Covering Sequences for $\ell$-Tuples

Sagi Marcovich; Tuvi Etzion; Eitan Yaakobi

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arXiv · 2206.03711

Covering Sequences for $\ell$-Tuples

Sagi Marcovich·Tuvi Etzion·Eitan Yaakobi

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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.

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Sagi Marcovich, Tuvi Etzion, Eitan Yaakobi. 2022-06-08. Covering Sequences for $\ell$-Tuples. https://arxiv.org/abs/2206.03711

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