Abstract. The goal of this paper is to establish links between computational complexity theory and the theory and practice of constrained block coding. In particular, the complexities of several fundamental problems in constrained block coding are precisely classified in terms of the existing complexity-theoretic structure. One type of problem studied is that of designing encoder and decoder circuits using minimum or approximately minimum hardware; for our purposes, an "input" to this problem is (i) a deterministic, irreducible finite state transition diagram (abbreviated DIF) defining a set of constrained binary sequences, and (ii) a desired rate p: q. Several of these minimumencoder and minimum-decoder problems are shown to be NP-hard, and more interestingly some are shown to be complete in the second and third levels of the polynomial hierarchy. Another fundamental problem is that of computing the maximum rate of a block code; that is, given a DIF and a codeword length q, find the maximum p such that a rate p: q block code exists for the constraint defined by the DIF. This problem is shown to be NP
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