176 research outputs found

    On Binary de Bruijn Sequences from LFSRs with Arbitrary Characteristic Polynomials

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    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)f(x). We study in detail the cycle structure of the set Ω(f(x))\Omega(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≈20n \approx 20. Many previously proposed constructions of de Bruijn sequences are shown to be special cases of our construction

    The Cycle Structure of LFSR with Arbitrary Characteristic Polynomial over Finite Fields

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    We determine the cycle structure of linear feedback shift register with arbitrary monic characteristic polynomial over any finite field. For each cycle, a method to find a state and a new way to represent the state are proposed.Comment: An extended abstract containing preliminary results was presented at SETA 201

    Quickest Sequence Phase Detection

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    A phase detection sequence is a length-nn cyclic sequence, such that the location of any length-kk contiguous subsequence can be determined from a noisy observation of that subsequence. In this paper, we derive bounds on the minimal possible kk in the limit of n→∞n\to\infty, and describe some sequence constructions. We further consider multiple phase detection sequences, where the location of any length-kk contiguous subsequence of each sequence can be determined simultaneously from a noisy mixture of those subsequences. We study the optimal trade-offs between the lengths of the sequences, and describe some sequence constructions. We compare these phase detection problems to their natural channel coding counterparts, and show a strict separation between the fundamental limits in the multiple sequence case. Both adversarial and probabilistic noise models are addressed.Comment: To appear in the IEEE Transactions on Information Theor

    Efficient indexing of necklaces and irreducible polynomials over finite fields

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    We study the problem of indexing irreducible polynomials over finite fields, and give the first efficient algorithm for this problem. Specifically, we show the existence of poly(n, log q)-size circuits that compute a bijection between {1, ... , |S|} and the set S of all irreducible, monic, univariate polynomials of degree n over a finite field F_q. This has applications in pseudorandomness, and answers an open question of Alon, Goldreich, H{\aa}stad and Peralta[AGHP]. Our approach uses a connection between irreducible polynomials and necklaces ( equivalence classes of strings under cyclic rotation). Along the way, we give the first efficient algorithm for indexing necklaces of a given length over a given alphabet, which may be of independent interest

    Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields

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    A maximal minor MM of the Laplacian of an nn-vertex Eulerian digraph Γ\Gamma gives rise to a finite group Zn−1/Zn−1M\mathbb{Z}^{n-1}/\mathbb{Z}^{n-1}M known as the sandpile (or critical) group S(Γ)S(\Gamma) of Γ\Gamma. We determine S(Γ)S(\Gamma) of the generalized de Bruijn graphs Γ=DB(n,d)\Gamma=\mathrm{DB}(n,d) with vertices 0,…,n−10,\dots,n-1 and arcs (i,di+k)(i,di+k) for 0≤i≤n−10\leq i\leq n-1 and 0≤k≤d−10\leq k\leq d-1, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs. Moreover, for a prime pp and an nn-cycle permutation matrix X∈GLn(p)X\in\mathrm{GL}_n(p) we show that S(DB(n,p))S(\mathrm{DB}(n,p)) is isomorphic to the quotient by ⟨X⟩\langle X\rangle of the centralizer of XX in PGLn(p)\mathrm{PGL}_n(p). This offers an explanation for the coincidence of numerical data in sequences A027362 and A003473 of the OEIS, and allows one to speculate upon a possibility to construct normal bases in the finite field Fpn\mathbb{F}_{p^n} from spanning trees in DB(n,p)\mathrm{DB}(n,p).Comment: I+24 page

    The algorithmics of solitaire-like games

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    One-person solitaire-like games are explored with a view to using them in teaching algorithmic problem solving. The key to understanding solutions to such games is the identification of invariant properties of polynomial arithmetic. We demonstrate this via three case studies: solitaire itself, tiling problems and a novel class of one-person games. The known classification of states of the game of (peg) solitaire into 16 equivalence classes is used to introduce the relevance of polynomial arithmetic. Then we give a novel algebraic formulation of the solution to a class of tiling problems. Finally, we introduce an infinite class of challenging one-person games, which we call ``replacement-set games'', inspired by earlier work by Chen and Backhouse on the relation between cyclotomic polynomials and generalisations of the seven-trees-in-one type isomorphism. We present an algorithm to solve arbitrary instances of replacement-set games and we show various ways of constructing infinite (solvable) classes of replacement-set games

    Number of cycles in the graph of 312-avoiding permutations

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    The graph of overlapping permutations is defined in a way analogous to the De Bruijn graph on strings of symbols. That is, for every permutation π=π1π2...πn+1\pi = \pi_{1} \pi_{2} ... \pi_{n+1} there is a directed edge from the standardization of π1π2...πn\pi_{1} \pi_{2} ... \pi_{n} to the standardization of π2π3...πn+1\pi_{2} \pi_{3} ... \pi_{n+1}. We give a formula for the number of cycles of length dd in the subgraph of overlapping 312-avoiding permutations. Using this we also give a refinement of the enumeration of 312-avoiding affine permutations and point out some open problems on this graph, which so far has been little studied.Comment: To appear in the Journal of Combinatorial Theory - Series
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