Abstract
Algorithm-Based Fault-Tolerance is a method of applying block error codes to array implementations of linear algebraic operations. When the underlying algebra is the real (or complex) number field, then numerical approximations to the reals cause certain fault-induced errors to be indistinguishable from roundoff errors, so that fault-induced errors may be either undetected or miscorrected. A worst case forward error analysis of floating-point implementations of minimum redundancy (SEC) codes shows that locating errors is inherently more difficult than detecting them. It is demonstrated that SEC Reed-Solomori (DFT) codes with equivalent theoretical power don't have equivalent numerical properties. A new minimum dynamic range SEC code is shown to be superior to all other minimum redundancy SEC codes. For integer affithmetic, the optimal method to add and to use available precision for encoding and decoding circuits is derived. Multiplierless SEC codes based on generalized Hamming codes and multi dimensional checksum codes are described.
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