Efficient self-testing/self-correction of linear recurrences

The authors consider the problem of designing self-testers/self-correctors for functions defined by linear recurrences. They present the first complete package of efficient and simple self-testers, self-correctors, and result-checkers for such functions. The results are proved by demonstrating an efficient reduction from this problem to the problem of testing linear functions over certain matrix groups. The tools include spectral analysis of matrices over finite fields, and various counting arguments that extend known techniques. The matrix twist yields simple and efficient self-testers for all linear recurrences. They also show a technique of using convolution identities to obtain very simple self-testers and self correctors. Their techniques promise new and efficient ways of testing VLSI chips for applications in control engineering, signal processing, etc. An interesting consequence of their methods is a completely new and randomness-efficient self-tester for polynomials over finite fields and rational domains. In particular the self-tester for polynomials over rational domains overcomes a main drawback of the result of Rubinfeld and Sudan (1992)-the need for a test domain of much larger size and of much finer precision.