Nested input-constrained codes

Abstract

An input-constrained channel, or simply a constraint, is a set S of words that is generated by a nite 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 nitestate 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. Speci cally, given two constraints S 1 and S 2 such that S 1 S 2 and two prescribed rates, conditions are provided for the existence of respective deterministic nite-state encoders E 1 and E 2 , at the given rates, such that (the state diagram of) E 1 is a subgraph of E 2. Such encoders are referred to as nested encoders. The provided conditions are also constructive in that they imply an algorithm for nding such encoders when they exist. The nesting structure allows to decode E 1 while using the decoder of E 2. Recent developments in optical recording suggest a potential application that can take a signi cant advantage of nested encoders.

Nested block decodable runlength-limited codes

“…Next we show (Proposition 3.1 below) that tables T and A that satisfy conditions 1{4 can indeed be constructed. We rst point out that if we deleted condition 4, then the existence of T and A would follow from the Gu- (q?i) : (5) Substituting q?1 for q, we get…”