Some aspects of the construction and implementation of error-correcting linear codes - Booth, Geoffrey L (Rhodes University, Faculty of Science, Department of Mathematics, 1978)
From Conclusion: The study of error-correcting codes is now approximately 25 years old. The first known publication on the subject was in 1949 by M. Golay, who later did much research into the subject of perfect codes. It has been recently established that all the perfect codes are known. R.W. Hamming presented his perfect single-error correcting codes in 1950, in ~n article in the Bell System Technical Journal. These codes turned out to be a special case of the powerful Bose-Chaudhuri codes which were discovered around 1960. Various work has been done on the theory of minimal redundancy of codes for a given error-correcting performance, by Plotkin, Gilbert, Varshamov and others, between 1950 and 1960. The binary BCH codes were found to be so close to the theoretical bounds that, to date, no better codes have been discovered. Although the BCH codes are extremely efficient in terms of ratio of information to check digits, they are not easily, decoded with a minimal amount of apparatus. Petersen in 1961 described an algorithm for d e coding BCH codes, but this was cumbersome compared with the majority-logic methods of Massey and others. Thus the search began for codes which are easily decoded with comparatively simple apparatus. The finite geometry codes which were described by Rudolph in a 1964 thesis were examples of codes which are easily decoded 58 by a small number of steps of majority logic. The simplicial codes of Saltzer are even better in this respect, since they can be decoded by a single step of majority logic, but are rather inefficient . The applications of coding theory have changed over the years, as well. The first computers were huge circuits of relays, which were unreliable and prone to errors. Error correcting codes were required to minimise the possibility of incorrect results. As vacuum tubes and later transistorised circuits made computers more reliable, the need for sophisticated and powerful codes in the computer world diminished. Other used presented themselves however, for example the control systems of unmanned space craft. Because of the difficulty of sending and receiving messages in this case, Â· very powerful codes were required. Other uses were found in transmission lines and telephone exchanges. The codes considered in this dissertation have, for the most part, been block codes for use on the binary symmetric channel. There are, however, several other applications, such as codes for use on an erasure channel, where bits are corrupted so as to be unrecognizable, rather than changed. There are also codes for burst-error correction, where chennel noise is not randomly distributed, but occurs in "bursts" a few bits long. Certain cyclic codes are of application in these cases. The theory of error correcting codes has risen from virtual non-existence in 1950 to a major and sophisticated part of communication theory. Judging from the articles in journals, it promises to be the subject of a great deal of research for some years to come.
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