We consider transmissions of secure messages over a burst erasure wiretap channel under decoding delay constraint. For block codes we introduce and study delay optimal secure burst erasure correcting (DO-SBE) codes that provide perfect security and recover a burst of erasures of a limited length with minimum possible delay. Our explicit constructions of DO-SBE block codes achieve maximum secrecy rate. We also consider a model of a burst erasure wiretap channel for the streaming setup, where in any sliding window of a given size, in a stream of encoded source packets, the eavesdropper is able to observe packets in an interval of a given size. For that model we obtain an information theoretic upper bound on the secrecy rate for delay optimal streaming codes. We show that our block codes can be used for construction of delay optimal burst erasure correcting streaming codes which provide perfect security and meet the upper bound for a certain class of code parameters.

Fractional repetition (FR) codes is a class of codes which were recently introduced for distributed storage systems. These codes are intended for exact uncoded repair of node failures, by downloading symbols from a suitable subset of surviving nodes. The repair procedure in FR codes is table based, unlike the regenerating codes, where the repair of a failed node is possible using arbitrary subset of a given size from surviving nodes. The advantage of this relaxation is that it allows to achieve low complexity in repair process, while these codes have minimum repair bandwidth like minimum bandwidth regenerating (MBR) codes. In this paper we give new and simple constructions for universally good FR codes based on partially ordered sets. These codes allow for efficient uncoded repair and the resulting designs are scalable and easy to implement. In particular, they allow to store larger files as compared to MBR codes. Furthermore, the constructions can be extended to FR codes for heterogeneous storage systems.

In this paper we consider a wiretap channel II with an active eavesdropper. Aggarwal et al (2009) were the first who studied the Ozarow-Wyner's binary wiretap channel II in the presence of an active eavesdropper. They derived achievable secrecy rates for two modification models where the eavesdropper can erase/replace the bits he observes. The existence of better achievable rates remains an open problem. Here we study a model with a less powerful eavesdropper. The eavesdropper is able now to observe any interval of μ symbol positions and erase the symbols in any interval of l positions of a transmitted codeword. We present an explicit construction of nested linear codes that achieve maximum secrecy rate for the finite length coding regime, with perfect secrecy and zero-error decoding, for any admissible code parameters.

Given a set of $n$ real numbers, if the sum of the elements of every subset of size larger than $k$ is negative, what is the maximum number of subsets of nonnegative sum? In this note we show that the answer is $\binom{n-1}{k-1} + \binom{n-1}{k-2} + \cdots + \binom{n-1}{0}+1$, settling a problem of Tsukerman. We provide two proofs; the first establishes and applies a weighted version of Hall's theorem, and the second is based on an extension of the nonuniform Erdös--Ko--Rado theorem.

Motivated by iterative decoding techniques for the binary erasure channel Hollmann and Tolhuizen introduced and studied the notion of generic erasure correcting sets for linear codes. A generic (r,s)-erasure correcting set generates for all codes of codimensionra parity check matrix that allows iterative decoding of all correctable erasure patterns of sizesor less. The problem is to derive bounds on the minimum sizeF(r,s) of generic erasure correcting sets and to find constructions for such sets. In this paper, we continue the study of these sets. We derive better lower and upper bounds. Hollmann and Tolhuizen also introduced the stronger notion of (r,s)-sets and derived bounds for their minimum sizeG(r,s) . Here also we improve these bounds. We observe that these two conceps are closely related to so calleds-wise intersecting codes, an area, in whichG(r,s) has been studied primarily with respect to ratewise performance. We derive connections. Finally, we observed that hypergraph covering can be used for both problems to derive good upper bounds.

A statistical database (SDB) is a database that is used to return statistical information derived from the records to user queries for statistical data analysis. Sometimes, by correlating enough statistics, confidential data (stored in an SDB) about an individual can be inferred. Examples of confidential information stored in an SDB might be salaries or data concerning the medical history of individuals. An important problem is to provide security to SDBs against the disclosure of confidential information. An SDB is said to be secure if no protected data can be inferred from the available queries. One of the security-control methods suggested in the literature consists of query restriction: the security problem is to limit the use of the SDB, introducing a control mechanism, such that no protected data can be obtained from the available queries. Chin and Ozsoyoglu [IEEE Trans. Software Engrg., 8 (1982), pp. 574–582] introduced a control mechanism, called AUDIT EXPERT, where only SUM queries, that is, only certain sums of individual records, are available for the users. This SUM query model leads to several challenging optimization problems. Assume there are n numeric records $\{z_1,\ldots,z_n\}$ stored in database. A natural problem is to maximize the number of answerable SUM queries, that is, the number of subset sums of $\{z_1,\ldots,z_n\}$ (possibly with some additional constraints), that can be returned, such that none of numbers $z_i$ (or sums of subsets of size not exceeding a specified number) can be inferred from these queries. In this paper we give tight bounds for this number under constraints on size and dimension of query subsets.

We consider transmission of messages encoded into binary matrices to be send over a parallel asymmetric channel. Two different channel models are introduced. They depend on the conditions on errors in the subchannels. For each channel model codes are constructed for error correction and for error detection and bounds are derived.

A statistical database (SDB) is a database that is used to provide statistical information (such as average, sums etc.), derived from the records, to user queries for statistical data analysis. Sometimes, by correlating enough statistics, protected data about an individual can be inferred. The security problem is to limit the use of the SDB such that only statistical information is available and no protected individual data can be obtained from the available queries. When users are able to infer protected information in the SDB from responses to queries, the SDB is said to be compromised. The goal is to maximize the number of available queries without compromise. One of the natural restrictions for the prevention of database compromise is to allow only SUM queries, that is, only certain sums of individual records are available for the users. In this paper we discuss security problems for databases where only SUM queries with certain constraints are allowed. Assume there are n numeric records {z1,..., zn} stored in a database. The problem is to find the largest number of subset sums of {z1,..., zn} (maybe with some other constraints) that can be disclosed such that none of numbers zi(or even sums of small subsets) can be determined from these sums. Some tight bounds for this number (under certain constraints on size and dimension of query subsets) are presented

We introduce and study error correcting codes for parallel synchronous communication via Z-channels. Constructions for error detecting/correcting codes are presented. More generally, we consider array codes for Z-channels