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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 the collections of de Bruijn sequences of the same order, k, that satisfy 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 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 being only required to cover all subsequences of length k whose maximum run length equals one.

A de Bruijn torus is the two dimensional generalization of a de Bruijn sequence. While methods exist to generate these tori, only a few methods of construction are known. We present a novel method to generate de Bruijn tori with rectangular windows by combining two variants of de Bruijn sequences.

We study the sequences generated by prefer-one rule with different initial vectors. Firstly, we give upper bounds of their periods and for initial vectors with Hamming weight one, we prove that the generated sequences are modified de Bruijn sequences. Moreover, for two of them, we give the truth tables of their feedback functions. We also investigate the feedback functions of prefer-one de Bruijn sequences. For order n prefer-one de Bruijn sequence, we give linear and quadratic terms in its feedback function and prove that the number of degree n - 2 terms has the same parity as n . The statistical result for small n shows that about half of all terms occur in the feedback functions.

We describe new, simple, recursive methods of construction for orientable sequences over an arbitrary finite alphabet, i.e. periodic sequences in which any sub-sequence of n consecutive elements occurs at most once in a period in either direction. In particular we establish how two variants of a generalised Lempel homomorphism can be used to recursively construct such sequences, generalising previous work on the binary case. We also derive an upper bound on the period of an orientable sequence.

A binary modified de Bruijn sequence is an infinite and periodic binary sequence derived by removing a zero from the longest run of zeros in a binary de Bruijn sequence. The minimal polynomial of the modified sequence is its unique least-degree characteristic polynomial. Leveraging a recent characterization, we devise a novel general approach to determine the minimal polynomial. We translate the characterization into a problem of identifying a Hamiltonian cycle in a specially constructed graph. The graph is isomorphic to the modified de Bruijn–Good graph. Along the way, we demonstrate the usefulness of some computational tools from the cycle joining method in the modified setup.

We investigate binary sequences generated by non-Markovian rules with memory length μ, similar to those adopted in elementary cellular automata. This generation procedure is equivalent to a shift register, and certain rules produce sequences with maximal periods, known as de Bruijn sequences. We introduce a novel methodology for generating de Bruijn sequences that combines (i) a set of derived properties that significantly reduce the space of feasible generating rules and (ii) a neural-network-based classifier that identifies which rules produce de Bruijn sequences. The experiments for some values of μ demonstrate the approach’s effectiveness and computational efficiency.

Defined by Borel, a real number is normal to an integer base ≥ 2 if in its base- expansion every block of digits occurs with the same limiting frequency as every other block of the same length. We consider the problem of in constructed base- normal expansions to obtain normality to base ( + 1).

Fuzzing is one of the most successful software testing techniques used to discover vulnerabilities in programs. Without seeds that fit the input format, existing runtime dependency recognition strategies are limited by incompleteness and high overhead. In this paper, for structured input applications, we propose a fast format-aware fuzzing approach to recognize dependencies from the specified input to the corresponding comparison instruction. We divided the dependencies into Input-to-State (I2S) and indirect dependencies. Our approach has the following advantages compared to existing works: (1) recognizing I2S dependencies more completely and swiftly using the input based on the de Bruijn sequence and its mapping structure; (2) obtaining indirect dependencies with a light dependency existence analysis on the input fragments. We implemented a fast format-aware fuzzing prototype, FFAFuzz, based on our method and evaluated FFAFuzz in real-world structured input applications. The evaluation results showed that FFAFuzz reduced the average time overhead by 76.49% while identifying more completely compared with Redqueen and by 89.10% compared with WEIZZ. FFAFuzz also achieved higher code coverage by 14.53% on average compared to WEIZZ.

De Bruijn sequence is an important kind of nonlinear shift register sequence, which has a very wide range of applications in the fields of communication and cryptography. From the perspective of the relationship between linear terms, quadratic terms and cubic terms in the feedback functions of de Bruijn sequences, some new necessary conditions for feedback functions of de Bruijn sequences are obtained. Some examples show that the known necessary conditions cannot deduce the proposed necessary conditions.

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.