Jianqin Zhou (Dept. of Computer Science, Anhui University of Technology, Ma’anshan 243002, P. R. China) (E-mail: [email protected]) Abstract: A fast algorithm is presented for determining the linear complexity and the minimal polynomial of periodic sequences over GF(q) with period q n p m , where p is a prime, q is a prime and a primitive root modulo p. The algorithm presented here generalizes...

Let y = f(x) be a continuous differentiable function on an interval J ⊂ R. In this paper we show that for any n ∈ N, n ≥ 2, sufficiently large integer Q and a real 0 < λ < 34 there exists a positive value c(n, f, J) such that all strips LJ(Q,λ) = { (x1, x2) ∈ R2 : |x2 − f(x1)| ≪ Q−λ, x1 ∈ J } contain at least c(n, f, J)Qn+1−λ points α = (α1, α2) with algebraically conjugate coordinates which mi...

A totally real polynomial in Z[x] with zeros α1 6 α2 6 · · · 6 αn has span αn − α1. Building on the classification of all characteristic polynomials of integer symmetric matrices having span less than 4, we obtain a classification of polynomials having span less than 4 that are the characteristic polynomial of a Hermitian matrix over some quadratic integer ring. We obtain as characteristic poly...

Binary sequences with optimal autocorrelation and large linear complexity have important applications in cryptography and communications. Very recently, a class of binary sequences of period 4p with optimal autocorrelation was proposed by interleaving four suitable Ding-Helleseth-Lam sequences (Des. Codes Cryptogr., DOI 10.1007/s10623-017-0398-5), where p is an odd prime with p ≡ 1( mod 4). The...

Let S = (s1, s2, . . . , sm, . . .) be a linear recurring sequence with terms in GF (q) and T be a linear transformation of GF (q) over GF (q). Denote T (S) = (T (s1), T (s2), . . . , T (sm), . . .). In this paper, we first present counter examples to show the main result in [A.M. Youssef and G. Gong, On linear complexity of sequences over GF (2), Theoretical Computer Science, 352(2006), 288-29...

This paper gives exact rates of quadratic approximations to an infinite class of cubic Pisot numbers. We show that for any cubic Pisot number q, with minimal polynomial p, such that p(0) = −1, and where p has only one real root, then there exists a C(q), explicitly given here, such that: (1) For all > 0, all but finitely many integer quadratics P satisfy |P (q)| ≥ C(q)− H(P )2 where H is the he...

We consider the β-expansion of 1 which encodes a rational rotation on R/Z under a certain partition. Via transforming its βpolynomial into a linear combination of Chebyshev polynomials, we specify the minimal polynomial of β over the field of rationals. At the end, some number theoretical conjectures are posed with computational evidence to support them.

Let L be a recursive algebraic extension of Q. Assume that, given α ∈ L, we can compute the roots in L of its minimal polynomial over Q and we can determine which roots are Aut(L)-conjugate to α. We prove there exists a pair of polynomials that characterizes the Aut(L)-conjugates of α, and that these polynomials can be effectively computed. Assume furthermore that L can be embedded in R, or in ...