A de Bruijn Sequence Construction by Concatenating Cycles of the Complemented Cycling Register

Gabric, Daniel; Sawada, Joe

doi:10.1007/978-3-319-66396-8_6

Cited by 7 publications

(5 citation statements)

References 17 publications

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“…The second result formalizes preliminary work presented in [15]. The asymptotic upper bound achieved in the third result was previously known [4,11], however no specific construction was known to attain this bound.…”

Section: Introductionsupporting

confidence: 72%

Investigating the discrepancy property of de Bruijn sequences

Gabric¹,

Sawada²

2020

Preprint

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The discrepancy of a binary string refers to the maximum (absolute) difference between the number of ones and the number of zeroes over all possible substrings of the given binary string. We provide an investigation of the discrepancy of known simple constructions of de Bruijn sequences. Furthermore, we demonstrate constructions that attain the lower bound of Θ(n) and a new construction that attains the previously known upper bound of Θ( 2 n √ n ). This extends the work of Cooper and Heitsch [Discrete Mathematics, 310 (2010)].

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Section: Introductionsupporting

confidence: 72%

Investigating the discrepancy property of de Bruijn sequences

Gabric¹,

Sawada²

2020

Preprint

Self Cite

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“…Dai et al [9] examined the binary case, and gave both a method of construction and a general upper bound for the period in this case. Much more recently, in 2022 an alternative method of construction for the binary case was described [16], using the Lempel Homomorphism [14]; this latter work has been further extended by Gabric and Sawada [11]. However, as far as we are aware, no authors have previously addressed the issue of constructing, or bounding the periods of, orientable sequences over non-binary alphabets, a shortcoming we address here.…”

Section: Introductionmentioning

confidence: 99%

Multi de Bruijn Sequences and the Cross-Join Method

Alhakim

Szmidt

2023

Mathematics

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We show a method to construct binary multi de Bruijn sequences using the cross-join method. We extend the proof given by Alhakim for ordinary de Bruijn sequences to the case of multi de Bruijn sequences. In particular, we establish that all multi de Bruijn sequences can be obtained by cross-joining an ordinary de Bruijn sequence concatenated with itself an appropriate number of times. We implemented the generation of all multi de Bruijn sequences of type C(2,2,2) and C(3,2,2). We experimentally confirm that some multi de Bruijn sequences can be generated by Galois Nonlinear Feedback Shift Registers (NLFSRs). It is supposed that all multi de Bruijn sequences can be generated using Galois NLFSRs.

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“…Beyond this enumeration, many DB sequences were discovered for k = 2 (e.g. [2,15,16,23,28,40]). However, for the non-binary case, the number of known constructions is smaller [5,6,14,22,41].…”

Section: Introductionmentioning

confidence: 99%

A Cycle Joining Construction of the Prefer-Max De Bruijn Sequence

Amram,

Rubin,

Weiss

2021

Preprint

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We propose a novel construction for the well-known prefer-max De Bruijn sequence, based on the cycle joining technique. We further show that the construction implies known results from the literature in a straightforward manner. First, it implies the correctness of the onion theorem, stating that, effectively, the reverse of prefer-max is in fact an infinite De Bruijn sequence. Second, it implies the correctness of recently discovered shift rules for prefer-max, prefer-min, and their reversals. Lastly, it forms an alternative proof for the seminal FKM-theorem.

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A de Bruijn Sequence Construction by Concatenating Cycles of the Complemented Cycling Register

Cited by 7 publications

References 17 publications

Investigating the discrepancy property of de Bruijn sequences

Investigating the discrepancy property of de Bruijn sequences

Multi de Bruijn Sequences and the Cross-Join Method

A Cycle Joining Construction of the Prefer-Max De Bruijn Sequence

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