On the Existence of Balanced Generalized de Bruijn Sequences

TL;DR

This paper establishes the precise conditions under which balanced generalized de Bruijn sequences, characterized by equal zeros and ones and limited substring repetitions, exist for given parameters.

Contribution

It provides necessary and sufficient conditions for the existence of balanced generalized de Bruijn sequences with specified parameters.

Findings

Derived exact existence criteria for these sequences.

Characterized the relationship between sequence parameters.

Extended understanding of sequence combinatorics.

Abstract

A balanced generalized de Bruijn sequence with parameters $(n,l,k)$ is a cyclic sequence of $n$ bits such that (a) the number of 0's equals the number of 1's, and (b) each substring of length $l$ occurs at most $k$ times. We determine necessary and sufficient conditions on $n,l$, and $k$ for the existence of such a sequence.