Construction of de Bruijn sequences from product of two irreducible polynomials

Chang, Zuling; Ezerman, Martianus Frederic; Ling, San; Wang, Huaxiong

doi:10.1007/s12095-017-0219-8

Cited by 9 publications

(7 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If 1 ∈ E 1 ∩ E 2 , i.e., a 1 = a ′ 1 = 1, then there is no conjugate pair between the corresponding pairs of cycles since n 1 = 1, making 2 n 1 − 2 = 0. Keeping p 1 (x) = x + 1, we prove a result that generalizes [7,Prop. 10].…”

Section: Properties Of the Adjacency Graph Of ω ( F (X))mentioning

confidence: 75%

“…We put forward a construction from LFSRs whose characteristic polynomials are products of two distinct irreducible polynomials and showed that it generates a large number of de Bruijn sequences in [7]. In another work [6], whose preliminary results were presented at SETA 2016, we discussed in detail how to determine the cycle structure and find a state for each cycle for an arbitrary polynomial f (x) ∈ F q [x] for any prime power q.…”

Section: Introductionmentioning

confidence: 99%

“…In generalizing the number of irreducible polynomial factors of f (x) from 2 in [7] to s ≥ 2 we introduce some modifications to stay efficient. Most notably, for an irreducible g(x), a state in Ω (g(x)) is computed based on the mapping ϕ in [7,Sect. 3], requiring computations over F 2 n .…”

mentioning

confidence: 99%

See 2 more Smart Citations

On binary de Bruijn sequences from LFSRs with arbitrary characteristic polynomials

Chang

Ezerman

Ling

et al. 2018

Des. Codes Cryptogr.

Self Cite

View full text Add to dashboard Cite

We propose a construction of de Bruijn sequences by the cycle joining method from linear feedback shift registers (LFSRs) with arbitrary characteristic polynomial f (x). We study in detail the cycle structure of the set Ω ( f (x)) that contains all sequences produced by a specific LFSR on distinct inputs and provide a fast way to find a state of each cycle. This leads to an efficient algorithm to find all conjugate pairs between any two cycles, yielding the adjacency graph. The approach is practical to generate a large class of de Bruijn sequences up to order n ≈ 20. Many previously proposed constructions of de Bruijn sequences are shown to be special cases of our construction.Keywords Binary periodic sequence · LFSR · de Bruijn sequence · cycle structure · adjacency graph · cyclotomic number Mathematics Subject Classification (2010) 11B50 · 94A55 · 94A60 1 Introduction A binary de Bruijn sequence of order n has period N = 2 n in which each n-tuple occurs exactly once in each period. There are 2 2 n−1 −n of them [5]. Some of their earliest applications are in communication systems. They are generated in a deterministic way, yet satisfy the randomness criteria in [14, Ch. 5] and are balanced, containing the same number of 1s and 0s. In cryptography, they have been used as a source of pseudo-random numbers and in keysequence generators of stream ciphers [29, Sect. 6.3]. In computational molecular biology, one of the three assembly paradigms in DNA sequencing is the de Bruijn graph assemblers model [31, Box 2]. Some roles of de Bruijn sequences in robust positioning patterns are discussed by Bruckstein et al. in [4]. They have numerous applications, e.g., in robotics,

show abstract

Section: Properties Of the Adjacency Graph Of ω ( F (X))mentioning

confidence: 75%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On binary de Bruijn sequences from LFSRs with arbitrary characteristic polynomials

Chang

Ezerman

Ling

et al. 2018

Des. Codes Cryptogr.

Self Cite

View full text Add to dashboard Cite

show abstract

“…2 and 3. Concatenated sequences are widely used in wireless communication as these sequences are derived from two PPs, having similar cross correlation and auto-correlations properties to those of the Gold codes and of long PN-sequences [45][46][47][48][49]. A pair of m-sequence generators are used for generating the concatenated sequence by XORing or performing modulo-2 addition of the appropriately shifted versions of the two m-sequences created by two different PPs having the length of 2 R − 1 and 2 S − 1 respectively, 3 Please note that c i over here will be feed to the channel when exploit the RSSE scheme of [18] and [20] to obtain synchronization for m-sequence.…”

Section: B Concatenated Sequencesmentioning

confidence: 99%

Primitive Polynomials for Iterative Recursive Soft Sequential Acquisition of Concatenated Sequences

et al. 2019

View full text Add to dashboard Cite

An iterative initial sequence acquisition technique is proposed for the pseudo-noise signal derived from a pair of m-sequences used for generating the concatenated sequence, which relies on the soft-in-soft-out detection to improve the acquisition performance. This recursive soft sequential estimation technique has a linearly increasing complexity with the number of chips in the concatenated sequence. Receiving as few as S consecutive chips of a (2 S − 1)-chip sequence is sufficient for the local concatenatedsequence generator of the receiver to synchronize. Hence, this initial synchronization technique is eminently suitable for long m-sequences and concatenated sequences. Another key result is the comparison of m-sequences and concatenated sequences regarding the acquisition time. It is also observed that loworder primitive polynomials (PPs) achieve better performances than higher-order polynomials for both the m-sequences and concatenated sequences. When considering PPs having a higher number of taps, the exploitation of concatenated sequences is capable of achieving in excess of 3-dB signal-to-noise ratio gains over m-sequences.

show abstract

“…This applies also for the last and first words in the ordering. DB sequences admit applications in various fields including cryptography [3,4,13,39,43], electrical engineering (mainly since they correspond to feedback-shift-registers) [7,10,11,30,31,32,36], molecular biology [33], and neuroscience [1], to name a few.…”

Section: Introductionmentioning

confidence: 99%

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

Amram,

Rubin,

Weiss

2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

Construction of de Bruijn sequences from product of two irreducible polynomials

Cited by 9 publications

References 21 publications

On binary de Bruijn sequences from LFSRs with arbitrary characteristic polynomials

On binary de Bruijn sequences from LFSRs with arbitrary characteristic polynomials

Primitive Polynomials for Iterative Recursive Soft Sequential Acquisition of Concatenated Sequences

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

Contact Info

Product

Resources

About