A Family of Binary Sequences with Optimal Correlation Property and Large Linear Span

A family of binary sequences is presented and proved to have optimal correlation property and large linear span. It includes the small set of Kasami sequences, No sequence set and TN sequence set as special cases. An explicit lower bound expression on the linear span of sequences in the family is given. With suitable choices of parameters, it is proved that the family has exponentially larger linear spans than both No sequences and TN sequences. A class of ideal autocorrelation sequences is also constructed and proved to have large linear span.Comment: 21 page

Construction of pp-ary Sequence Families of Period (pn−1)/2(p^n-1)/2 and Cross-Correlation of pp-ary m-Sequences and Their Decimated Sequences

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2015. 2. 노종선.This dissertation includes three main contributions: a construction of a new family of $p$-ary sequences of period $\frac{p^n-1}{2}$ with low correlation, a derivation of the cross-correlation values of decimated $p$-ary m-sequences and their decimations, and an upper bound on the cross-correlation values of ternary m-sequences and their decimations. First, for an odd prime $p = 3 \mod 4$ and an odd integer $n$, a new family of $p$-ary sequences of period $N = \frac{p^n-1}{2}$ with low correlation is proposed. The family is constructed by shifts and additions of two decimated m-sequences with the decimation factors 2 and $d = N-p^{n-1}$. The upper bound on the maximum value of the magnitude of the correlation of the family is shown to be $2\sqrt{N+1/2} = \sqrt{2p^n}$ by using the generalized Kloosterman sums. The family size is four times the period of sequences, $2(p^n-1)$. Second, based on the work by Helleseth \cite{Helleseth1}, the cross-correlation values between two decimated m-sequences by 2 and $4p^{n/2}-2$ are derived, where $p$ is an odd prime and $n = 2m$ is an integer. The cross-correlation is at most 4-valued and their values are $\{\frac{-1\pm p^{n/2}}{2}, \frac{-1+3p^{n/2}}{2}, \frac{-1+5p^{n/2}}{2}\}$. As a result, for $p^m \neq 2 \mod 3$, a new sequence family with the maximum correlation value $\frac{5}{\sqrt{2}} \sqrt{N}$ and the family size $4N$ is obtained, where $N = \frac{p^n-1}{2}$ is the period of sequences in the family. Lastly, the upper bound on the cross-correlation values of ternary m-sequences and their decimations by $d = \frac{3^{4k+2}-3^{2k+1}+2}{4}+3^{2k+1}$ is investigated, where $k$ is an integer and the period of m-sequences is $N = 3^{4k+2}-1$. The magnitude of the cross-correlation is upper bounded by $\frac{1}{2} \cdot 3^{2k+3}+1 = 4.5 \sqrt{N+1}+1$. To show this, the quadratic form technique and Bluher's results \cite{Bluher} are employed. While many previous results using quadratic form technique consider two quadratic forms, four quadratic forms are involved in this case. It is proved that quadratic forms have only even ranks and at most one of four quadratic forms has the lowest rank $4k-2$.Abstract i Contents iii List of Tables vi List of Figures vii 1. Introduction 1 1.1. Background 1 1.2. Overview of Dissertation 9 2. Sequences with Low Correlation 11 2.1. Trace Functions and Sequences 11 2.2. Sequences with Low Autocorrelation 13 2.3. Sequence Families with Low Correlation 17 3. A New Family of p-ary Sequences of Period (p^n−1)/2 with Low Correlation 21 3.1. Introduction 22 3.2. Characters 24 3.3. Gaussian Sums and Kloosterman Sums 26 3.4. Notations 28 3.5. Definition of Sequence Family 29 3.6. Correlation Bound 30 3.7. Size of Sequence Family 35 3.8. An Example 38 3.9. Related Work 40 3.10. Conclusion 41 4. On the Cross-Correlation between Two Decimated p-ary m-Sequences by 2 and 4p^{n/2}−2 44 4.1. Introduction 44 4.2. Decimated Sequences of Period (p^n−1)/2 49 4.3. Correlation Bound 53 4.4. Examples 59 4.5. A New Sequence Family of Period (p^n−1)/2 60 4.6. Discussions 61 4.7. Conclusion 67 5. On the Cross-Correlation of Ternary m-Sequences of Period 3^{4k+2} − 1 with Decimation (3^{4k+2}−3^{2k+1}+2)/4 + 3^{2k+1} 69 5.1. Introduction 69 5.2. Quadratic Forms and Linearized Polynomials 71 5.3. Number of Solutions of x^{p^s+1} − cx + c 78 5.4. Notations 79 5.5. Quadratic Form Expression of the Cross-Correlation Function 80 5.6. Ranks of Quadratic Forms 83 5.7. Upper Bound on the Cross-Correlation Function 89 5.8. Examples 93 5.9. Related Works 94 5.10. Conclusion 94 6. Conclusions 96 Bibliography 98 초록 109Docto

NEW CONSTRUCTION FOR FAMILIES OF BINARY SEQUENCES WITH OPTIMAL CORRELATION PROPERTIES

In this correspondence, we present a construction, in a closed form, for an optimal family of 2(m) binary sequences of period 2(2m) - 1 with respect to Welch's bound, whenever there exists a balanced binary sequence of period 2(m) - 1 with ideal autocorrelation property using the trace function. This construction enables us to reinterpret a small set of Kasami and No sequences as a family constructed from m-sequences. New optimal families of binary sequences are constructed from the Legendre sequences of Mersenne prime period, Hall's sextic residue sequences, and miscellaneous sequences of unknown type. In addition, we enumerate the number of distinct families of binary sequences, which are constructed from a given binary sequence by this method.X1118sciescopu

Optimisation de l'infrastructure d'un système de positionnement indoor à base de transmetteurs GNSS

Dans le but de fournir un service GNSS (Global Navigation Satellite System) de localisation continu et disponible partout, les systèmes utilisant des pseudolites et des répéteurs semblent être des solutions pertinentes pour la localisation en indoor. Le système à répélites, inspiré de ces deux méthodes (répéteurs et pseudolites), est aussi proposé pour résoudre cette problématique. Les répélites sont des transmetteurs locaux qui, installés en intérieur, formeront une constellation locale. Ils émettent tous un signal GNSS unique mais déphasé par un délai spécifique à chacun d eux. Ces délais sont nécessaires pour distinguer les différents signaux reçus au niveau du récepteur. Les travaux de cette thèses sont réalisés dans le cadre du système à répélites et dans l objectif d améliorer son architecture et de réduire ses interférences inter-système. En effet, l architecture du système (un peu encombrante) et les interférences éventuelles avec les signaux satellitaires reçus par un récepteur placé à l extérieur font partie des inconvénients de ce système. On cherche donc à traiter ces deux difficultés de façon à minimiser leurs effets. Dans une première partie, on étudie les différents codes GNSS existants dans la littérature ainsi que les techniques de modulation employées. Ceci nous mène à proposer des codes ayant un niveau d interférence équivalent à la référence GPS (obtenue entre deux codes GPS) pour les bandes L1 de GPS et G1 de Glonass. Dans une seconde étape, on développe la modulation IMBOC (Indoor Modified Binary Offset Carrier) pour générer de nouveaux codes caractérisés par des niveaux d interférence réduits (comparés à la référence GPS). Parmi ces codes il y a deux catégories : ceux qui sont adaptés aux systèmes à répélites (émettant un code unique) et ceux qui sont adaptés aux systèmes pseudolites. Une étude théorique et des simulations des niveaux d interférences pour les codes émis dans la bande GPS et Glonass sont réalisées pour déterminer les gains en termes de niveaux de bruit. Ce gain (par rapport à la référence GPS) en puissance d interférence s élève à 16 dB pour Glonass et 20 dB pour GPS. Pour valider les performances de ces codes, on génère les signaux IMBOC et on observe les interférences réelles qu ils induisent sur un récepteur GPS recevant un signal satellitaire. Dans la deuxième partie, on utilise la fibre optique pour transmettre le signal du générateur jusqu aux répélites et pour créer les délais initiaux par propagation du signal dans des bobines de fibre. Ainsi on remplace les câbles coaxiaux et les montages électroniques (de déphasage) par des bobines de fibres plus légères, facile à installer et à faible perte de puissance. Il reste cependant à évaluer avec une précision centimétrique les délais réels induits sur chaque signal dans le but de garantir une précision de localisation inférieure au mètre. Cette précision semble en effet représenter un bon compromis entre complexité globale du système de localisation et réponse à un ensemble suffisant de besoins des utilisateurs potentiels. On développe alors une technique d estimation des délais basée sur la mesure de déphasage (entre deux signaux sinusoïdaux) et une analyse statistique des séries de mesures. Pour finir, on présente quelques résultats de localisation obtenus avec notre système à répélites déployé dans un environnement indoor typiqueIn order to make the GNSS positioning service continuous and available when going from an outdoor to an indoor environment, pseudolite and repeater based systems have been developed. A new system called repealite is a combination of both pseudolites and repeaters. It is based on transmitting a single signal through a set of transmitters (thus creating the local constellation). In order to avoid interference between the repealite signals and to distinguish between them at the receiver s end, each signal is shifted with a specific delay. The research carried out in this PhD aims at optimizing two aspects of the repealite based system. Firstly, we need to mitigate the effect of the interference caused on the satellite signals received outdoors. So we decided to design new codes characterized by low interference levels with outdoor signals. Secondly, we worked on the infrastructure part in order to simplify it and to make it easier to install: this is mainly achieved through the use of optical fibers. In the first part, we study the codes and the modulation techniques currently used in the GNSS systems. Then, we propose a few codes having an interference level equivalent to that of the GPS (obtained when computing two GPS codes). These new codes are compatible with the GPS L1 or the Glonass G1 bands. In a second step, we focus on the modulation techniques and create the so-called IMBOC (Indoor Modified Binary Offset Carrier) that aims at minimizing the interference levels with outdoor signals. With this modulation, we propose new IMBOC codes capable of much lower interference levels than the GPS reference. In order to evaluate the performance of the proposed codes, we carried out a theoretical study, simulations and experimental tests. The interference gain reached about 20 dB on the GPS band and 16 dB on the Glonass one. The proposed codes are divided into two categories: those reserved to the repealite system (using a single code) and families of codes suited to pseudolite based systems. Finally, we generated the IMBOC signals modulated by the new codes and tested the real interference induced on an outdoor receiver tracking the satellite signals. In the second part, we use optical fibers in order to replace the coaxial cables used to transmit signals from the GNSS-like signal generator to the repealites. In addition, the initial delay needed for each repealite is added by propagating the signals through rolls of fibers. Indeed, optical fiber offers advantages such as lightness, flexibility and low power loss that make it suitable to simplify the infrastructure of the system. In order to evaluate the real delays of these various fibers, we develop an estimating method based on phase shift measurements (between two sinusoidal signals) and statistical analysis of the series of measurements. This method should have uncertainties lower than one centimeter in order to insure a sub-meter precision (in absolute positioning with the repealite positioning system). In order to validate this method, we compare it to a GNSS based calibration approach. Finally, we carry out a few positioning tests with the repealite positioning system deployed in a typical indoor environment. These tests deal with absolute and relative positioning and give an idea about the system s performanceEVRY-INT (912282302) / SudocSudocFranceF