A group-theoretic framework for fault-tolerant computation

A general technique for protecting computation with systematic-separate codes is presented. These codes use parity symbols to check the result of computation. A group-theoretic approach and model computation are used as operations in an algebraic group. It is shown that in order for a code to commute with computation, it must define a homomorphism between the original group and the group of parity symbols. A quotient group isomorphism is applied, and the problem of finding coding schemes is reduced to that of finding normal subgroups. In many instances, the method can be shown to identify all possible systematic-separate codes. For a given code, conditions on errors are given so that they may be detected and corrected. The extension of the technique to other algebraic systems is discussed, and two examples are included