A shift of finite type (SFT) is a set of all bi-infinite sequences over some alphabet which is characterized by a finite set of forbidden words. It is a typical example of sofic shifts and has been used in media storage area, such as CD's or DVD's. The study of sofic shifts is based on graph theory, and the irreducibility of shifts is an important property to be considered for the study. In this paper, we will provide some sufficient conditions for an SFT to be irreducible from the perspective of the antidictionary of a word and the number of forbidden words. We also present a necessary and sufficient condition for an SFT to be irreducible when the number of forbidden words is one less than the alphabet size.

The antidictionary of a string is the set of all words of minimal length that never appear in this string. Antidictionaries are in particular useful for source coding. We present a fast and memory-efficient algorithm to construct an antidictionary using a suffix tree. It is proved that the complexity of this algorithm is linear in space and time, and its effectiveness is demonstrated by simulation results.