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In the literature on diagnosable multiprocessor systems, regular testing structures which isolate up to t1 faulty processors to within a set of t1 processors have been studied. These structures are called D(n, t0, X) systems and are said to be t1/t1-diagnosable. In this correspondence, the problem of designing regular testing structures which isolate the largest number of faulty processors is shown to be equivalent to determining full-difference triangles. These triangles have been used to design error-correcting convolutional codes and transmission systems without third-order intermodulation interference, and have been studied extensively. Recognizing this relationship allows results from these areas to be applied directly to the design of diagnosable systems. A list of the best-known systems is given.