPC codes. Luckily, implementing such a helpful genie is easy. Our list decoder doubles the number of decoding paths at each decoding step, and then uses a pruning procedure to discard all but the $L$ "best" paths. %In order to implement this algorithm, we introduce a natural pruning criterion that can be easily evaluated. Nevertheless, a straightforward implementation still requires $Ω(L \cdot n^2)$ time, which is in stark contrast with the $O(n \log n)$ complexity of the original successive-cancellation decoder. We utilize the structure of polar codes to overcome this problem. Specifically, we devise an efficient, numerically stable, implementation taking only $O(L \cdot n \log n)$ time and $O(L \cdot n)$ space.

cs.IT↗

arXiv2012

List-decoding of Subspace Codes and Rank-Metric Codes up to Singleton Bound

Subspace codes and rank-metric codes can be used to correct errors and erasures in network, with linear network coding. Subspace codes were introduced by Koetter and Kschischang to correct errors and erasures in networks where topology is unknown (the noncoherent case). In a previous work, we have developed a family of subspace codes, based upon the Koetter-Kschichang construction, which are efficiently list decodable. Using these codes, we achieved a better decoding radius than Koetter-Kschischiang codes at low rates. Herein, we introduce a new family of subspace codes based upon a different approach which leads to a linear-algebraic list-decoding algorithm. The resulting error correction radius can be expressed as follows: for any integer $s$, our list-decoder using $s+1$-interpolation polynomials guarantees successful recovery of the message subspace provided the normalized dimension of errors is at most $s(1-sR)$. The same list-decoding algorithm can be used to correct erasures as well as errors. The size of output list is at most $Q^{s-1}$, where $Q$ is the size of the field that message symbols are chosen from. Rank-metric codes are suitable for error correction in the case where the network topology and the underlying network code are known (the coherent case). Gabidulin codes are a well-known class of algebraic rank-metric codes that meet the Singleton bound on the minimum rank metric of a code. In this paper, we introduce a folded version of Gabidulin codes analogous to the folded Reed-Solomon codes of Guruswami and Rudra along with a list-decoding algorithm for such codes. Our list-decoding algorithm makes it possible to recover the message provided that the normalized rank of error is at most $1-R-ε$, for any $ε> 0$. Notably this achieves the information theoretic bound on the decoding radius of a rank-metric code.

cs.IT↗

arXiv2012

Algebraic List-decoding of Subspace Codes

Subspace codes were introduced in order to correct errors and erasures for randomized network coding, in the case where network topology is unknown (the noncoherent case). Subspace codes are indeed collections of subspaces of a certain vector space over a finite field. The Koetter-Kschischang construction of subspace codes are similar to Reed-Solomon codes in that codewords are obtained by evaluating certain (linearized) polynomials. In this paper, we consider the problem of list-decoding the Koetter-Kschischang subspace codes. In a sense, we are able to achieve for these codes what Sudan was able to achieve for Reed-Solomon codes. In order to do so, we have to modify and generalize the original Koetter-Kschischang construction in many important respects. The end result is this: for any integer $L$, our list-$L$ decoder guarantees successful recovery of the message subspace provided that the normalized dimension of the error is at most $ L - \frac{L(L+1)}{2}R $ where $R$ is the normalized packet rate. Just as in the case of Sudan's list-decoding algorithm, this exceeds the previously best known error-correction radius $1-R$, demonstrated by Koetter and Kschischang, for low rates $R$.

cs.IT↗

arXiv2012

A Cryptographic Treatment of the Wiretap Channel

The wiretap channel is a setting where one aims to provide information-theoretic privacy of communicated data based solely on the assumption that the channel from sender to adversary is "noisier" than the channel from sender to receiver. It has been the subject of decades of work in the information and coding (I&C) community. This paper bridges the gap between this body of work and modern cryptography with contributions along two fronts, namely metrics (definitions) of security, and schemes. We explain that the metric currently in use is weak and insufficient to guarantee security of applications and propose two replacements. One, that we call mis-security, is a mutual-information based metric in the I&C style. The other, semantic security, adapts to this setting a cryptographic metric that, in the cryptography community, has been vetted by decades of evaluation and endorsed as the target for standards and implementations. We show that they are equivalent (any scheme secure under one is secure under the other), thereby connecting two fundamentally different ways of defining security and providing a strong, unified and well-founded target for designs. Moving on to schemes, results from the wiretap community are mostly non-constructive, proving the existence of schemes without necessarily yielding ones that are explicit, let alone efficient, and only meeting their weak notion of security. We apply cryptographic methods based on extractors to produce explicit, polynomial-time and even practical encryption schemes that meet our new and stronger security target.

cs.IT↗

arXiv2011

Hardware Implementation of Successive Cancellation Decoders for Polar Codes

The recently-discovered polar codes are seen as a major breakthrough in coding theory; they provably achieve the theoretical capacity of discrete memoryless channels using the low complexity successive cancellation (SC) decoding algorithm. Motivated by recent developments in polar coding theory, we propose a family of efficient hardware implementations for SC polar decoders. We show that such decoders can be implemented with O(n) processing elements, O(n) memory elements, and can provide a constant throughput for a given target clock frequency. Furthermore, we show that SC decoding can be implemented in the logarithm domain, thereby eliminating costly multiplication and division operations and reducing the complexity of each processing element greatly. We also present a detailed architecture for an SC decoder and provide logic synthesis results confirming the linear growth in complexity of the decoder as the code length increases.

cs.AR↗

arXiv2011

Achieving the Secrecy Capacity of Wiretap Channels Using Polar Codes

Suppose Alice wishes to send messages to Bob through a communication channel C_1, but her transmissions also reach an eavesdropper Eve through another channel C_2. The goal is to design a coding scheme that makes it possible for Alice to communicate both reliably and securely. Reliability is measured in terms of Bob's probability of error in recovering the message, while security is measured in terms of the mutual information between the message and Eve's observations. Wyner showed that the situation is characterized by a single constant C_s, called the secrecy capacity, which has the following meaning: for all $ε> 0$, there exist coding schemes of rate $R \ge C_s - ε$ that asymptotically achieve both the reliability and the security objectives. However, his proof of this result is based upon a nonconstructive random-coding argument. To date, despite a considerable research effort, the only case where we know how to construct coding schemes that achieve secrecy capacity is when Eve's channel C_2 is an erasure channel, or a combinatorial variation thereof. Polar codes were recently invented by Arikan; they approach the capacity of symmetric binary-input discrete memoryless channels with low encoding and decoding complexity. Herein, we use polar codes to construct a coding scheme that achieves the secrecy capacity of general wiretap channels. Our construction works for any instantiation of the wiretap channel model, as originally defined by Wyner, as long as both C_1 and C_2 are symmetric and binary-input. Moreover, we show how to modify our construction in order to achieve strong security, as defined by Maurer, while still operating at a rate that approaches the secrecy capacity. In this case, we cannot guarantee that the reliability condition will be satisfied unless the main channel C_1 is noiseless, although we believe it can be always satisfied in practice.

cs.IT↗

arXiv2011

Achieving the Secrecy Capacity of Wiretap Channels Using Polar Codes

Suppose Alice wishes to send messages to Bob through a communication channel C_1, but her transmissions also reach an eavesdropper Eve through another channel C_2. The goal is to design a coding scheme that makes it possible for Alice to communicate both reliably and securely. Reliability is measured in terms of Bob's probability of error in recovering the message, while security is measured in terms of Eve's equivocation ratio. Wyner showed that the situation is characterized by a single constant C_s, called the secrecy capacity, which has the following meaning: for all $ε> 0$, there exist coding schemes of rate $R \ge C_s - ε$ that asymptotically achieve both the reliability and the security objectives. However, his proof of this result is based upon a nonconstructive random-coding argument. To date, despite a considerable research effort, the only case where we know how to construct coding schemes that achieve secrecy capacity is when Eve's channel C_2 is an erasure channel, or a combinatorial variation thereof. Polar codes were recently invented by Arikan; they approach the capacity of symmetric binary-input discrete memoryless channels with low encoding and decoding complexity. Herein, we use polar codes to construct a coding scheme that achieves the secrecy capacity for a wide range of wiretap channels. Our construction works for any instantiation of the wiretap channel model, as long as both C_1 and C_2 are symmetric and binary-input, and C_2 is degraded with respect to C_1. Moreover, we show how to modify our construction in order to provide strong security, in the sense defined by Maurer, while still operating at a rate that approaches the secrecy capacity. In this case, we cannot guarantee that the reliability condition will be satisfied unless the main channel C_1 is noiseless, although we believe it can be always satisfied in practice.

cs.IT↗

arXiv2011

Linearity and Complements in Projective Space

The projective space of order $n$ over the finite field $\Fq$, denoted here as $\Ps$, is the set of all subspaces of the vector space $\Fqn$. The projective space can be endowed with distance function $d_S(X,Y) = \dim(X) + \dim(Y) - 2\dim(X\cap Y)$ which turns $\Ps$ into a metric space. With this, \emph{an $(n,M,d)$ code $\C$ in projective space} is a subset of $\Ps$ of size $M$ such that the distance between any two codewords (subspaces) is at least $d$. Koetter and Kschischang recently showed that codes in projective space are precisely what is needed for error-correction in networks: an $(n,M,d)$ code can correct $t$ packet errors and $ρ$ packet erasures introduced (adversarially) anywhere in the network as long as $2t + 2ρ< d$. This motivates new interest in such codes. In this paper, we examine the two fundamental concepts of \myemph{complements} and \myemph{linear codes} in the context of $\Ps$. These turn out to be considerably more involved than their classical counterparts. These concepts are examined from two different points of view, coding theory and lattice theory. Our discussion reveals some surprised phenomena of these concepts in $\Ps$ and leaves some interesting problems for further research.

cs.IT↗

arXiv2010

A Quadratic Time-Space Tradeoff for Unrestricted Deterministic Decision Branching Programs

For a decision problem from coding theory, we prove a quadratic expected time-space tradeoff of the form $\eT\eS=Ω(\tfrac{n^2}{q})$ for $q$-way deterministic decision branching programs, where $q\geq 2$. Here $\eT$ is the expected computation time and $\eS$ is the expected space, when all inputs are equally likely. This bound is to our knowledge, the first such to show an exponential size requirement whenever $\eT = O(n^2)$. Previous exponential size tradeoffs for Boolean decision branching programs were valid for time-restricted models with $T=o(n\log_2{n})$. Proving quadratic time-space tradeoffs for unrestricted time decision branching programs has been a major goal of recent research -- this goal has already been achieved for multiple-output branching programs two decades ago. We also show the first quadratic time-space tradeoffs for Boolean decision branching programs verifying circular convolution, matrix-vector multiplication and discrete Fourier transform. Furthermore, we demonstrate a constructive Boolean decision function which has a quadratic expected time-space tradeoff in the Boolean deterministic decision branching program model. When $q$ is a constant the tradeoff results derived here for decision functions verifying various functions are order-comparable to previously known tradeoff bounds for calculating the corresponding multiple-output functions.

cs.CC↗

arXiv2010

Hardware architectures for Successive Cancellation Decoding of Polar Codes

The recently-discovered polar codes are widely seen as a major breakthrough in coding theory. These codes achieve the capacity of many important channels under successive cancellation decoding. Motivated by the rapid progress in the theory of polar codes, we propose a family of architectures for efficient hardware implementation of successive cancellation decoders. We show that such decoders can be implemented with O(n) processing elements and O(n) memory elements, while providing constant throughput. We also propose a technique for overlapping the decoding of several consecutive codewords, thereby achieving a significant speed-up factor. We furthermore show that successive cancellation decoding can be implemented in the logarithmic domain, thereby eliminating the multiplication and division operations and greatly reducing the complexity of each processing element.

cs.AR↗

arXiv2010

The Re-Encoding Transformation in Algebraic List-Decoding of Reed-Solomon Codes

The main computational steps in algebraic soft-decoding, as well as Sudan-type list-decoding, of Reed-Solomon codes are bivariate polynomial interpolation and factorization. We introduce a computational technique, based upon re-encoding and coordinate transformation, that significantly reduces the complexity of the bivariate interpolation procedure. This re-encoding and coordinate transformation converts the original interpolation problem into another reduced interpolation problem, which is orders of magnitude smaller than the original one. A rigorous proof is presented to show that the two interpolation problems are indeed equivalent. An efficient factorization procedure that applies directly to the reduced interpolation problem is also given.

cs.IT↗

arXiv2010

Dense Error-Correcting Codes in the Lee Metric

Several new applications and a number of new mathematical techniques have increased the research on error-correcting codes in the Lee metric in the last decade. In this work we consider several coding problems and constructions of error-correcting codes in the Lee metric. First, we consider constructions of dense error-correcting codes in relatively small dimensions over small alphabets. The second problem we solve is construction of diametric perfect codes with minimum distance four. We will construct such codes over various lengths and alphabet sizes. The third problem is to transfer an n-dimensional Lee sphere with large radius into a shape, with the same volume, located in a relatively small box. Hadamard matrices play an essential role in the solutions for all three problems. A construction of codes based on Hadamard matrices will start our discussion. These codes approach the sphere packing bound for very high rate range and appear to be the best known codes over some sets of parameters.

cs.IT↗

arXiv2010

$q$-Analogs for Steiner Systems and Covering Designs

The $q$-analogs of basic designs are discussed. It is proved that the existence of any unknown Steiner structures, the $q$-analogs of Steiner systems, implies the existence of unknown Steiner systems. Optimal $q$-analogs covering designs are presented. Some lower and upper bounds on the sizes of $q$-analogs covering designs are proved.

math.CO↗

arXiv2009

Storage Coding for Wear Leveling in Flash Memories

Flash memory is a non-volatile computer memory comprised of blocks of cells, wherein each cell is implemented as either NAND or NOR floating gate. NAND flash is currently the most widely used type of flash memory. In a NAND flash memory, every block of cells consists of numerous pages; rewriting even a single page requires the whole block to be erased and reprogrammed. Block erasures determine both the longevity and the efficiency of a flash memory. Therefore, when data in a NAND flash memory are reorganized, minimizing the total number of block erasures required to achieve the desired data movement is an important goal. This leads to the flash data movement problem studied in this paper. We show that coding can significantly reduce the number of block erasures required for data movement, and present several optimal or nearly optimal data-movement algorithms based upon ideas from coding theory and combinatorics. In particular, we show that the sorting-based (non-coding) schemes require at least O(nlogn) erasures to move data among n blocks, whereas coding-based schemes require only O(n) erasures. Furthermore, coding-based schemes use only one auxiliary block, which is the best possible, and achieve a good balance between the number of erasures in each of the n+1 blocks.

cs.IT↗