arXiv subjects

Tuvi Etzion

Publications and source records attributed to Tuvi Etzion.

At least 91 records · Page 5Linked to original sources

arXiv2013

Partial k-Parallelisms in Finite Projective Spaces

In this paper we consider the following question. What is the maximum number of pairwise disjoint $k$-spreads which exist in PG(n,q)? We prove that if k+1 divides n+1 and n>k then there exist at least two disjoint k-spreads in PG(n,q) and there exist at least $2^{k+1}-1$ pairwise disjoint $k$-spreads in PG(n,2). We also extend the known results on parallelism in a projective geometry from which the points of a given subspace were removed.

math.CO↗

arXiv2013

Problems on q-Analogs in Coding Theory

The interest in $q$-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.

cs.IT↗

arXiv2013

Existence of $q$-Analogs of Steiner Systems

Let $\F_q^n$ be a vector space of dimension $n$ over the finite field $\F_q$. A $q$-analog of a Steiner system (briefly, a $q$-Steiner system), denoted $S_q[t,k,n]$, is a set $S$ of $k$-dimensional subspaces of $\F_q^n$ such that each $t$-dimensional subspace of $\F_q^n$ is contained in exactly one element of $S$. Presently, $q$-Steiner systems are known only for $t=1$, and in the trivial cases $t = k$ and $k = n$. Invthis paper, the first nontrivial $q$-Steiner systems with $t >= 2$ are constructed. Specifically, several nonisomorphic $q$-Steiner systems $S_2[2,3,13]$ are found by requiring that their automorphism groups contain the normalizer of a Singer subgroup of $\GL(13,2)$. This approach leads to an instance of the exact cover problem, which turns out to have many solutions.

math.CO↗

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

Tilings with $n$-Dimensional Chairs and their Applications to Asymmetric Codes

An $n$-dimensional chair consists of an $n$-dimensional box from which a smaller $n$-dimensional box is removed. A tiling of an $n$-dimensional chair has two nice applications in coding for write-once memories. The first one is in the design of codes which correct asymmetric errors with limited-magnitude. The second one is in the design of $n$ cells $q$-ary write-once memory codes. We show an equivalence between the design of a tiling with an integer lattice and the design of a tiling from a generalization of splitting (or of Sidon sequences). A tiling of an $n$-dimensional chair can define a perfect code for correcting asymmetric errors with limited-magnitude. We present constructions for such tilings and prove cases where perfect codes for these type of errors do not exist.

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

Tilings by $(0.5,n)$-Crosses and Perfect Codes

The existence question for tiling of the $n$-dimensional Euclidian space by crosses is well known. A few existence and nonexistence results are known in the literature. Of special interest are tilings of the Euclidian space by crosses with arms of length one, known also as Lee spheres with radius one. Such a tiling forms a perfect code. In this paper crosses with arms of length half are considered. These crosses are scaled by two to form a discrete shape. We prove that an integer tiling for such a shape exists if and only if $n=2^t-1$ or $n=3^t-1$, $t>0$. A strong connection of these tilings to binary and ternary perfect codes in the Hamming scheme is shown.

math.CO↗

arXiv2012

Covering of Subspaces by Subspaces

Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph $\cG_q(n,r)$ by subspaces from the Grassmann graph $\cG_q(n,k)$, $k \geq r$, are discussed. The problem is of interest from four points of view: coding theory, combinatorial designs, $q$-analogs, and projective geometry. In particular we examine coverings based on lifted maximum rank distance codes, combined with spreads and a recursive construction. New constructions are given for $q=2$ with $r=2$ or $r=3$. We discuss the density for some of these coverings. Tables for the best known coverings, for $q=2$ and $5 \leq n \leq 10$, are presented. We present some questions concerning possible constructions of new coverings of smaller size.

math.CO↗

arXiv2012

Codes and Designs Related to Lifted MRD Codes

Lifted maximum rank distance (MRD) codes, which are constant dimension codes, are considered. It is shown that a lifted MRD code can be represented in such a way that it forms a block design known as a transversal design. A slightly different representation of this design makes it similar to a $q-$analog of a transversal design. The structure of these designs is used to obtain upper bounds on the sizes of constant dimension codes which contain a lifted MRD code. Codes which attain these bounds are constructed. These codes are the largest known codes for the given parameters. These transversal designs can be also used to derive a new family of linear codes in the Hamming space. Bounds on the minimum distance and the dimension of such codes are given.

cs.IT↗

arXiv2011

Simple and Robust Binary Self-Location Patterns

A simple method to generate a two-dimensional binary grid pattern, which allows for absolute and accurate self-location in a finite planar region, is proposed. The pattern encodes position information in a local way so that reading a small number of its black or white pixels at any place provides sufficient data from which the location can be decoded both efficiently and robustly.

cs.IT↗

arXiv2011

On the Existence of Perfect Codes for Asymmetric Limited-Magnitude Errors

Block codes, which correct asymmetric errors with limited-magnitude, are studied. These codes have been applied recently for error correction in flash memories. The codes will be represented by lattices and the constructions will be based on a generalization of Sidon sequences. In particular we will consider perfect codes for these type of errors.

cs.IT↗

arXiv2011

Perfect Codes for Uniform Chains Poset Metrics

The class of poset metrics is very large and contains some interesting families of metrics. A family of metrics, based on posets which are formed from disjoint chains which have the same size, is examined. A necessary and sufficient condition, for the existence of perfect single-error-correcting codes for such poset metrics, is proved.

cs.IT↗

arXiv2011

The asymptotic behavior of Grassmannian codes

The iterated Johnson bound is the best known upper bound on a size of an error-correcting code in the Grassmannian $\mathcal{G}_q(n,k)$. The iterated Schönheim bound is the best known lower bound on the size of a covering code in $\mathcal{G}_q(n,k)$. We use probabilistic methods to prove that both bounds are asymptotically attained for fixed $k$ and fixed radius, as $n$ approaches infinity. We also determine the asymptotics of the size of the best Grassmannian codes and covering codes when $n-k$ and the radius are fixed, as $n$ approaches infinity.

cs.DM↗

arXiv2011

Product Constructions for Perfect Lee Codes

A well known conjecture of Golomb and Welch is that the only nontrivial perfect codes in the Lee and Manhattan metrics have length two or minimum distance three. This problem and related topics were subject for extensive research in the last forty years. In this paper two product constructions for perfect Lee codes and diameter perfect Lee codes are presented. These constructions yield a large number of nonlinear perfect codes and nonlinear diameter perfect codes in the Lee and Manhattan metrics. A short survey and other related problems on perfect codes in the Lee and the Manhattan metrics are also discussed.

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↗

arXiv2011

Sidon Sequences and Doubly Periodic Two-Dimensional Synchronization Patterns

Sidon sequences and their generalizations have found during the years and especially recently various applications in coding theory. One of the most important applications of these sequences is in the connection of synchronization patterns. A few constructions of two-dimensional synchronization patterns are based on these sequences. In this paper we present sufficient conditions that a two-dimensional synchronization pattern can be transformed into a Sidon sequence. We also present a new construction for Sidon sequences over an alphabet of size q(q-1), where q is a power of a prime.

cs.IT↗

arXiv2010

Enumerative Coding for Grassmannian Space

The Grassmannian space $\Gr$ is the set of all $k-$dimensional subspaces of the vector space~\smash{$\F_q^n$}. Recently, codes in the Grassmannian have found an application in network coding. The main goal of this paper is to present efficient enumerative encoding and decoding techniques for the Grassmannian. These coding techniques are based on two different orders for the Grassmannian induced by different representations of $k$-dimensional subspaces of $\F_q^n$. One enumerative coding method is based on a Ferrers diagram representation and on an order for $\Gr$ based on this representation. The complexity of this enumerative coding is $O(k^{5/2} (n-k)^{5/2})$ digit operations. Another order of the Grassmannian is based on a combination of an identifying vector and a reduced row echelon form representation of subspaces. The complexity of the enumerative coding, based on this order, is $O(nk(n-k)\log n\log\log n)$ digits operations. A combination of the two methods reduces the complexity on average by a constant factor.

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