Asymptotic Behavior and Typicality Properties of Runlength-Limited Sequences

We study properties of binary runlength-limited sequences with additional restrictions on their weight and/or the number of runs of identical symbols they contain. An algebraic and a probabilistic (entropic) characterization of the exponential growth rate of the number of such sequences, i.e., their information capacity, are obtained, and properties of the capacity as a function of its parameters are stated. The second-order term in the asymptotic expansion of the rate of these sequences is also given, and the typical values of the relevant quantities are derived. Several applications of the results are illustrated, including bounds on codes for the run-preserving insertion-deletion channel in the fixed-number-of-errors regime, a sphere-packing bound for sparse-noise channels in the fixed-fraction-of errors regime, and the asymptotics of constant-weight sub-block constrained sequences. In addition, the asymptotics of a closely related notion -- $ q $-ary sequences with fixed Manhattan weight -- is briefly discussed, and an application in coding for molecular timing channels is illustrated.