A MATHEMATICAL FRAMEWORK FOR ALGORITHM-BASED FAULT-TOLERANT COMPUTINGOVER A RING OF INTEGERS

In this work, we establish a complete mathematical framework for algor ithm-based fault-tolerant computing for data vectors defined over a ri ng of integers. The ring of integers consists of integers {0, 1, ..., M-1} and all the arithmetic operations are performed modulo the intege r M, which is assumed to be composite. The importance of the work lies in the suitability of modulo arithmetic in certain computational envi ronments. Lack of an underlying Galois field, GF(q), presents a unique challenge to this framework. We develop the theory and algorithms for single as well as multiple fault correction and detection. We also an alyze the parallel and serial nature of the encoder and decoder config urations. Certain known but rather old results in the theory of number s dealing with linear congruences and matrix algebra are also describe d and extended further using mathematical terminology that modern-day researchers are expected to be familiar with.