On the classification of deBruijn sequences

Hauge, Erik R.; Mykkeltveit, Johannes

doi:10.1016/0012-365x(94)00265-k

Cited by 25 publications

(13 citation statements)

References 4 publications

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“…Definition 1 (From [22]) The adjacency graph G of an FSR with feedback function h is an undirected multigraph whose vertices correspond to the cycles in (h) . There exists an edge between two vertices if and only if they are adjacent.…”

Section: Basic Notions and Known Resultsmentioning

confidence: 99%

Binary de Bruijn Sequences via Zech’s Logarithms

et al. 2021

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The focus of this work is to show how to combine Zech's logarithms and each of the cycle joining and cross-join pairing methods to construct binary de Bruijn sequences. Basic implementations are supplied as proofs of concept. The cycles, in the cycle joining method, are typically generated by a linear feedback shift register. We prove a crucial characterization that determining Zech's logarithms is equivalent to identifying conjugate pairs shared by any two distinct cycles. This speeds up the task of building a connected adjacency subgraph that contains all vertices of the complete adjacency graph. Distinct spanning trees in either graph correspond to cyclically inequivalent de Bruijn sequences. As the cycles are being joined, guided by the conjugate pairs, we track the changes in the feedback function. We show how to produce certificates of existence for spanning trees of certain types to conveniently handle large order cases. The characterization of conjugate pairs via Zech's logarithms, as positional markings, is then adapted to identify cross-join pairs. A modified m-sequence is initially used, for ease of generation. The process can be repeated on each of the resulting de Bruijn sequences. Most prior constructions in the literature measure the complexity of the corresponding bit-by-bit algorithms. Our approach is different. We aim first to build a connected adjacency subgraph that is certified to contain all of the cycles as vertices. The ingredients are computed just once and concisely stored. Simple strategies are offered to keep the complexities low as the order grows.

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Section: Basic Notions and Known Resultsmentioning

confidence: 99%

Binary de Bruijn Sequences via Zech’s Logarithms

et al. 2021

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“…This leads us to the definition of adjacency graph. Definition 1 [7,14] For an FSR, its adjacency graph is an undirected graph where the vertexes correspond to the cycles in it, and there exists an edge labeled with an integer m > 0 between two vertexes if and only if the two vertexes share m conjugate pairs.…”

Section: Adjacency Graphsmentioning

confidence: 99%

The adjacency graphs of some feedback shift registers

Jiang

Lin

2016

Des. Codes Cryptogr.

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In this paper, we consider the adjacency graphs of some feedback shift registers (FSRs), namely, the FSRs with characteristic functions of the form g = (x 0 + x 1 ) * f . Firstly, we give some properties about these FSRs. We prove that these FSRs generate only prime cycles, and these cycles can be divided into two sets such that each set contains no adjacent cycles. When f is a linear function, more properties about these FSRs are derived. For example, it is shown that, when f contains an odd number of terms, the adjacency graph of FSR((x 0 + x 1 ) * f ) can be determined directly from the adjacency graph of FSR( f ). Then, as an application of these results, we continue the work of Li et al. (IEEE Trans Inf Theory 60(5):3052-3061, 2014) to determine the adjacency graphs of FSR((1 + x) 4 p(x)) and FSR((1 + x) 5 p(x)), where p(x) is a primitive polynomial, and construct a large class of De Bruijn sequences from them.

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“…Definition 1 (adjacency graph) [10] For an FSR, its adjacency graph is an undirected graph where the vertexes correspond to the cycles in it, and there exist m > 0 edges between two vertexes if and only if their corresponding cycles share m conjugate pairs.…”

Section: Adjacency Graphs For a Statementioning

confidence: 99%

“…In this method, the knowledge of the distribution of conjugate pairs between the cycles of FSRs is needed. The concept of adjacency graph [10] is then introduced to explain the distribution of conjugate pairs. In the early study of maximum-length NFSRs, pure cycling registers and pure summing registers were used to generate new de Bruijn sequences since they have special cycle structures [1,7,11,12] .…”

Section: Introductionmentioning

confidence: 99%