How to determine linear complexity and kk-error linear complexity in some classes of linear recurring sequences

Several fast algorithms for the determination of the linear complexity of $d$-periodic sequences over a finite field \F_q, i.e. sequences with characteristic polynomial $f(x) = x^d-1$, have been proposed in the literature. In this contribution fast algorithms for determining the linear complexity of binary sequences with characteristic polynomial $f(x) = (x-1)^d$ for an arbitrary positive integer $d$, and $f(x) = (x^2+x+1)^{2^v}$ are presented. The result is then utilized to establish a fast algorithm for determining the $k$-error linear complexity of binary sequences with characteristic polynomial $(x^2+x+1)^{2^v}$

Zero-patterns of polynomials and Newton polytopes

We present an upper bound on the number of regions into which affine space or the torus over a field may be partitioned by the vanishing and non-vanishing of a finite collection of multivariate polynomials. The bound is related to the number of lattice points in the Newton polytopes of the polynomials, and is optimal to within a factor depending only on the dimension (assuming suitable inequalities hold amongst the relevant parameters). This refines previous work by different authors

Random krylov spaces over finite fields

Motivated by a connection with block iterative methods for solving linear systems over finite fields, we consider the probability that the Krylov space generated by a fixed linear mapping and a random set of elements in a vector space over a finite fiel

Hensel Lifting And Bivariate Polynomial Factorisation Over Finite Fields

. This paper presents an average time analysis of a Hensel lifting based factorisation algorithm for bivariate polynomials over finite fields. It is shown that the average running time is almost linear in the input size. This explains why the Hensel lifting technique is fast in practice for most polynomials. 1. Introduction It is well known that the Hensel lifting technique provides practical methods for factoring polynomials over various fields. Such methods are known to run in exponential time in the worst case, but seem fast for most polynomials. The latter phenomenon has not been fully understood and calls for an average running time analysis. The only analysis we know of is that of Collins 1979 [4] for univariate integral polynomials (factoring over the rational numbers). He shows, under some reasonable number theoretic conjectures, that the average running time is indeed polynomial. In this paper, we present a rigorous analysis for bivariate polynomials over finite fields. We sh..

Hydrodynamic modelling of aquatic suction performance and intra-oral pressures: limitations for comparative studies

The magnitude of sub-ambient pressure inside the bucco-pharyngeal cavity of aquatic animals is generally considered a valuable metric of suction feeding performance. However, these pressures do not provide a direct indication of the effect of the suction act on the movement of the prey item. Especially when comparing suction performance of animals with differences in the shape of the expanding bucco-pharyngeal cavity, the link between speed of expansion, water velocity, force exerted on the prey and intra-oral pressure remains obscure. By using mathematical models of the heads of catfishes, a morphologically diverse group of aquatic suction feeders, these relationships were tested. The kinematics of these models were fine-tuned to transport a given prey towards the mouth in the same way. Next, the calculated pressures inside these models were compared. The results show that no simple relationship exists between the amount of generated sub-ambient pressure and the force exerted on the prey during suction feeding, unless animals of the same species are compared. Therefore, for evaluating suction performance in aquatic animals in future studies, the focus should be on the flow velocities in front of the mouth, for which a direct relationship exists with the hydrodynamic force exerted on prey