We derive the linear complexity and the minimal polynomial over the finite fields of order two and p of series of binary sequences with a period 4p and optimal autocorrelation value/magnitude. These sequences are constructed by cyclotomic classes of order two, four and six by methods proposed by K.T. Arasu et al. and Y. Sun et al. We define the parameters of sequences with optimal autocorrelati...

The notion of a separable extension is an important concept in Galois theory. Traditionally, this concept is introduced using the minimal polynomial and the formal derivative. In this work, we present an alternative approach to this classical concept. Based on our approach, we will give new proofs of some basic results about separable extensions (such as the existence of the separable closure, ...

We develop a new p-adic algorithm to compute the minimal polynomial of a class invariant. Our approach works for virtually any modular function yielding class invariants. The main algorithmic tool is modular polynomials, a concept which we generalize to functions of higher level.

These groups are isomorphic to the free product of two finite cyclic groups of orders  and q. The first few Hecke groups are H(λ) = = PSL(,Z) (the modular group), H(λ) = H( √ ), H(λ) = H( + √   ), and H(λ) = H( √ ). It is clear from the above that H(λq) ⊂ PSL(,Z[λq]), but unlike in the modular group case (the case q = ), the inclusion is strict and the index [PSL(,Z[λq]) :H(λq)] i...

The mth Chebyshev polynomial of a square matrix A is the monic polynomial that minimizes the matrix 2-norm of p(A) over all monic polynomials p(z) of degree m. This polynomial is uniquely defined if m is less than the degree of the minimal polynomial of A. We study general properties of Chebyshev polynomials of matrices, which in some cases turn out to be generalizations of well known propertie...

There exist various well-known characterizations of sets of numbers recognizable by a nite automaton, when they are represented in some integer base p 2. We show how to modify these characterizations, when integer bases p are replaced by linear numeration systems whose characteristic polynomial is the minimal polynomial of a Pisot number. We also prove some related interesting properties.

Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I . Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in P/I . It is well known that minimal polynomials may be computed via elimination, therefore this is considered to be a “resolved problem”. But being the key of...

Let n1 and n2 be two distinct primes with gcd(n1 − 1, n2 − 1) = 4. In this paper, we compute the autocorrelation values of generalized cyclotomic sequence of order 4. Our results show that this sequence can have very good autocorrelation property. We determine the linear complexity and minimal polynomial of the generalized cyclotomic sequence over GF(q) where q = pm and p is an odd prime. Our r...

We investigate the linear complexity and the minimal polynomial over the finite fields of the characteristic sequences of cubic and biquadratic residue classes. Also we find the linear complexity and the minimal polynomial of the balanced cyclotomic sequences of order three. Keywords—linear complexity, finite field, cubic residue classes, biquadratic residue classes

Let V be a vector space over some field k, and let α : V V be a linear map (an ‘endomorphism of V ’). Given any polynomial p with coefficients in k, there is an endomorphism p(α) of V , and we say that p is an annihilating polynomial for α if p(α) = 0. Our first major goal is to see that for any α, the annihilating polynomials can easily be classified: they’re precisely the multiples of a certa...