arXiv subjects
Publications and source records attributed to Tuvi Etzion.
In this paper we consider the existence of nontrivial perfect codes in the Johnson graph J(n,w). We present combinatorial and number theory techniques to provide necessary conditions for existence of such codes and reduce the range of parameters in which 1-perfect and 2-perfect codes may exist.
In this paper we construct multidimensional codes with high dimension. The codes can correct high dimensional errors which have the form of either small clusters, or confined to an area with a small radius. We also consider small number of errors in a small area. The clusters which are discussed are mainly spheres such as semi-crosses and crosses. Also considered are clusters with small number of errors such as 2-bursts, two errors in various clusters, and three errors on a line. Our main focus is on the redundancy of the codes when the most dominant parameter is the dimension of the code.
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.
Constant dimension codes, with a prescribed minimum distance, have found recently an application in network coding. All the codewords in such a code are subspaces of $\F_q^n$ with a given dimension. A computer search for large constant dimension codes is usually inefficient since the search space domain is extremely large. Even so, we found that some constant dimension lexicodes are larger than other known codes. We show how to make the computer search more efficient. In this context we present a formula for the computation of the distance between two subspaces, not necessarily of the same dimension.
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.
Folding a sequence $S$ into a multidimensional box is a well-known method which is used as a multidimensional coding technique. The operation of folding is generalized in a way that the sequence $S$ can be folded into various shapes and not just a box. The new definition of folding is based on a lattice tiling for the given shape $\cS$ and a direction in the $D$-dimensional integer grid. Necessary and sufficient conditions that a lattice tiling for $\cS$ combined with a direction define a folding of a sequence into $\cS$ are derived. The immediate and most impressive application is some new lower bounds on the number of dots in two-dimensional synchronization patterns. This can be also generalized for multidimensional synchronization patterns. The technique and its application for two-dimensional synchronization patterns, raise some interesting problems in discrete geometry. We will also discuss these problems. It is also shown how folding can be used to construct multidimensional error-correcting codes. Finally, by using the new definition of folding, multidimensional pseudo-random arrays with various shapes are generated.
A two-dimensional grid with dots is called a \emph{configuration with distinct differences} if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many applications such as radar, sonar, physical alignment, and time-position synchronization. Rather than restricting dots to lie in a square or rectangle, as previously studied, we restrict the maximum distance between dots of the configuration; the motivation for this is a new application of such configurations to key distribution in wireless sensor networks. We consider configurations in the hexagonal grid as well as in the traditional square grid, with distances measured both in the Euclidean metric, and in the Manhattan or hexagonal metrics. We note that these configurations are confined inside maximal anticodes in the corresponding grid. We classify maximal anticodes for each diameter in each grid. We present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes. Our bounds settle (in the negative) a question of Golomb and Taylor on the existence of honeycomb arrays of arbitrarily large size. We present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various two-dimensional shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid.
A distinct difference configuration is a set of points in $\mathbb{Z}^2$ with the property that the vectors (\emph{difference vectors}) connecting any two of the points are all distinct. Many specific examples of these configurations have been previously studied: the class of distinct difference configurations includes both Costas arrays and sonar sequences, for example. Motivated by an application of these structures in key predistribution for wireless sensor networks, we define the $k$-hop coverage of a distinct difference configuration to be the number of distinct vectors that can be expressed as the sum of $k$ or fewer difference vectors. This is an important parameter when distinct difference configurations are used in the wireless sensor application, as this parameter describes the density of nodes that can be reached by a short secure path in the network. We provide upper and lower bounds for the $k$-hop coverage of a distinct difference configuration with $m$ points, and exploit a connection with $B_{h}$ sequences to construct configurations with maximal $k$-hop coverage. We also construct distinct difference configurations that enable all small vectors to be expressed as the sum of two of the difference vectors of the configuration, an important task for local secure connectivity in the application.
Folding a sequence $S$ into a multidimensional box is a method that is used to construct multidimensional codes. The well known operation of folding is generalized in a way that the sequence $S$ can be folded into various shapes. The new definition of folding is based on lattice tiling and a direction in the $D$-dimensional grid. There are potentially $\frac{3^D-1}{2}$ different folding operations. Necessary and sufficient conditions that a lattice combined with a direction define a folding are given. The immediate and most impressive application is some new lower bounds on the number of dots in two-dimensional synchronization patterns. This can be also generalized for multidimensional synchronization patterns. We show how folding can be used to construct multidimensional error-correcting codes and to generate multidimensional pseudo-random arrays.
Coding in the projective space has received recently a lot of attention due to its application in network coding. Reduced row echelon form of the linear subspaces and Ferrers diagram can play a key role for solving coding problems in the projective space. In this paper we propose a method to design error-correcting codes in the projective space. We use a multilevel approach to design our codes. First, we select a constant weight code. Each codeword defines a skeleton of a basis for a subspace in reduced row echelon form. This skeleton contains a Ferrers diagram on which we design a rank-metric code. Each such rank-metric code is lifted to a constant dimension code. The union of these codes is our final constant dimension code. In particular the codes constructed recently by Koetter and Kschischang are a subset of our codes. The rank-metric codes used for this construction form a new class of rank-metric codes. We present a decoding algorithm to the constructed codes in the projective space. The efficiency of the decoding depends on the efficiency of the decoding for the constant weight codes and the rank-metric codes. Finally, we use puncturing on our final constant dimension codes to obtain large codes in the projective space which are not constant dimension.
Codes in the Grassmannian space have found recently application in network coding. Representation of $k$-dimensional subspaces of $\F_q^n$ has generally an essential role in solving coding problems in the Grassmannian, and in particular in encoding subspaces of the Grassmannian. Different representations of subspaces in the Grassmannian are presented. We use two of these representations for enumerative encoding of the Grassmannian. One enumerative encoding is based on Ferrers diagrams representation of subspaces; and another is based on identifying vector and reduced row echelon form representation of subspaces. A third method which combine the previous two is more efficient than the other two enumerative encodings.
A constant-rate encoder--decoder pair is presented for a fairly large family of two-dimensional (2-D) constraints. Encoding and decoding is done in a row-by-row manner, and is sliding-block decodable. Essentially, the 2-D constraint is turned into a set of independent and relatively simple one-dimensional (1-D) constraints; this is done by dividing the array into fixed-width vertical strips. Each row in the strip is seen as a symbol, and a graph presentation of the respective 1-D constraint is constructed. The maxentropic stationary Markov chain on this graph is next considered: a perturbed version of the corresponding probability distribution on the edges of the graph is used in order to build an encoder which operates in parallel on the strips. This perturbation is found by means of a network flow, with upper and lower bounds on the flow through the edges. A key part of the encoder is an enumerative coder for constant-weight binary words. A fast realization of this coder is shown, using floating-point arithmetic.
In this work we present error-correcting codes for random network coding based on rank- metric codes, Ferrers diagrams, and puncturing. For most parameters, the constructed codes are larger than all previously known codes.
In this paper we present several constructions to generate codes for correcting a multidimensional cluster-error. The goal is to correct a cluster-error whose shape can be a box-error, a Lee sphere error, or an error with an arbitrary shape. Our codes have very low redundancy, close to optimal, and large range of parameters of arrays and clusters. Our main results are summarized as follows: 1) A construction of two-dimensional codes capable to correct a rectangular-error with considerably more flexible parameters from previously known constructions. Another advantage of this construction is that it is easily generalized for D dimensions. 2) A novel method based on D colorings of the D-dimensional space for constructing D-dimensional codes correcting D-dimensional cluster-error of various shapes. This method is applied efficiently to correct a D-dimensional cluster error of parameters not covered efficiently by previous onstructions. 3) A transformation of the D-dimensional space into another D-dimensional space such that a D-dimensional Lee sphere is transformed into a shape located in a D-dimensional box of a relatively small size. We use the previous constructions to correct a D-dimensional error whose shape is a D-dimensional Lee sphere. 4) Applying the coloring method to correct more efficiently a two-dimensional error whose shape is a Lee sphere. The D-dimensional case is also discussed. 5) A construction of one-dimensional codes capable to correct a burst-error of length b in which the number of erroneous positions is relatively small compared to b. This construction is generalized for D-dimensional codes. 6) Applying the constructions correcting a Lee sphere error and a cluster-error with small number of erroneous positions, to correct an arbitrary cluster-error.
Upper bounds on the maximum number of codewords in a binary code of a given length and minimum Hamming distance are considered. New bounds are derived by a combination of linear programming and counting arguments. Some of these bounds improve on the best known analytic bounds. Several new record bounds are obtained for codes with small lengths.
For $z_1,z_2,z_3 \in \Z^n$, the \emph{tristance} $d_3(z_1,z_2,z_3)$ is a generalization of the $L_1$-distance on $\Z^n$ to a quantity that reflects the relative dispersion of three points rather than two. A tristance anticode $\cA_d$ of diameter $d$ is a subset of $\Z^n$ with the property that $d_3(z_1,z_2,z_3) \leq d$ for all $z_1,z_2,z_3 \in \cA_d$. An anticode is optimal if it has the largest possible cardinality for its diameter $d$. We determine the cardinality and completely classify the optimal tristance anticodes in $\Z^2$ for all diameters $d \ge 1$. We then generalize this result to two related distance models: a different distance structure on $\Z^2$ where $d(z_1,z_2) = 1$ if $z_1,z_2$ are adjacent either horizontally, vertically, or diagonally, and the distance structure obtained when $\Z^2$ is replaced by the hexagonal lattice $A_2$. We also investigate optimal tristance anticodes in $\Z^3$ and optimal quadristance anticodes in $\Z^2$, and provide bounds on their cardinality. We conclude with a brief discussion of the applications of our results to multi-dimensional interleaving schemes and to connectivity loci in the game of Go.