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

TL;DR

This paper analyzes the asymptotic limits of (d,k)-constrained sequences used in error correction for channels with bit-shift errors, providing bounds and properties relevant for storage and communication systems.

Contribution

It derives bounds on the size of optimal (d,k)-constrained codes correcting bit-shifts and characterizes their growth rate, introducing new insights into their combinatorial properties.

Findings

Bounds on code sizes for large block lengths

Exponential growth rate of constrained sequences

Properties of (d,k)-constrained sequences

Abstract

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.