Constrained systems with unconstrained positions

Abstract

We develop methods for analyzing and constructing combined modulation/error-correcting codes (ECC codes), in particular codes that employ some form of reversed concatenation and whose ECC decoding scheme requires easy access to soft information (e.g., turbo codes, low-density parity.check (LDPC) codes or parity codes). We expand on earlier work of Immink and Wijngaarden and also of Fan, in which certain bit positions are reserved for ECC parity, in the sense that the bit values in these positions can be changed without violating the constraint. Earlier work has focused more on block codes for specific modulation constraints. While our treatment is completely general, we focus on finite-state codes for maximum transition run (MTR) constraints. We 1) obtain some improved constructions for MTR codes based on short block lengths, 2) specify an asymptotic lower bound for MTR constraints, which is tight in very special cases, for the maximal code rate achievable for an MTR code with a given density of unconstrained positions, and 3) show how to compute the capacity of the set of sequences that satisfy a completely arbitrary constraint with a specified set of hit positions unconstrained.