Cut-down de Bruijn sequences

A cut-down de Bruijn sequence is a cyclic string of length L, where $1\le L\le k^n$, such that every substring of length n appears at most once. Etzion [Theor. Comp. Sci 44 (1986)] introduced an algorithm to construct binary cut-down de Bruijn sequences requiring $o\left(n\right)$ simple n-bit operations per symbol generated. In this paper, we simplify the algorithm and improve the running time to $O\left(n\right)$ time per symbol generated using $O\left(n\right)$ space. Additionally, we develop the first successor-rule approach for constructing a binary cut-down de Bruijn sequence by leveraging recent ranking/unranking algorithms for fixed-density Lyndon words. Finally, we develop an algorithm to generate cut-down de Bruijn sequences for $k>2$ that runs in $O\left(n\right)$ time per symbol using $O\left(n\right)$ space after some initialization.