Asymptotic Behavior and Typicality Properties of Runlength-Limited Sequences

TL;DR

This paper analyzes the asymptotic properties and typical behaviors of runlength-limited binary sequences with constraints, providing algebraic and probabilistic characterizations of their capacity and applications in coding theory.

Contribution

It introduces new algebraic and probabilistic methods to characterize the capacity and typical properties of constrained sequences, with applications in coding and information theory.

Findings

Derived the exponential growth rate of constrained sequences.

Established the second-order asymptotic expansion of the sequence rate.

Demonstrated applications in bounds for specialized coding channels.

Abstract

This paper studies properties of binary runlength-limited sequences with additional constraints on their Hamming weight and/or their number of runs of identical symbols. 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 by using the methods of multivariate analytic combinatorics, 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 weight-preserving and run-preserving channels (e.g., the run-preserving insertion-deletion channel), a sphere-packing bound for channels with sparse error patterns, 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.