Maximum Length RLL Sequences in de Bruijn Graph

Abstract: Free-space quantum key distribution requires to synchronize the transmitted and received signals. A timing and synchronization system for this purpose based on a de Bruijn sequence has been proposed and studied recently for a channel associated with quantum communication that requires reliable synchronization. To avoid a long period of no-pulse in such a system on-off pulses are used to simulate a \emph{zero} and on-on pulses are used to simulate a \emph{one}. However, these sequences have high redundancy and low rate. To reduce the redundancy and increase the rate, run-length limited sequences in the de Bruijn graph are proposed for the same purpose. The maximum length of such sequences in the de Bruijn graph is studied and an efficient algorithm to construct a large set of these sequences is presented. Based on known algorithms and enumeration methods, maximum length sequence for which the position of each window can be computed efficiently is presented and an enumeration on the number of such sequences is given.