Salah A. Aly

On Quantum and Classical BCH Codes

Classical Bose-Chaudhuri-Hocquenghem (BCH) codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown that a BCH code of length n can contain its dual code only if its designed distance delta=O(radicn), and the converse is proved in the case of narrow-sense codes. Furthermore, the dimension of narrow-sense BCH codes with small design distance is completely determined, and - consequently - the bounds on their minimum distance are improved. These results make it possible to determine the parameters of quantum BCH codes in terms of their design parameters

Decentralized Coding Algorithms for Distributed Storage in Wireless Sensor Networks

We consider large-scale wireless sensor networks with n nodes, out of which k are in possession, (e.g., have sensed or collected in some other way) k information packets. In the scenarios in which network nodes are vulnerable because of, for example, limited energy or a hostile environment, it is desirable to disseminate the acquired information throughout the network so that each of the n nodes stores one (possibly coded) packet so that the original k source packets can be recovered, locally and in a computationally simple way from any k(1 + ¿) nodes for some small ¿ > 0. We develop decentralized Fountain codes based algorithms to solve this problem. Unlike all previously developed schemes, our algorithms are truly distributed, that is, nodes do not know n, k or connectivity in the network, except in their own neighborhoods, and they do not maintain any routing tables.

Fountain Codes Based Distributed Storage Algorithms for Large-Scale Wireless Sensor Networks

We consider large-scale networks with n nodes, out of which k are in possession, (e.g., have sensed or collected in some other way) k information packets. In the scenarios in which network nodes are vulnerable because of, for example, limited energy or a hostile environment, it is desirable to disseminate the acquired information throughout the network so that each of the n nodes stores one (possibly coded) packet and the original k source packets can be recovered later in a computationally simple way from any (1 + isin)k nodes for some small isin > 0. We developed two distributed algorithms for solving this problem based on simple random walks and Fountain codes. Unlike all previously developed schemes, our solution is truly distributed, that is, nodes do not know n, k or connectivity in the network, except in their own neighborhoods, and they do not maintain any routing tables. In the first algorithm, all the sensors have the knowledge of n and k. In the second algorithm, each sensor estimates these parameters through the random walk dissemination. We present analysis of the communication/transmission and encoding/decoding complexity of these two algorithms, and provide extensive simulation results as well.

A Class of Quantum LDPC Codes Constructed From Finite Geometries

Low-density parity check (LDPC) codes are a significant class of classical codes with many applications. Several good LDPC codes have been constructed using random, algebraic, and finite geometries approaches, with containing cycles of length at least six in their Tanner graphs. However, it is impossible to design a self-orthogonal parity check matrix of an LDPC code without introducing cycles of length four. In this paper, a new class of quantum LDPC codes based on lines and points of finite geometries is constructed. The parity check matrices of these codes are adapted to be self- orthogonal with containing only one cycle of length four in each pair of two rows. Also, the column and row weights, and bounds on the minimum distance of these codes are given. As a consequence, these codes can be encoded using shift-register encoding algorithms and can be decoded using iterative decoding algorithms over various quantum depolarizing channels.

Raptor codes based distributed storage algorithms for wireless sensor networks

We consider a distributed storage problem in a large-scale wireless sensor network with n nodes among which k acquire (sense) independent data. The goal is to disseminate the acquired information throughout the network so that each of the n sensors stores one possibly coded packet and the original k data packets can be recovered later in a computationally simple way from any (1 + isin)k of nodes for some small isin Gt 0. We propose two Raptor codes based distributed storage algorithms for solving this problem. In the first algorithm, all the sensors have the knowledge of n and k. In the second one, we assume that no sensor has such global information.

Secure hop-by-hop aggregation of end-to-end concealed data in wireless sensor networks

In-network data aggregation is an essential technique in mission critical wireless sensor networks (WSNs) for achieving effective transmission and hence better power conservation. Common security protocols for aggregated WSNs are either hop-by-hop or end-to-end, each of which has its own encryption schemes considering different security primitives. End-to-end encrypted data aggregation protocols introduce maximum data secrecy with in-efficient data aggregation and more vulnerability to active attacks, while hop-by-hop data aggregation protocols introduce maximum data integrity with efficient data aggregation and more vulnerability to passive attacks. In this paper, we propose a secure aggregation protocol for aggregated WSNs deployed in hostile environments in which dual attack modes are present. Our proposed protocol is a blend of flexible data aggregation as in hop-by-hop protocols and optimal data confidentiality as in end-to-end protocols. Our protocol introduces an efficient O(1) heuristic for checking data integrity along with cost-effective heuristic-based divide and conquer attestation process which is O(ln n) in average -O(n) in the worst scenario- for further verification of aggregated results.

Primitive Quantum BCH Codes over Finite Fields

An attractive feature of BCH codes is that one can infer valuable information from their design parameters (length, size of the finite field, and designed distance), such as bounds on the minimum distance and dimension of the code. In this paper, it is shown that one can also deduce from the design parameters whether or not a primitive, narrow-sense BCH contains its Euclidean or Hermitian dual code. This information is invaluable in the construction of quantum BCH codes. A new proof is provided for the dimension of BCH codes with small designed distance, and simple bounds on the minimum distance of such codes and their duals are derived as a consequence. These results allow us to derive the parameters of two families of primitive quantum BCH codes as a function of their design parameters

Quantum Convolutional BCH Codes

Quantum convolutional codes can be used to protect a sequence of qubits of arbitrary length against decoherence. We introduce two new families of quantum convolutional codes. Our construction is based on an algebraic method which allows to construct classical convolutional codes from block codes, in particular BCH codes. These codes have the property that they contain their Euclidean, respectively Hermitian, dual codes. Hence, they can be used to define quantum convolutional codes by the stabilizer code construction. We compute BCH-like bounds on the free distances which can be controlled as in the case of block codes, and establish that the codes have non-catastrophic encoders.

Network Protection Codes Against Link Failures Using Network Coding

Protecting against link failures in communication networks is essential to increase robustness, accessibility, and reliability of data transmission. Recently, network coding has been proposed as a solution to provide agile and cost efficient network protection against link failures, which does not require data rerouting, or packet retransmission. To achieve this, separate paths have to be provisioned to carry encoded packets, hence requiring either the addition of extra links, or reserving some of the resources for this purpose. In this paper, we propose network protection codes against a single link failure using network coding, where a separate path using reserved links is not needed. In this case portions of the link capacities are used to carry the encoded packets. The scheme is extended to protect against multiple link failures and can be implemented at an overlay layer. Although this leads to reducing the network capacity, the network capacity reduction is asymptotically small in most cases of practical interest. We demonstrate that such network protection codes are equivalent to error correcting codes for erasure channels. Finally, we study the encoding and decoding operations of such codes over the binary field.

Subsystem code constructions

Subsystem codes are the most versatile class of quantum error-correcting codes known to date that combine the best features of all known passive and active error-control schemes. The subsystem code is a subspace of the quantum state space that is decomposed into a tensor product of two vector spaces: the subsystem and the co-subsystem. A generic method to derive subsystem codes from existing subsystem codes is given that allows one to trade the dimensions of subsystem and co-subsystem while maintaining or improving the minimum distance. As a consequence, it is shown that all pure MDS subsystem codes are derived from MDS stabilizer codes. The existence of numerous families of MDS subsystem codes is established. Propagation rules are derived that allow one to obtain longer and shorter subsystem codes from given subsystem codes. Furthermore, propagation rules are derived that allow one to construct a new subsystem code by combining two given subsystem codes.