The competing demands of speed and fault tolerance in finite field Fourier transform implementations have been optimally balanced here by using the chord property in finite field.

This paper addresses fault tolerance of a processor array that is reconfigurable by replacing faulty processors with spare processors. The fault tolerance of such a reconfigurable array depends on not only an algorithm for spare processor assignment but also the folloving factor of an organization of spare processors in the reconfigurable array: the number of spare processors; the number of processors that can be replaced by each spare processor; and how spare processors are connected with processors. We discuss a relationship between fault tolerance of reconfigurable arrays and their organizations of spare processors in terms of the smallest size of fatal sets and the reliability function. The smallest size of fatal sets is the smallest number of faulty processors for which the reconfigurable array cannot be failure-free as a processor array system no matter what reconfiguration is used. The reliability function is a function of time t whose value is the probability that the reconfigurable array is failure-free as a processor array system by time t when the best possible reconfiguration is used. First, we show that the larger smallest size of fatal sets a reconfigurable array has, the larger reliability function it has by some time. It suggests that it is important to maximize the smallest size of fatal sets in orer to improve the reliability function as well. Second, we present the best possible smallest size of fatal sets for nn reconfigurable arrays using 2n spare processor each of which is connected with n processors. Third, we show that the nn reconfigurable array previously presented in a literature achieves the best smallest size of fatal sets. That is, it is optimum with respect to the smallest size of fatal sets. Fourth, we present an uppr bound of the reliability function of the optimum nn reconfigurable array using 2n spare processors.