On the computation of the linear complexity and the k-error linear complexity of binary sequences with period a power of two

The linear Games-Chan algorithm for computing the linear complexity c(s) of a binary sequence s of period ℓ = 2n requires the knowledge of the full sequence, while the quadratic Berlekamp-Massey algorithm only requires knowledge of 2c(s) terms. We show that we can modify the Games-Chan algorithm so that it computes the complexity in linear time knowing only 2c(s) terms. The algorithms of Stamp-Martin and Lauder-Paterson can also be modified, without loss of efficiency, to compute analogues of the k-error linear complexity for finite binary sequences viewed as initial segments of infinite sequences with period a power of two. We also develop an algorithm which, given a constant c and an infinite binary sequence s with period ℓ = 2n, computes the minimum number k of errors (and the associated error sequence) needed over a period of s for bringing the linear complexity of s below c. The algorithm has a time and space bit complexity of O(ℓ). We apply our algorithm to decoding and encoding binary repeated-root cyclic codes of length ℓ in linear, O(ℓ), time and space. A previous decoding algorithm proposed by Lauder and Paterson has O(ℓ(logℓ)2) complexity

Secuencias binarias y sus aplicaciones

ABSTRACT: The generation of families of sequences with a bounded correlation, among other properties, is of interest in several fields such as cryptography, wireless communications and digital watermarks. Commercial applications such as GPS or military uses such as Radar have been developed and improved thanks to the search of these sequences and the analysis of their autocorrelation function and the correlation between members of the same family. In this project, we focus on a technique for the algebraic construction of sequences where a sequence of shifts and a sequence with good properties generate a new sequence. The aim is the exhaustive search of longer sequences then the ones already present in the literature. To do so, a software intended to provide support to a designer has been developed to assist in the search of said sequences and check their properties, which can be deployed in a supercomputer. The applications of these sequences are, among others, radars and location systems with higher spatial resolution.RESUMEN: Generar familias de secuencias con correlación acotada, entre otras propiedades, es de interés en diversas áreas como criptografía, comunicaciones inalámbricas y marcas de agua digitales. Aplicaciones comerciales como el GPS, o con usos militares como el Radar se han desarrollado y mejorado a partir de la búsqueda de estas secuencias mediante el estudio de su función tanto de autocorrelación como de la correlación entre los miembros de una misma familia. En este proyecto, nos centramos en una técnica para la construcción algebraica de secuencias donde, a través de una secuencia de desplazamientos y una secuencia con buenas propiedades, se genera una nueva secuencia. El objetivo es la búsqueda exhaustiva de secuencias de mayor longitud que las ya existentes en la literatura. Para este fin, se ha desarrollado un software de apoyo al diseñador con capacidad de ser desplegado en un nodo de supercomputación para asistir a la búsqueda de dichas secuencias y la comprobación de sus propiedades. La finalidad de estas secuencias, entre otros posibles usos, son radares y sistemas de localización con mayor resolución espacial.Grado en Ingeniería Informátic

Synthetic in vitro transcriptional oscillators

The construction of synthetic biochemical circuits from simple components illuminates how complex behaviors can arise in chemistry and builds a foundation for future biological technologies. A simplified analog of genetic regulatory networks, in vitro transcriptional circuits, provides a modular platform for the systematic construction of arbitrary circuits and requires only two essential enzymes, bacteriophage T7 RNA polymerase and Escherichia coli ribonuclease H, to produce and degrade RNA signals. In this study, we design and experimentally demonstrate three transcriptional oscillators in vitro. First, a negative feedback oscillator comprising two switches, regulated by excitatory and inhibitory RNA signals, showed up to five complete cycles. To demonstrate modularity and to explore the design space further, a positive-feedback loop was added that modulates and extends the oscillatory regime. Finally, a three-switch ring oscillator was constructed and analyzed. Mathematical modeling guided the design process, identified experimental conditions likely to yield oscillations, and explained the system's robust response to interference by short degradation products. Synthetic transcriptional oscillators could prove valuable for systematic exploration of biochemical circuit design principles and for controlling nanoscale devices and orchestrating processes within artificial cells