Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms

Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical programming used in the context of decoding include linear, integer, and nonlinear programming, network flows, notions of duality as well as matroid and polyhedral theory. This survey article reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory. Published July 201

Transmission and coding of information - problem list

2018/201

Abstract Algebra: Theory and Applications

Tom Judson\u27s Abstract Algebra: Theory and Applications is an open source textbook designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications. Rob Beezer has contributed complementary material using the open source system, Sage.An HTML version on the PreText platform is available here. The first half of the book presents group theory, through the Sylow theorems, with enough material for a semester-long course. The second-half is suitable for a second semester and presents rings, integral domains, Boolean algebras, vector spaces, and fields, concluding with Galois Theory.https://scholarworks.sfasu.edu/ebooks/1022/thumbnail.jp

Pseudorandom Constructions: Computing in Parallel and Applications to Edit Distance Codes

The thesis focuses on two problems about pseudorandom constructions. The first problem is how to compute pseudorandom constructions by constant depth circuits. Pseudorandom constructions are deterministic functions which are used to substitute random constructions in various computational tasks. Constant depth circuits here refer to the computation model which can compute functions using circuits of \AND, \OR and negation gates, with constant depth, unbounded fan-in, taking function inputs by input wires and giving function outputs by output wires. They can be simulated by fast parallel algorithms. We study such constructions mainly for randomness extractors, secret sharing schemes and their applications. Randomness extractors are functions which transform biased random bits to uniform ones. They can be used to recycle random bits in computations if there are some entropies remaining. Secret sharing schemes efficiently share secrets among multi-parties s.t. the collusion of a bounded number of parties cannot recover any information of the secret while a certain larger number of parties can recover the secret. Our work constructs these objects with near optimal parameters and explores their applications. The second problem is about applying pseudorandom constructions to build error correcting codes (ECCs) for edit distance. ECCs project messages to codewords in a metric space s.t. one can recover the codewords even if there are bounded number of errors which can drive the codeword away by some bounded distance. They are widely used in both the theoretical and practical part of computer science. Classic errors are hamming errors which are substitutions and erasures of symbols. They are well studied by numerous literatures before. We consider one kind of more general errors i.e. edit errors, consists of insertions and deletions that may change the positions of symbols. Our work give explicit constructions of binary ECCs for edit errors with redundancy length near optimal. The constructions utilize document exchange protocols which can let two party synchronize their strings with bounded edit distance, by letting one party send a short sketch of its string to the other. We apply various pseudorandom constructions to get deterministic document exchange protocols from randomized ones. Then we construct ECCs using them. We also extend these constructions to handle block insertions/deletions and transpositions. All these constructions have near optimal parameters

Vector Signal Processors in Data Compression and Image Processing

The objective is to evaluate the applicability of the Vector Signal Processor to real time signal processing for data compression or manipulation. Particular emphasis has been placed on its role as a co-processor and the contribution that it might be expected to make during joint activities with the host. These activities would have the combination used as the embedded computing subsystem of a FAX machine or as an image processing unit in desk top publishing. In these cases the hypothesis is that the Vector Signal Processor would act as an accelerator for many computationally intensive applicable processes. After a review of current data compression techniques and of specialised architectures which may also be appropriate it is concluded that the Vector Signal Processor is the best option available. The operational details are then discussed. In order to be able to approximately compare experimental results with other workers a benchmarking exercise is undertaken. Following this is the core of the study which details schemes for data compression of data sources involving character symbols, line drawings, and grey scale pictures. This involves pattern matching and substitution,Transform coding and quadtrees. New encoding procedures are suggested based on Morse code for the secondary encoding of symbols and on Delta modulation for quadtrees. Image entity manipulation is discussed followed by some speculative work on neural networks and error control coding. It is concluded that some processes are well served by the Vector Signal Processor but that the lack of conditional decision making and the difficulty of performing certain arithmetic functions make the processor unwieldy in its necessary host interactions