Some General Results of Abstract Coding Theory with Applications to the Study of Codes for the Correction of Synchronization Errors

Abstract

This paper, written during the last months of the study in 1968 and revised in 1969, presents the results of rumination about past endeavors and concentrates on the three important code properties: error-correction, error-decoding, and error-limitation. Attempts to generalize the results of Chapter 11 forced a logical analysis of the relationships involved. As usual, such an analysis was initially cast into a highly symbolic form amenable to manipulation. Somewhat surprisingly, it became apparent exactly what was involved in error-correction irrespective of the model: it was possible to characterize the property of error-correction for arbitrary models. With minor restrictions on the models, decodability was likewise characterized after it became clear precisely what the real definition of decodability should be. Once this was done, it was obvious how synchronizability should be defined for these now essentially arbitrary models. All of the critical implications established for the classical models in Chapters 2–6 held for the general models with modest restrictions and the classical models appeared, as they indeed should, as a special case of the general model. The final result is a simple, conceptual, and adequate frame-work for a consideration of the four basic problems of Coding Theory.