Coding Theory

Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete mathematics and combinatorics, complexity theory, and statistical physics. The workshop on coding theory covered many facets of the recent research advances

Design and Analysis of Graph-based Codes Using Algebraic Lifts and Decoding Networks

Error-correcting codes seek to address the problem of transmitting information efficiently and reliably across noisy channels. Among the most competitive codes developed in the last 70 years are low-density parity-check (LDPC) codes, a class of codes whose structure may be represented by sparse bipartite graphs. In addition to having the potential to be capacity-approaching, LDPC codes offer the significant practical advantage of low-complexity graph-based decoding algorithms. Graphical substructures called trapping sets, absorbing sets, and stopping sets characterize failure of these algorithms at high signal-to-noise ratios. This dissertation focuses on code design for and analysis of iterative graph-based message-passing decoders. The main contributions of this work include the following: the unification of spatially-coupled LDPC (SC-LDPC) code constructions under a single algebraic graph lift framework and the analysis of SC-LDPC code construction techniques from the perspective of removing harmful trapping and absorbing sets; analysis of the stopping and absorbing set parameters of hypergraph codes and finite geometry LDPC (FG-LDPC) codes; the introduction of multidimensional decoding networks that encode the behavior of hard-decision message-passing decoders; and the presentation of a novel Iteration Search Algorithm, a list decoder designed to improve the performance of hard-decision decoders. Adviser: Christine A. Kelle

Combinatorial Methods in Coding Theory

This thesis is devoted to a range of questions in applied mathematics and signal processing motivated by applications in error correction, compressed sensing, and writing on non-volatile memories. The underlying thread of our results is the use of diverse combinatorial methods originating in coding theory and computer science. The thesis addresses three groups of problems. The first of them is aimed at the construction and analysis of codes for error correction. Here we examine properties of codes that are constructed using random and structured graphs and hypergraphs, with the main purpose of devising new decoding algorithms as well as estimating the distribution of Hamming weights in the resulting codes. Some of the results obtained give the best known estimates of the number of correctable errors for codes whose decoding relies on local operations on the graph. In the second part we address the question of constructing sampling operators for the compressed sensing problem. This topic has been the subject of a large body of works in the literature. We propose general constructions of sampling matrices based on ideas from coding theory that act as near-isometric maps on almost all sparse signal. This matrices can be used for dimensionality reduction and compressed sensing. In the third part we study the problem of reliable storage of information in non-volatile memories such as flash drives. This problem gives rise to a writing scheme that relies on relative magnitudes of neighboring cells, known as rank modulation. We establish the exact asymptotic behavior of the size of codes for rank modulation and suggest a number of new general constructions of such codes based on properties of finite fields as well as combinatorial considerations

Dekodovanje kodova sa malom gustinom provera parnosti u prisustvu grešaka u logičkim kolima

Sve ve´ca integracija poluprovodniˇckih tehnologija, varijacije nastale usled nesavršenosti procesa proizvodnje, kao zahtevi za smanjenjem napona napajanja cˇine elektronske ured¯aje inherentno nepouzdanim. Agresivno skaliranje napona smanjuje otpornost na šum i dovodi do nepouzdanog rada ured¯aja. Široko je prihvac´ena paradigma prema kojoj se naredne generacije digitalnih elektronskih ured¯aja moraju opremiti logikom za korekciju hardverskih grešaka...Due to huge density integration increase, lower supply voltages, and variations in technological process, complementary metal-oxide-semiconductor (CMOS) and emerging nanoelectronic devices are inherently unreliable. Moreover, the demands for energy efficiency require reduction of energy consumption by several orders of magnitude, which can be done only by aggressive supply voltage scaling. Consequently, the signal levels are much lower and closer to the noise level, which reduces the component noise immunity and leads to unreliable behavior. It is widely accepted that future generations of circuits and systems must be designed to deal with unreliable components..

On codes from hypergraphs

We propose a new family of asymptotically good binary codes, generalizing previous constructions of expander codes to ¢-uniform hypergraphs. We also describe an efficient decoding algorithm for these codes, that for a certain region of rates improves the known results for decoding distance of expander codes. The construction is based on hypergraphs with a certain “expansion” property called herein£-homogeneity. For t-uniform t-partite ¤-regular hypergraphs, the expansion property required is roughly as follows: Given ¢ sets,¥¦§¨¨¨§¥©, one at each side, the number of hyper-edges with one vertex in each set is approximately what is expected, had the edges been chosen at random. We show that in an appropriate random model, almost all hypergraphs have this property, and also present an explicit construction of such hypergraphs. Having a family of such hypergraphs, and a small code � � � ��§���, with relative distance � � and rate ��, we construct “Hypergraph Codes”. These rate � ¢� � have ��¢���, and distance � � � � � relative � �����. When ¢ � � � we also suggest a decoding algorithm, and prove that the fraction of errors it decodes correctly is at � least ��� ¦ � � �� � �� � � � � � ��¦��� ����. In both cases, the ��� � is an additive term that tends to � as the length of the hypergraph-code tends to infinity