Runlength-Limited Sequences and Shift-Correcting Codes: Asymptotic Analysis

This work is motivated by the problem of error correction in bit-shift channels with the so-called $(d,k)$ input constraints (where successive $1$'s are required to be separated by at least $d$ and at most $k$ zeros, $0 \leq d < k \leq \infty$). Bounds on the size of optimal $(d,k)$-constrained codes correcting a fixed number of bit-shifts are derived, with a focus on their asymptotic behavior in the large block-length limit. The upper bound is obtained by a packing argument, while the lower bound follows from a construction based on a family of integer lattices. Several properties of $(d, k)$-constrained sequences that may be of independent interest are established as well; in particular, the exponential growth-rate of the number of $(d, k)$-constrained constant-weight sequences is characterized. The results are relevant for magnetic and optical information storage systems, reader-to-tag RFID channels, and other communication models where bit-shift errors are dominant and where $(d, k)$-constrained sequences are used for modulation.