arXiv subjects

Tuvi Etzion

Publications and source records attributed to Tuvi Etzion.

At least 73 records · Page 4Linked to original sources

arXiv2016

PIR Array Codes with Optimal PIR Rates

There has been much recent interest in Private information Retrieval (PIR) in models where a database is stored across several servers using coding techniques from distributed storage, rather than being simply replicated. In particular, a recent breakthrough result of Fazelli, Vardy and Yaakobi introduces the notion of a PIR code and a PIR array code, and uses this notion to produce efficient protocols. In this paper we are interested in designing PIR array codes. We consider the case when we have $m$ servers, with each server storing a fraction $(1/\omegaR)$ of the bits of the database; here $\omegaR$ is a fixed rational number with $\omegaR > 1$. We study the maximum PIR rate of a PIR array code with the $k$-PIR property (which enables a $k$-server PIR protocol to be emulated on the $m$ servers), where the PIR rate is defined to be $k/m$. We present upper bounds on the achievable rate, some constructions, and ideas how to obtain PIR array codes with the highest possible PIR rate. In particular, we present constructions that asymptotically meet our upper bounds, and the exact largest PIR rate is obtained when $1 < \omegaR \leq 2$.

cs.IT↗

arXiv2016

Regular Graphs with Forbidden Subgraphs of $K_n$ with $k$ Edges

In this paper we raise a variant of a classic problem in extremal graph theory, which is motivated by a design of fractional repetition codes, a model in distributed storage systems. For any feasible positive integers $d\geq 3$, $n \geq 3$, and $k$, where $n-1 \leq k \leq \binom{n}{2}$, what is the minimum possible number of vertices in a $d$-regular undirected graph whose subgraphs with $n$ vertices contain at most $k$ edges? The goal of this paper is to give the exact number of vertices for each instance of the problem and also to provide some bounds for general values of $n$, $d$, and $k$. A few general bounds with some exact values, for this Turán-type problem, are given. We present an almost complete solution for $3 \leq n \leq 5$.

math.CO↗

arXiv2016

Vector Network Coding Based on Subspace Codes Outperforms Scalar Linear Network Coding

This paper considers vector network coding based on rank-metric codes and subspace codes. Our main result is that vector network coding can significantly reduce the required field size compared to scalar linear network coding in the same multicast network. The achieved gap between the field size of scalar and vector network coding is in the order of $q^{(\ell-1)t^2/\ell}-q^t$, for any $q \geq 2$, where $t$ denotes the dimension of the vector solution, and the number of messages is $2 \ell$, ${\ell \geq 2}$. Previously, only a gap of constant size had been shown. This implies also the same gap between the field size in linear and non-linear scalar network coding for multicast networks. Similar results are given for any number of odd messages greater than two. The results are obtained by considering several multicast networks which are variations of the well-known combination network.

cs.IT↗

arXiv2016

Constructions of High-Rate MSR Codes over Small Fields

A novel technique for construction of minimum storage regenerating (MSR) codes is presented. Based on this technique, three explicit constructions of MSR codes are given. The first two constructions provide access-optimal MSR codes, with two and three parities, respectively, which attain the sub-packetization bound for access-optimal codes. The third construction provides longer MSR codes with three parities, which are not access-optimal, and do not necessarily attain the sub-packetization bound. In addition to a minimum storage in a node, all three constructions allow the entire data to be recovered from a minimal number of storage nodes. That is, given storage $\ell$ in each node, the entire stored data can be recovered from any $2\log_2 \ell$ for 2 parity nodes, and either $3\log_3\ell$ or $4\log_3\ell$ for 3 parity nodes. Second, in the first two constructions, a helper node accesses the minimum number of its symbols for repair of a failed node (access-optimality). The generator matrix of these codes is based on perfect matchings of complete graphs and hypergraphs, and on a rational canonical form of matrices. The goal of this paper is to provide a construction of such optimal codes over the smallest possible finite fields. For two parities, the field size is reduced by a factor of two for access-optimal codes compared to previous constructions. For three parities, in the first construction the field size is $6\log_3 \ell+1$ (or $3\log_3 \ell+1$ for fields with characteristic 2), and in the second construction the field size is larger, yet linear in $\log_3\ell$. Both constructions with 3 parities provide a significant improvement over existing previous works, since only non-explicit constructions with exponential field size (in $\log_3\ell$) were known so far.

cs.IT↗

arXiv2015

On the Structure of the $q$-Fano Plane

The $q$-Fano plane is the $q$-analog of the Steiner system $S(2,3,7)$. For any given prime power $q$ it is not known whether the $q$-Fano plane exists or not. We consider the structure of the $q$-Fano plane for any given $q$ and conclude that most of its structure is known. Even so, we were unable to determine whether it exists or not. A special attention is given for the case $q=2$ which was considered by most researchers before.

math.CO↗

arXiv2015

Equidistant Codes in the Grassmannian

Equidistant codes over vector spaces are considered. For $k$-dimensional subspaces over a large vector space the largest code is always a sunflower. We present several simple constructions for such codes which might produce the largest non-sunflower codes. A novel construction, based on the Plücker embedding, for 1-intersecting codes of $k$-dimensional subspaces over $\F_q^n$, $n \geq \binom{k+1}{2}$, where the code size is $\frac{q^{k+1}-1}{q-1}$ is presented. Finally, we present a related construction which generates equidistant constant rank codes with matrices of size $n \times \binom{n}{2}$ over $\F_q$, rank $n-1$, and rank distance $n-1$.

math.CO↗

arXiv2015

Subspace Polynomials and Cyclic Subspace Codes

Subspace codes have received an increasing interest recently due to their application in error-correction for random network coding. In particular, cyclic subspace codes are possible candidates for large codes with efficient encoding and decoding algorithms. In this paper we consider such cyclic codes and provide constructions of optimal codes for which their codewords do not have full orbits. We further introduce a new way to represent subspace codes by a class of polynomials called subspace polynomials. We present some constructions of such codes which are cyclic and analyze their parameters.

cs.IT↗

arXiv2015

Binary Polarization Kernels from Code Decompositions

In this paper, code decompositions (a.k.a. code nestings) are used to design binary polarization kernels. The proposed kernels are in general non-linear. They provide a better polarization exponent than the previously known kernels of the same dimensions. In particular, non-linear kernels of dimensions 14, 15, and 16 are constructed and are shown to have optimal asymptotic error-correction performance. The optimality is proved by showing that the exponents of these kernels achieve a new upper bound that is developed in this paper.

cs.IT↗

arXiv2015

Optimal Fractional Repetition Codes based on Graphs and Designs

Fractional repetition (FR) codes is a family of codes for distributed storage systems that allow for uncoded exact repairs having the minimum repair bandwidth. However, in contrast to minimum bandwidth regenerating (MBR) codes, where a random set of a certain size of available nodes is used for a node repair, the repairs with FR codes are table based. This usually allows to store more data compared to MBR codes. In this work, we consider bounds on the fractional repetition capacity, which is the maximum amount of data that can be stored using an FR code. Optimal FR codes which attain these bounds are presented. The constructions of these FR codes are based on combinatorial designs and on families of regular and biregular graphs. These constructions of FR codes for given parameters raise some interesting questions in graph theory. These questions and some of their solutions are discussed in this paper. In addition, based on a connection between FR codes and batch codes, we propose a new family of codes for DSS, namely fractional repetition batch codes, which have the properties of batch codes and FR codes simultaneously. These are the first codes for DSS which allow for uncoded efficient exact repairs and load balancing which can be performed by several users in parallel. Other concepts related to FR codes are also discussed.

cs.IT↗

arXiv2015

Optimal Fractional Repetition Codes and Fractional Repetition Batch Codes

Fractional repetition (FR) codes is a family of codes for distributed storage systems (DSS) that allow uncoded exact repairs with minimum repair bandwidth. In this work, we consider a bound on the maximum amount of data that can be stored using an FR code. Optimal FR codes which attain this bound are presented. The constructions of these FR codes are based on families of regular graphs, such as Turán graphs and graphs with large girth; and on combinatorial designs, such as transversal designs and generalized polygons. In addition, based on a connection between FR codes and batch codes, we propose a new family of codes for DSS, called fractional repetition batch codes, which allow uncoded efficient exact repairs and load balancing which can be performed by several users in parallel.

cs.IT↗

arXiv2014

Constructions of Snake-in-the-Box Codes for Rank Modulation

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 obtain a snake of length $M_{2n+1}=((2n+1)(2n)-1)M_{2n-1}$ for permutations of $S_{2n+1}$, from a snake of length $M_{2n-1}$ for permutations of~$S_{2n-1}$. Thus, we have $\lim\limits_{n\to \infty} \frac{M_{2n+1}}{S_{2n+1}}\approx 0.4338$, improving on the previous known ratio of $\lim\limits_{n\to \infty} \frac{1}{\sqrt{πn}}$. By using the general method we also present a direct construction. This direct construction is based on necklaces and it might yield snakes of length $\frac{(2n+1)!}{2} -2n+1$ for permutations of $S_{2n+1}$. The direct construction was applied successfully for $S_7$ and $S_9$, and hence $\lim\limits_{n\to \infty} \frac{M_{2n+1}}{S_{2n+1}}\approx 0.4743$.

cs.IT↗

arXiv2014

A New Construction for Constant Weight Codes

A new construction for constant weight codes is presented. The codes are constructed from $k$-dimensional subspaces of the vector space $\F_q^n$. These subspaces form a constant dimension code in the Grassmannian space $\cG_q(n,k)$. Some of the constructed codes are optimal constant weight codes with parameters not known before. An efficient algorithm for error-correction is given for the constructed codes. If the constant dimension code has an efficient encoding and decoding algorithms then also the constructed constant weight code has an efficient encoding and decoding algorithms.

cs.IT↗

arXiv2014

Distributed Storage Systems based on Equidistant Subspace Codes

Distributed storage systems based on equidistant constant dimension codes are presented. These equidistant codes are based on the Plücker embedding, which is essential in the repair and the reconstruction algorithms. These systems posses several useful properties such as high failure resilience, minimum bandwidth, low storage, simple algebraic repair and reconstruction algorithms, good locality, and compatibility with small fields.

cs.IT↗

arXiv2014

Optimal Ferrers Diagram Rank-Metric Codes

Optimal rank-metric codes in Ferrers diagrams are considered. Such codes consist of matrices having zeros at certain fixed positions and can be used to construct good codes in the projective space. Four techniques and constructions of Ferrers diagram rank-metric codes are presented, each providing optimal codes for different diagrams and parameters.

cs.IT↗

arXiv2014

Perfect Permutation Codes with the Kendall's $τ$-Metric

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 Kendall's $τ$-metric and consider the related codes and specifically we prove the existence of a perfect single-error-correcting code in $S_5$. Finally, we examine the existence problem of diameter perfect codes in $S_n$ and obtain a new upper bound on the size of a code in $S_n$ with even minimum Kendall's $τ$-distance.

cs.IT↗

arXiv2014

Systematic Codes for Rank Modulation

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 symbols than the number of redundancy symbols in the codes of the known constructions. In particular, for a given $t$ and for sufficiently large $k$ we can obtain $r=t+1$. The same construction is also applied to obtain related systematic error-correcting codes for multi-permutations.

cs.IT↗

arXiv2014

Local Rank Modulation for Flash Memories II

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 techniques which are suggested have some generalizations for $(1,t,n)$-LRM, $t > 3$, but the proofs will become more complicated. The enumeration problem is presented also as a purely combinatorial problem. Finally, we prove the conjecture that the size of a constant weight $(1,2,n)$-LRM Gray code with weight two is at most $2n$.

cs.IT↗

arXiv2014

The q-Analog of the Middle Levels Problem

The well-known middle levels problem is to find a Hammiltonian cycle in the graph induced from the binary Hamming graph $\cH_2(2k+1)$ by the words of weight $k$ or $k+1$. In this paper we define the $q$-analog of the middle levels problem. Let $n=2k+1$ and let $q$ be a power of a prime number. Consider the set of $(k+1)$-dimensional subspaces and the set of $k$-dimensional subspaces of $\F_q^n$. Can these subspaces be ordered in a way that for any two adjacent subspaces $X$ and $Y$, either $X \subset Y$ or $Y \subset X$? A construction method which yields many Hamiltonian cycles for any given $q$ and $k=2$ is presented.

math.CO↗