This thesis attempts to construct a sequential t-diagnosable system with fewer number of tests than the previously best known class when $2t+1 < n < (t+2)^2 /4$. A class of diagnosable systems S(n,m,a) is proposed. And for the class necessary and sufficient conditions of sequential t-diagnosability are obtainted. The class of system S(n,m,a) requires more tests than the class of optimal designs  by one or two tests when t = [(n-1)/2]. But when t approaches to [2n]-3 the number of tests reduces significantly. A conjecture is given with some evidence for its validity and correctness which, if true, determines an optimal sequential t-diagnosable system for every t.