On Network Coding Capacity - Matroidal Networks and Network Capacity Regions

One fundamental problem in the field of network coding is to determine the network coding capacity of networks under various network coding schemes. In this thesis, we address the problem with two approaches: matroidal networks and capacity regions. In our matroidal approach, we prove the converse of the theorem which states that, if a network is scalar-linearly solvable then it is a matroidal network associated with a representable matroid over a finite field. As a consequence, we obtain a correspondence between scalar-linearly solvable networks and representable matroids over finite fields in the framework of matroidal networks. We prove a theorem about the scalar-linear solvability of networks and field characteristics. We provide a method for generating scalar-linearly solvable networks that are potentially different from the networks that we already know are scalar-linearly solvable. In our capacity region approach, we define a multi-dimensional object, called the network capacity region, associated with networks that is analogous to the rate regions in information theory. For the network routing capacity region, we show that the region is a computable rational polytope and provide exact algorithms and approximation heuristics for computing the region. For the network linear coding capacity region, we construct a computable rational polytope, with respect to a given finite field, that inner bounds the linear coding capacity region and provide exact algorithms and approximation heuristics for computing the polytope. The exact algorithms and approximation heuristics we present are not polynomial time schemes and may depend on the output size.Comment: Master of Engineering Thesis, MIT, September 2010, 70 pages, 10 figure

High-Speed Visible Light Indoor Networks Based on Optical Orthogonal Codes and Combinatorial Designs

Interconnecting devices in an indoor environment using the illumination system and white light emitting diodes (LED) requires adaptive networking techniques that can provide network access for multiple users. Two techniques based on multilevel signaling and optical orthogonal codes (OOC) are explored in this paper in order to provide simultaneous multiple access in an indoor multiuser network. Balanced incomplete block designs (BIBD) are used to construct multilevel symbols for M-ary signaling. Using these multilevel symbols we are able to control the optical peak to average power ratio (PAPR) in the system, and hereby control the dimming level. In the first technique, the M-ary data of each user is first encoded using the OOC codeword that is assigned to that user, and then it is fed into a BIBD encoder to generate a multilevel signal. The second multiple access method uses sub-sets of a BIBD code to apply multilevel expurgated pulse-position modulation (MEPPM) to the data of each user. While the first approach has a larger Hamming distance between the symbols of each user, the latter can provide higher bit-rates for users in VLC systems with bandwidth-limited LEDs

Some Applications of Coding Theory in Computational Complexity

Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable error-correcting codes, and their applications to complexity theory and to cryptography. Locally decodable codes are error-correcting codes with sub-linear time error-correcting algorithms. They are related to private information retrieval (a type of cryptographic protocol), and they are used in average-case complexity and to construct ``hard-core predicates'' for one-way permutations. Locally testable codes are error-correcting codes with sub-linear time error-detection algorithms, and they are the combinatorial core of probabilistically checkable proofs

Combinatorial Channel Signature Modulation for Wireless ad-hoc Networks

In this paper we introduce a novel modulation and multiplexing method which facilitates highly efficient and simultaneous communication between multiple terminals in wireless ad-hoc networks. We term this method Combinatorial Channel Signature Modulation (CCSM). The CCSM method is particularly efficient in situations where communicating nodes operate in highly time dispersive environments. This is all achieved with a minimal MAC layer overhead, since all users are allowed to transmit and receive at the same time/frequency (full simultaneous duplex). The CCSM method has its roots in sparse modelling and the receiver is based on compressive sampling techniques. Towards this end, we develop a new low complexity algorithm termed Group Subspace Pursuit. Our analysis suggests that CCSM at least doubles the throughput when compared to the state-of-the art.Comment: 6 pages, 7 figures, to appear in IEEE International Conference on Communications ICC 201

Optimal Deterministic Polynomial-Time Data Exchange for Omniscience

We study the problem of constructing a deterministic polynomial time algorithm that achieves omniscience, in a rate-optimal manner, among a set of users that are interested in a common file but each has only partial knowledge about it as side-information. Assuming that the collective information among all the users is sufficient to allow the reconstruction of the entire file, the goal is to minimize the (possibly weighted) amount of bits that these users need to exchange over a noiseless public channel in order for all of them to learn the entire file. Using established connections to the multi-terminal secrecy problem, our algorithm also implies a polynomial-time method for constructing a maximum size secret shared key in the presence of an eavesdropper. We consider the following types of side-information settings: (i) side information in the form of uncoded fragments/packets of the file, where the users' side-information consists of subsets of the file; (ii) side information in the form of linearly correlated packets, where the users have access to linear combinations of the file packets; and (iii) the general setting where the the users' side-information has an arbitrary (i.i.d.) correlation structure. Building on results from combinatorial optimization, we provide a polynomial-time algorithm (in the number of users) that, first finds the optimal rate allocations among these users, then determines an explicit transmission scheme (i.e., a description of which user should transmit what information) for cases (i) and (ii)

Statistical mechanics of error exponents for error-correcting codes

Error exponents characterize the exponential decay, when increasing message length, of the probability of error of many error-correcting codes. To tackle the long standing problem of computing them exactly, we introduce a general, thermodynamic, formalism that we illustrate with maximum-likelihood decoding of low-density parity-check (LDPC) codes on the binary erasure channel (BEC) and the binary symmetric channel (BSC). In this formalism, we apply the cavity method for large deviations to derive expressions for both the average and typical error exponents, which differ by the procedure used to select the codes from specified ensembles. When decreasing the noise intensity, we find that two phase transitions take place, at two different levels: a glass to ferromagnetic transition in the space of codewords, and a paramagnetic to glass transition in the space of codes.Comment: 32 pages, 13 figure