Distinct difference configurations: multihop paths and key predistribution in sensor networks

A distinct difference configuration is a set of points in Z2 with the property that the vectors (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 Bh 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

Two-dimensional patterns with distinct differences; constructions, bounds, and maximal anticodes

A two-dimensional (2-D) grid with dots is called a 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 2-D shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid

Two Optimal One-Error-Correcting Codes of Length 13 That Are Not Doubly Shortened Perfect Codes

The doubly shortened perfect codes of length 13 are classified utilizing the classification of perfect codes in [P.R.J. \"Osterg{\aa}rd and O. Pottonen, The perfect binary one-error-correcting codes of length 15: Part I - Classification, IEEE Trans. Inform. Theory, to appear]; there are 117821 such (13,512,3) codes. By applying a switching operation to those codes, two more (13,512,3) codes are obtained, which are then not doubly shortened perfect codes.Comment: v2: a correction concerning shortened codes of length 1

PIR Array Codes with Optimal Virtual Server Rate

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 PIR 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 $(1/s)$ of the bits of the database; here $s$ is a fixed rational number with $s > 1$. A PIR array code with the $k$-PIR property enables a $k$-server PIR protocol (with $k\leq m$) to be emulated on $m$ servers, with the overall storage requirements of the protocol being reduced. The communication complexity of a PIR protocol reduces as $k$ grows, so the virtual server rate, defined to be $k/m$, is an important parameter. We study the maximum virtual server rate of a PIR array code with the $k$-PIR property. We present upper bounds on the achievable virtual server rate, some constructions, and ideas how to obtain PIR array codes with the highest possible virtual server rate. In particular, we present constructions that asymptotically meet our upper bounds, and the exact largest virtual server rate is obtained when $1 < s \leq 2$. A $k$-PIR code (and similarly a $k$-PIR array code) is also a locally repairable code with symbol availability $k-1$. Such a code ensures $k$ parallel reads for each information symbol. So the virtual server rate is very closely related to the symbol availability of the code when used as a locally repairable code. The results of this paper are discussed also in this context, where subspace codes also have an important role

New Upper Bounds on Codes via Association Schemes and Linear Programming

Let A(n, d) denote the maximum number of codewords in a binary code of length n and minimum Hamming distance d. Upper and lower bounds on A(n, d) have been a subject for extensive research. In this paper we examine upper bounds on A(n, d) as a special case of bounds on the size of subsets in metric association scheme. We will first obtain general bounds on the size of such subsets, apply these bounds to the binary Hamming scheme, and use linear programming to further improve the bounds. We show that the sphere packing bound and the Johnson bound as well as other bounds are special cases of one of the bounds obtained from association schemes. Specific bounds on A(n, d) as well as on the sizes of constant weight codes are also discussed

A Complete Characterization of Irreducible Cyclic Orbit Codes and their Pl\"ucker Embedding

Constant dimension codes are subsets of the finite Grassmann variety. The study of these codes is a central topic in random linear network coding theory. Orbit codes represent a subclass of constant dimension codes. They are defined as orbits of a subgroup of the general linear group on the Grassmannian. This paper gives a complete characterization of orbit codes that are generated by an irreducible cyclic group, i.e. a group having one generator that has no non-trivial invariant subspace. We show how some of the basic properties of these codes, the cardinality and the minimum distance, can be derived using the isomorphism of the vector space and the extension field. Furthermore, we investigate the Pl\"ucker embedding of these codes and show how the orbit structure is preserved in the embedding.Comment: submitted to Designs, Codes and Cryptograph

Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound

We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show relations between them. A slightly improved version of the so-called linkage construction is presented which is e.g. used to construct constant dimension codes with subspace distance $d=4$, dimension $k=3$ of the codewords for all field sizes $q$, and sufficiently large dimensions $v$ of the ambient space, that exceed the MRD bound, for codes containing a lifted MRD code, by Etzion and Silberstein.Comment: 30 pages, 3 table

PIR schemes with small download complexity and low storage requirements

Shah, Rashmi and Ramchandran recently considered a model for Private Information Retrieval (PIR) where a user wishes to retrieve one of several Ä-bit messages from a set of n non-colluding servers. Their security model is information-theoretic. Their paper is the first to consider a model for PIR in which the database is not necessarily replicated, so allowing distributed storage techniques to be used. Shah et al. show that at least Ä+1 bits must be downloaded from servers, and describe a scheme with linear total storage (in R) that downloads between 2R and 3R bits. For any positive e, we provide a construction with the same storage property, that requires at most (1 + e)R bits to be downloaded; moreover one variant of our scheme only requires each server to store a bounded number of bits (in the sense of being bounded by a function that is independent of R). We also provide variants of a scheme of Shah et al which downloads exactly R +1 bits and has quadratic total storage. Finally, we simplify and generalise a lower bound due to Shah et al. on the download complexity of a PIR scheme. In a natural model, we show that an n-server PIR scheme requires at least nR/(n - 1) download bits in many cases, and provide a scheme that meets this bound. This paper provides various bounds on the download complexity of a PIR scheme, generalising those of Shah et al.\ to the case when the number $n$ of servers is bounded, and providing links with classical techniques due to Chor et al. The paper also provides a range of constructions for PIR schemes that are either simpler or perform better than previously known schemes. These constructions include explicit schemes that achieve the best asymptotic download complexity of Sun and Jafar with significantly lower upload complexity, and general techniques for constructing a scheme with good worst case download complexity from a scheme with good download complexity on average

PIR schemes with small download complexity and low storage requirements

Shah, Rashmi and Ramchandran recently considered a model for Private Information Retrieval (PIR) where a user wishes to retrieve one of several Ä-bit messages from a set of n non-colluding servers. Their security model is information-theoretic. Their paper is the first to consider a model for PIR in which the database is not necessarily replicated, so allowing distributed storage techniques to be used. Shah et al. show that at least Ä+1 bits must be downloaded from servers, and describe a scheme with linear total storage (in R) that downloads between 2R and 3R bits. For any positive e, we provide a construction with the same storage property, that requires at most (1 + e)R bits to be downloaded; moreover one variant of our scheme only requires each server to store a bounded number of bits (in the sense of being bounded by a function that is independent of R). We also provide variants of a scheme of Shah et al which downloads exactly R +1 bits and has quadratic total storage. Finally, we simplify and generalise a lower bound due to Shah et al. on the download complexity of a PIR scheme. In a natural model, we show that an n-server PIR scheme requires at least nR/(n - 1) download bits in many cases, and provide a scheme that meets this bound. This paper provides various bounds on the download complexity of a PIR scheme, generalising those of Shah et al.\ to the case when the number $n$ of servers is bounded, and providing links with classical techniques due to Chor et al. The paper also provides a range of constructions for PIR schemes that are either simpler or perform better than previously known schemes. These constructions include explicit schemes that achieve the best asymptotic download complexity of Sun and Jafar with significantly lower upload complexity, and general techniques for constructing a scheme with good worst case download complexity from a scheme with good download complexity on average