Mark G. Karpovsky

Bio: Mark G. Karpovsky is an academic researcher from Boston University. The author has contributed to research in topics: Error detection and correction & Linear code. The author has an hindex of 32, co-authored 134 publications receiving 3681 citations. Previous affiliations of Mark G. Karpovsky include Binghamton University & Alcatel-Lucent.

On a new class of codes for identifying vertices in graphs

Abstract: We investigate a new class of codes for the optimal covering of vertices in an undirected graph G such that any vertex in G can be uniquely identified by examining the vertices that cover it. We define a ball of radius t centered on a vertex /spl upsi/ to be the set of vertices in G that are at distance at most t from /spl upsi/. The vertex /spl upsi/ is then said to cover itself and every other vertex in the ball with center /spl upsi/. Our formal problem statement is as follows: given an undirected graph G and an integer t/spl ges/1, find a (minimal) set C of vertices such that every vertex in G belongs to a unique set of balls of radius t centered at the vertices in C. The set of vertices thus obtained constitutes a code for vertex identification. We first develop topology-independent bounds on the size of C. We then develop methods for constructing C for several specific topologies such as binary cubes, nonbinary cubes, and trees. We also describe the identification of sets of vertices using covering codes that uniquely identify single vertices. We develop methods for constructing optimal topologies that yield identifying codes with a minimum number of codewords. Finally, we describe an application of the theory developed in this paper to fault diagnosis of multiprocessor systems.

Covering radius---Survey and recent results

Abstract: All known results on covering radius are presented, as well as some new results. There are a number of upper and lower bounds, including asymptotic results, a few exact determinations of covering radius, some extensive relations with other aspects of coding theory through the Reed-Muller codes, and new results on the least covering radius of any linear [n,k] code. There is also a recent result on the complexity of computing the covering radius.

Robust protection against fault-injection attacks on smart cards implementing the advanced encryption standard

Abstract: We present a method of protecting a hardware implementation of the advanced encryption standard (AES) against a side-channel attack known as differential fault analysis attack The method uses systematic nonlinear (cubic) robust error detecting codes Error-detecting capabilities of these codes depend not just on error patterns (as in the case of linear codes) but also on data at the output of the device which is protected by the code and this data is unknown to the attacker since it depends on the secret key In addition to this, the proposed nonlinear (n,k)-codes reduce the fraction of undetectable errors from 2/sup -r/ to 2/sup -2r/ as compared to the corresponding (n,k) linear code (where n - k = r and k >= r) We also present results on a FPGA implementation of the proposed protection scheme for AES as well as simulation results on efficiency of the robust codes

A data compression technique for built-in self-test

Abstract: A data compression technique called self-testable and error-propagating space compression is proposed and analyzed. Faults in a realization of Exclusive-OR and Exclusive-NOR gates are analyzed, and the use of these gates in the design of self-testing and error propagating space compressors is discussed. It is argued that the proposed data-compression technique reduce the hardware complexity in built-in self-test (BIST) logic designs using external tester environments. >

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes

Abstract: Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson (1967), Kerdock (1972), Preparata (1968), Goethals (1974), and Delsarte-Goethals (1975). It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z/sub 4/, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z/sub 4/ domain implies that the binary images have dual weight distributions. The Kerdock and "Preparata" codes are duals over Z/sub 4/-and the Nordstrom-Robinson code is self-dual-which explains why their weight distributions are dual to each other. The Kerdock and "Preparata" codes are Z/sub 4/-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z/sub 4/, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the "Preparata" code and a Hadamard-transform soft-decision decoding algorithm for the I(Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over Z/sub 4/, but extended Hamming codes of length n/spl ges/32 and the Golay code are not. Using Z/sub 4/-linearity, a new family of distance regular graphs are constructed on the cosets of the "Preparata" code. >

The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes

Abstract: Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals. It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z_4, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z_4 domain implies that the binary images have dual weight distributions. The Kerdock and "Preparata" codes are duals over Z_4 -- and the Nordstrom-Robinson code is self-dual -- which explains why their weight distributions are dual to each other. The Kerdock and "Preparata" codes are Z_4-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z_4, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the "Preparata" code and a Hadamard-transform soft-decision decoding algorithm for the Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over Z_4, but extended Hamming codes of length n >= 32 and the Golay code are not. Using Z_4-linearity, a new family of distance regular graphs are constructed on the cosets of the "Preparata" code.

Grid coverage for surveillance and target location in distributed sensor networks

Abstract: We present novel grid coverage strategies for effective surveillance and target location in distributed sensor networks. We represent the sensor field as a grid (two or three-dimensional) of points (coordinates) and use the term target location to refer to the problem of locating a target at a grid point at any instant in time. We first present an integer linear programming (ILP) solution for minimizing the cost of sensors for complete coverage of the sensor field. We solve the ILP model using a representative public-domain solver and present a divide-and-conquer approach for solving large problem instances. We then use the framework of identifying codes to determine sensor placement for unique target location, We provide coding-theoretic bounds on the number of sensors and present methods for determining their placement in the sensor field. We also show that grid-based sensor placement for single targets provides asymptotically complete (unambiguous) location of multiple targets in the grid.