Unordered Error-Correcting Codes and their Applications

We give efficient constructions for error correcting unordered {ECU) codes, i.e., codes such that any pair of codewords are at a certain minimal distance apart and at the same time they are unordered. These codes are used for detecting a predetermined number of (symmetric) errors and for detecting all unidirectional errors. We also give an application in parallel asynchronous communications

Some new EC/AUED codes

A novel construction that differs from the traditional way of constructing systematic EC/AUED/(error-correcting/all unidirectional error-detecting) codes is presented. The usual method is to take a systematic t-error-correcting code and then append a tail so that the code can detect more than t errors when they are unidirectional. In the authors' construction, the t-error-correcting code is modified in such a way that the weight distribution of the original code is reduced. The authors then have to add a smaller tail. Frequently they have less redundancy than the best available systematic t-EC/AUED codes

A Computational Framework for Efficient Error Correcting Codes Using an Artificial Neural Network Paradigm.

The quest for an efficient computational approach to neural connectivity problems has undergone a significant evolution in the last few years. The current best systems are far from equaling human performance, especially when a program of instructions is executed sequentially as in a von Neuman computer. On the other hand, neural net models are potential candidates for parallel processing since they explore many competing hypotheses simultaneously using massively parallel nets composed of many computational elements connected by links with variable weights. Thus, the application of modeling of a neural network must be complemented by deep insight into how to embed algorithms for an error correcting paradigm in order to gain the advantage of parallel computation. In this dissertation, we construct a neural network for single error detection and correction in linear codes. Then we present an error-detecting paradigm in the framework of neural networks. We consider the problem of error detection of systematic unidirectional codes which is assumed to have double or triple errors. The generalization of network construction for the error-detecting codes is discussed with a heuristic algorithm. We also describe models of the code construction, detection and correction of t-EC/d-ED/AUED (t-Error Correcting/d-Error Detecting/All Unidirectional Error Detecting) codes which are more general codes in the error correcting paradigm

Unidirectional error correcting/detecting codes

An extensive theory of symmetric error control coding has been developed in the last few decades. The recently developed VLSI circuits, ROM, and RAM memories have given an impetus to the extension of error control coding to include asymmetric and unidirectional types of error control. The maximal numbers of unidirectional errors which can be detected by systematic codes using r checkbits are investigated. They are found for codes with k, the number of information bits, being equal to 2[superscript r] and 2[superscript r] + 1. The importance of their characteristic in unidirectional error detection is discussed. A new method of constructing a systematic t-error correcting/all-unidirectional error detecting(t-EC/AUED) code, which uses fewer checkbits than any of the previous methods, is developed. It is constructed by appending t + 1 check symbols to a systematic t-error correcting and (t+l)-error detecting code. Its decoding algorithm is developed. A bound on the number of checkbits for a systematic t-EC/AUED code is also discussed. Bose-Rao codes, which are the best known single error correcting/all-unidirectional error detecting(SEC/AUED) codes, are completely analyzed. The maximal Bose-Rao codes for a fixed weight and for all weights are found. Of course, the base group and the group element which make the Bose-Rao code maximal are found, too. The bounds on the size of SEC/AUED codes are discussed. Nonsystematic single error correcting/d-unidirectional error detecting codes are constructed. Three methods for constructing the systematic t-error correcting/d-unidirectional error detecting(t-EC/d-UED) codes are developed. From these, simple and efficient t-EC/(t+2)-UED codes are derived. The decoding algorithm for one of these methods, which can be applied to the other two methods with slight modification, is described. A lower bound on the number of checkbits for a systematic t-EC/d-UED code is derived. Finally, future research efforts are proposed

Optimal unidirectional error-detecting codes

Various problems related to systematic error-detecting and error-correcting unidirectional codes are discussed. Systematic codes with r check bits are the main topic of the thesis. Classes of codes are presented which work for specific numbers of information bits and then a class of codes is given which detects the same number of unidirectional errors for any number of information bits. This class is shown to be optimal under certain conditions. A result that indicates it is optimal under wider conditions follows. Finally, a class of error correcting codes is presented

New techniques for constructing EC/AUED codes

The most common method to construct a t-error correcting/all unidirectional error detecting (EC/AUED) code is to choose a t-error correcting (EC) code and then to append a tail in such a way that the new code can detect more than t errors when they are unidirectional. The tail is a function of the weight of the codeword. We present two new techniques for constructing t-EC/AUED codes. The first technique modifies the t-EC code in such a way that the weight distribution of the original code is reduced. So, a smaller tail is needed. Frequently, this technique gives less overall redundancy than the best available t-EC/AUED codes

Diversity combining ARQ over the m(> 2)-ary unidirectional channel

In diversity combining automatic repeat request (ARQ), erroneous packets are combined together forming a single, more reliable, packet. In this thesis, we give a diversity combining scheme for the m-ary unidirectional channel. A system using the given scheme with a t-unidirectional error detecting code is able to correct up to Emax = floor(t/2) unidirectional errors. To use the given scheme, the decoder should be able to decide the error type (increasing or decreasing). Hence, we give simple techniques to make this decision for various unidirectional error detecting codes

Error control coding for semiconductor memories

All modern computers have memories built from VLSI RAM chips. Individually, these devices are highly reliable and any single chip may perform for decades before failing. However, when many of the chips are combined in a single memory, the time that at least one of them fails could decrease to mere few hours. The presence of the failed chips causes errors when binary data are stored in and read out from the memory. As a consequence the reliability of the computer memories degrade. These errors are classified into hard errors and soft errors. These can also be termed as permanent and temporary errors respectively. In some situations errors may show up as random errors, in which both 1-to-O errors and 0-to-l errors occur randomly in a memory word. In other situations the most likely errors are unidirectional errors in which 1-to-O errors or 0-to-l errors may occur but not both of them in one particular memory word. To achieve a high speed and highly reliable computer, we need large capacity memory. Unfortunately, with high density of semiconductor cells in memory, the error rate increases dramatically. Especially, the VLSI RAMs suffer from soft errors caused by alpha-particle radiation. Thus the reliability of computer could become unacceptable without error reducing schemes. In practice several schemes to reduce the effects of the memory errors were commonly used. But most of them are valid only for hard errors. As an efficient and economical method, error control coding can be used to overcome both hard and soft errors. Therefore it is becoming a widely used scheme in computer industry today. In this thesis, we discuss error control coding for semiconductor memories. The thesis consists of six chapters. Chapter one is an introduction to error detecting and correcting coding for computer memories. Firstly, semiconductor memories and their problems are discussed. Then some schemes for error reduction in computer memories are given and the advantages of using error control coding over other schemes are presented. In chapter two, after a brief review of memory organizations, memory cells and their physical constructions and principle of storing data are described. Then we analyze mechanisms of various errors occurring in semiconductor memories so that, for different errors different coding schemes could be selected. Chapter three is devoted to the fundamental coding theory. In this chapter background on encoding and decoding algorithms are presented. In chapter four, random error control codes are discussed. Among them error detecting codes, single* error correcting/double error detecting codes and multiple error correcting codes are analyzed. By using examples, the decoding implementations for parity codes, Hamming codes, modified Hamming codes and majority logic codes are demonstrated. Also in this chapter it was shown that by combining error control coding and other schemes, the reliability of the memory can be improved by many orders. For unidirectional errors, we introduced unordered codes in chapter five. Two types of the unordered codes are discussed. They are systematic and nonsystematic unordered codes. Both of them are very powerful for unidirectional error detection. As an example of optimal nonsystematic unordered code, an efficient balanced code are analyzed. Then as an example of systematic unordered codes Berger codes are analyzed. Considering the fact that in practice random errors still may occur in unidirectional error memories, some recently developed t-random error correcting/all unidirectional error detecting codes are introduced. Illustrative examples are also included to facilitate the explanation. Chapter six is the conclusions of the thesis. The whole thesis is oriented to the applications of error control coding for semiconductor memories. Most of the codes discussed in the thesis are widely used in practice. Through the thesis we attempt to provide a review of coding in computer memories and emphasize the advantage of coding. It is obvious that with the requirement of higher speed and higher capacity semiconductor memories, error control coding will play even more important role in the future