Covering Sequences for $\ell$-Tuples

TL;DR

This paper investigates sequences that cover all possible $ ext{l}$-tuples at least once, providing bounds on their rate, redundancy, and efficient encoding/decoding schemes, aiming to improve upon the low-rate de Bruijn sequences.

Contribution

It introduces bounds on the rate and redundancy of $ ext{l}$-tuples covering sequences and presents efficient encoding and decoding methods for these sequences.

Findings

Derived asymptotic bounds on the rate of covering sequences.

Established an upper bound on $ ext{l}$ for minimal redundancy.

Developed efficient encoding and decoding schemes meeting the redundancy bound.

Abstract

de Bruijn sequences of order $\ell$, i.e., sequences that contain each $\ell$-tuple as a window exactly once, have found many diverse applications in information theory and most recently in DNA storage. This family of binary sequences has rate of $1/2$. To overcome this low rate, we study $\ell$-tuples covering sequences, which impose that each $\ell$-tuple appears at least once as a window in the sequence. The cardinality of this family of sequences is analyzed while assuming that $\ell$ is a function of the sequence length $n$. Lower and upper bounds on the asymptotic rate of this family are given. Moreover, we study an upper bound for $\ell$ such that the redundancy of the set of $\ell$-tuples covering sequences is at most a single symbol. Lastly, we present efficient encoding and decoding schemes for $\ell$-tuples covering sequences that meet this bound.