Nested input-constrained codes

An input-constrained channel, or simply a constraint, is a set S of words that is generated by a finite labeled directed graph. An encoder for S maps, in a lossless manner, sequences of unconstrained input blocks into sequences of channel blocks, the latter sequences being words of S. In most applications, the encoders are finite-state machines and, thus, presented by state diagrams. In the special case where the state diagram of the encoder is (output) deterministic, only the current encoder state and the current channel block are needed for the decoding of the current input block. In this work, the problem of designing coding schemes that can serve two constraints simultaneously is considered. Specifically, given two constraints S₁ and S ₂ such that S₁⊆S₂ and two described rates, conditions are provided for the existence of respective deterministic finite-state encoders ε₁ and ε₂ , at the given rates, such that (the state diagram of) ε₁ is a subgraph of ε₂ Such encoders are referred to as nested encoders. The provided conditions are also constructive in that they imply an algorithm for finding such encoders when they exist. The nesting structure allows to decode ε₁ while using the decoder of ε₂. Developments in optical recording suggest a potential application that can take a significant advantage of nested encoders