A shorthand universal cycle for permutations is a cyclic string in which each shorthand permutation appears exactly once as a unique substring. In this paper, we present three simple and efficient methods of generating shorthand universal cycles for permutations, and these three methods generate a total of 2 + ∏ t = 2 n − 2 t ! shift inequivalent shorthand universal cycles for permutations. In addition, each of the three new methods can be used to generate a shorthand universal cycle for permutations in O ( 1 ) -amortized time per symbol using O ( n ) space.

In this paper, we study a class of binary nonlinear feedback shift register sequences generated by cascaded feedback registers, one is an LFSR and the other one generates a de Bruijn sequence. The cycle structure (in particular, the initial state of each cycle) is determined by solving a system of linear equations. As an application, we can generate de Bruijn sequences of large period algorithmically.

A recent study by one of the authors has demonstrated the importance of profile vectors in DNA-based data storage. We provide exact values and lower bounds on the number of profile vectors for finite values of alphabet size q, read length 1, and word length n. Consequently, we demonstrate that for q ≥ 2 and n ≤ q1/2-1, the number of profile vectors is at least qκnwith κ very close to 1. In addition to enumeration results, we provide a set of efficient encoding and decoding algorithms for certain families of profile vectors.

A recent study by one of the authors has demonstrated the relevance of profile vectors in DNA-based data storage. We provide exact values and lower bounds on the number of profile vectors for finite values of alphabet size q, read length ℓ, and word length n. Consequently, we demonstrate that for q ≥ 3 and n = qaℓ, a = o(ℓ), the number of profile vectors is at least qκnfor some constant 0 < κ ≤ 1. In addition to enumeration results, we provide a set of efficient encoding and decoding algorithms for a family of profile vectors.

Sequences with high linear complexity play a fundamental part in cryptography. In this study, the authors construct general forms of Whiteman's generalised cyclotomic quaternary sequences with period 2 p m +1 q n +1 of order two over 𝔽 4 and give the linear complexity of the proposed sequences. The conclusions reveal that such sequences have good balance property and high linear complexity.

SUMMARY In this paper, a new class of generalized cyclotomic binary sequences with period of 2pq is established. The linear complexity and minimal polynomials of proposed sequences of length 2pq are determined. The results show that such sequences of length 2pq have high linear complexity. Copyright © 2013 John Wiley & Sons, Ltd.