Codes Andn-ary Relations

The aim of this thesis is to develop a general mechanism for the construction of codes and to extract general properties of classes of codes. This mechanism makes it unnecessary to study various classes of codes separately--at least to some extent--by different constructions and properties.;To achieve this goal, the mechanism of characterizing classes of languages by binary relations is studied. Some general properties related to binary relations and languages are obtained. Moreover, three new classes of codes, n-shuffle codes, solid codes, and intercodes are constructed. Solid codes and intercodes have the synchronous decoding property which is very useful in the design of circuits of coders and decoders.;The studies of codes, n-codes, and intercodes indicate that these three classes of codes cannot be characterized by binary relations. We introduce a more general mechanism, that is, to characterize classes of languages by finitary relations. This mechanism can be used to characterize more classes of languages, such as the classes of n-codes and intercodes. Sometimes, it is difficult to show inclusion relations between classes of codes and hierarchy properties of classes of codes. Results derived in this thesis provide a mechanism which can simplify this task

Using Short Synchronous WOM Codes to Make WOM Codes Decodable

In the framework of write-once memory (WOM) codes, it is important to distinguish between codes that can be decoded directly and those that require that the decoder knows the current generation to successfully decode the state of the memory. A widely used approach to construct WOM codes is to design first nondecodable codes that approach the boundaries of the capacity region, and then make them decodable by appending additional cells that store the current generation, at an expense of a rate loss. In this paper, we propose an alternative method to make nondecodable WOM codes decodable by appending cells that also store some additional data. The key idea is to append to the original (nondecodable) code a short synchronous WOM code and write generations of the original code and of the synchronous code simultaneously. We consider both the binary and the nonbinary case. Furthermore, we propose a construction of synchronous WOM codes, which are then used to make nondecodable codes decodable. For short-to-moderate block lengths, the proposed method significantly reduces the rate loss as compared to the standard method.Comment: To appear in IEEE Transactions on Communications. The material in this paper was presented in part at the 2012 IEEE International Symposium on Information Theory, Cambridge, MA, July 201

Interactive Consistency Algorithms Based on Voting and Error-Correding Codes

This paper presents a new class of synchronous deterministic non authenticated algorithms for reaching interactive consistency (Byzantine agreement). The algorithms are based on voting and error correcting codes and require considerably less data communication than the original algorithm, whereas the number of rounds and the number of modules meet the minimum bounds. These algorithms based on voting and coding are defined and proved on the basis of a class of algorithms, called the dispersed joined communication algorithm

On Frame Asynchronous Coded Slotted ALOHA: Asymptotic, Finite Length, and Delay Analysis

We consider a frame asynchronous coded slotted ALOHA (FA-CSA) system for uncoordinated multiple access, where users join the system on a slot-by-slot basis according to a Poisson random process and, in contrast to standard frame synchronous CSA (FS-CSA), users are not frame-synchronized. We analyze the performance of FA-CSA in terms of packet loss rate and delay. In particular, we derive the (approximate) density evolution that characterizes the asymptotic performance of FA-CSA when the frame length goes to infinity. We show that, if the receiver can monitor the system before anyone starts transmitting, a boundary effect similar to that of spatially-coupled codes occurs, which greatly improves the iterative decoding threshold. Furthermore, we derive tight approximations of the error floor (EF) for the finite frame length regime, based on the probability of occurrence of the most frequent stopping sets. We show that, in general, FA-CSA provides better performance in both the EF and waterfall regions as compared to FS-CSA. Moreover, FA-CSA exhibits better delay properties than FS-CSA.Comment: 13 pages, 12 figures. arXiv admin note: substantial text overlap with arXiv:1604.0629

The Parametric Ordinal-Recursive Complexity of Post Embedding Problems

Post Embedding Problems are a family of decision problems based on the interaction of a rational relation with the subword embedding ordering, and are used in the literature to prove non multiply-recursive complexity lower bounds. We refine the construction of Chambart and Schnoebelen (LICS 2008) and prove parametric lower bounds depending on the size of the alphabet.Comment: 16 + vii page