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 <; d <; k <; ∞). 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.