Generalized Orthogonal de Bruijn and Kautz Sequences

TL;DR

This paper introduces generalized orthogonal de Bruijn and Kautz sequences, exploring relaxations of classical constraints, providing bounds, and extending concepts to nonbinary sequences with potential applications in coding theory and synthetic biology.

Contribution

It presents new generalizations of orthogonal de Bruijn and Kautz sequences, including relaxations of constraints and bounds on fixed-weight sequences, expanding their theoretical understanding and practical applicability.

Findings

Derived bounds on the number of fixed-weight orthogonal de Bruijn sequences.

Extended concepts to orthogonal nonbinary Kautz sequences.

Explored relaxations of classical sequence constraints.

Abstract

A de Bruijn sequence of order $k$ over a finite alphabet is a cyclic sequence with the property that it contains every possible $k$-sequence as a substring exactly once. Orthogonal de Bruijn sequences are collections of de Bruijn sequences of the same order, $k$, satisfying the joint constraint that every $(k+1)$-sequence appears as a substring in at most one of the sequences in the collection. Both de Bruijn and orthogonal de Bruijn sequences have found numerous applications in synthetic biology, although the latter remain largely unexplored in the coding theory literature. Here we study three relevant practical generalizations of orthogonal de Bruijn sequences where we relax either the constraint that every $(k+1)$-sequence appears exactly once, or that the sequences themselves are de Bruijn rather than balanced de Bruijn sequences. We also provide lower and upper bounds on the number of fixed-weight orthogonal de Bruijn sequences. The paper concludes with parallel results for orthogonal nonbinary Kautz sequences, which satisfy similar constraints as de Bruijn sequences except for only being required to cover all subsequences of length $k$ whose maximum runlength equals to one.