Multiplication of ternary complementary pairs

We present a computer-search method for concatenating or multiplying binary or ternary complementary pairs. All multiplications by a particular number m are considered. The computer-search method is new and leads to a large set of new results. The results and equivalences are discussed and some applications and numerical consequences are shown

A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes

Some combinatorial designs, such as Hadamard matrices, have been extensively researched and are familiar to readers across the spectrum of Science and Engineering. They arise in diverse fields such as cryptography, communication theory, and quantum computing. Objects like this also lend themselves to compelling mathematics problems, such as the Hadamard conjecture. However, complex generalized weighing matrices, which generalize Hadamard matrices, have not received anything like the same level of scrutiny. Motivated by an application to the construction of quantum error-correcting codes, which we outline in the latter sections of this paper, we survey the existing literature on complex generalized weighing matrices. We discuss and extend upon the known existence conditions and constructions, and compile known existence results for small parameters. Some interesting quantum codes are constructed to demonstrate their value.Comment: 33 pages including appendi

Efficient complementary sequences-based architectures and their application to ranging measurements

Premio Extraordinario de Doctorado de la UAH en 2015En las últimas décadas, los sistemas de medición de distancias se han beneficiado de los avances en el área de las comunicaciones inalámbricas. En los sistemas basados en CDMA (Code-Division Multiple-Access), las propiedades de correlación de las secuencias empleadas juegan un papel fundamental en el desarrollo de dispositivos de medición de altas prestaciones. Debido a las sumas ideales de correlaciones aperiódicas, los conjuntos de secuencias complementarias, CSS (Complementary Sets of Sequences), son ampliamente utilizados en sistemas CDMA. En ellos, es deseable el uso de arquitecturas eficientes que permitan generar y correlar CSS del mayor número de secuencias y longitudes posibles. Por el término eficiente se hace referencia a aquellas arquitecturas que requieren menos operaciones por muestra de entrada que con una arquitectura directa. Esta tesis contribuye al desarrollo de arquitecturas eficientes de generación/correlación de CSS y derivadas, como son las secuencias LS (Loosely Synchronized) y GPC (Generalized Pairwise Complementary), que permitan aumentar el número de longitudes y/o de secuencias disponibles. Las contribuciones de la tesis pueden dividirse en dos bloques: En primer lugar, las arquitecturas eficientes de generación/correlación para CSS binarios, derivadas en trabajos previos, son generalizadas al alfabeto multinivel (secuencias con valores reales) mediante el uso de matrices de Hadamard multinivel. Este planteamiento tiene dos ventajas: por un lado el aumento del número de longitudes que pueden generarse/correlarse y la eliminación de las limitaciones de las arquitecturas previas en el número de secuencias en el conjunto. Por otro lado, bajo ciertas condiciones, los parámetros de las arquitecturas generalizadas pueden ajustarse para generar/correlar eficientemente CSS binarios de mayor número de longitudes que con las arquitecturas eficientes previas. En segundo lugar, las arquitecturas propuestas son usadas para el desarrollo de nuevos algoritmos de generación/correlación de secuencias derivadas de CSS que reducen el número de operaciones por muestra de entrada. Finalmente, se presenta la aplicación de las secuencias estudiadas en un nuevo sistema de posicionamiento local basado en Ultra-Wideband y en un sistema de posicionamiento local basado en ultrasonidos

On Ternary Complementary Pairs

Let A = {a0, ... aℓ-1 }, B = {b0, . . . , bℓ-1 } be two finite sequences of length ℓ. Their nonperiodic autocorrelation function NA,B (s) is defined as: NA,B(s) = ∑aiai+1 + ∑b1bi+1, s = 0,…,ℓ-1 where x* is the complex conjugate of x. If NA,B (s) = 0 for s = 1, ... , ℓ - 1 then A, B is called a complementary pair. If, furthermore, ai, bi ε {-1,1}, i = 0,...,ℓ-1 , or, ai, bi ε {-1,0,1}, i = 0, ... , ℓ-1, then A, B is called a binary complementary pair (BCP), or, a ternary complementary pair (TCP), respectively. A BCP is also called Golay sequences. A TCP is a generalisation of a BCP. Since Golay sequences are only known to exist for lengths n = 2a10b26c, a, b, c ≥ 0, recent papers have focused on TCP\u27s. The purpose of this paper is to give an overview of existing constructions and techniques and present a variety of new constuctions, new restrictions on the deficiences and new computational results for TCP\u27s. In particular: We give new constructions which concatenate shorter group of sequences to obtain longer sequences. Many of these constructions can be applied recursively and lead to infinite families of TCP\u27s. We give many new restrictions on TCP\u27s of lengths ℓ and deficiencies ∂ = 2x, where x ≡ ℓmod 4. We settle all the cases for existence/non existence of TCP\u27s of lengths ℓ ≤ 20 and weights w ≤ 40. We give TCP\u27s with minimum deficiencies for all lengths ℓ ≤ 22

A theory of Ternary complementary pairs

AbstractSequences with zero autocorrelation are of interest because of their use in constructing orthogonal matrices and because of applications in signal processing, range finding devices, and spectroscopy. Golay sequences, which are pairs of binary sequences (i.e., all entries are ±1) with zero autocorrelation, have been studied extensively, yet are known only in lengths 2a10b26c. Ternary complementary pairs are pairs of (0, ±1)-sequences with zero autocorrelation (thus, Golay pairs are ternary complementary pairs with no 0's). Other kinds of pairs of sequences with zero autocorrelation, such as those admitting complex units for nonzero entries, are studied in similar contexts. Work on ternary complementary pairs is scattered throughout the combinatorics and engineering literature where the majority approach has been to classify pairs first by length and then by deficiency (the number of 0's in a pair); however, we adopt a more natural classification, first by weight (the number of nonzero entries) and then by length. We use this perspective to redevelop the basic theory of ternary complementary pairs, showing how to construct all known pairs from a handful of initial pairs we call primitive. We display all primitive pairs up to length 14, more than doubling the number that could be inferred from the existing literature