In this paper, we construct fault-tolerant linear finite-state machines (LFSMs) in which error. detection and correction can be performed nonconcurrently (e.g., periodically). More specifically, by jointly choosing the state encoding constraints and the redundant dynamics of the fault-tolerant LFSM we enable an external checker to detect and identify errors due to past faults based on the current, possibly corrupted state of. the LFSM. The paper presents systematic constructions of fault-tolerant LFSMs based on a characterization of nonconcurrent error detection/correction in terms of state encoding constraints and redundant dynamics. In particular, we develop a scheme that uses Bose-Chaudhuri-Hocquenghem (BCH) coding and obtains fault-tolerant LFSMs that require 2D additional state variables and have the ability to correct up to D errors in any state variable at any time step in the time interval consisting of the latest N time steps of operations The construction uses the minimum possible number of additional state variables and requires an error detecting/correcting mechanism with computational complexity that is only linear in N.