The cross-correlation measure for families of binary sequences

Large families of binary sequences of the same length are considered and a new measure, the cross-correlation measure of order $k$ is introduced to study the connection between the sequences belonging to the family. It is shown that this new measure is related to certain other important properties of families of binary sequences. Then the size of the cross-correlation measure is studied. Finally, the cross-correlation measures of two important families of pseudorandom binary sequences are estimated

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

Low Ambiguity Zone: Theoretical Bounds and Doppler-Resilient Sequence Design in Integrated Sensing and Communication Systems

In radar sensing and communications, designing Doppler resilient sequences (DRSs) with low ambiguity function for delay over the entire signal duration and Doppler shift over the entire signal bandwidth is an extremely difficult task. However, in practice, the Doppler frequency range is normally much smaller than the bandwidth of the transmitted signal, and it is relatively easy to attain quasi-synchronization for delays far less than the entire signal duration. Motivated by this observation, we propose a new concept called low ambiguity zone (LAZ) which is a small area of the corresponding ambiguity function of interest defined by the certain Doppler frequency and delay. Such an LAZ will reduce to a zero ambiguity zone (ZAZ) if the maximum ambiguity values of interest are zero. In this paper, we derive a set of theoretical bounds on periodic LAZ/ZAZ of unimodular DRSs with and without spectral constraints, which include the existing bounds on periodic global ambiguity function as special cases. These bounds may be used as theoretical design guidelines to measure the optimality of sequences against Doppler effect. We then introduce four optimal constructions of DRSs with respect to the derived ambiguity lower bounds based on some algebraic tools such as characters over finite field and cyclic difference sets

On the cross-combined measure of families of binary lattices and sequences

The cross-combined measure (which is a natural extension of cross-correlation measure) is introduced and important constructions of large families of binary lattices with optimal or nearly optimal cross-combined measures are presented. These results are also strongly related to the one-dimensional case: An easy method is showed obtaining strong constructions of families of binary sequences with nearly optimal cross-correlation measures based on the previous constructions of families of lattices. The important feature of this result is that so far there exists only one type of constructions of very large families of binary sequences with small cross-correlation measure, and this only type of constructions was based on one-variable irreducible polynomials. Since it is very complicated to construct one-variable irreducible polynomials over $\mathbb F_p$, it became necessary to show other types of constructions where the generation of sequences is much faster. Using binary lattices based on two-variable irreducible polynomials this problem can be avoided. (Since, contrary to one-variable polynomials, using Sch\"oneman-Eisenstein criteria it is possible to generate two-variable irreducible polynomials over $\mathbb F_p$ fast.

두 p진 데시메이션 수열 간의 상호상관도

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2017. 2. 노종선.In this dissertation, the cross-correlation between two differently decimated sequences of a $p$-ary m-sequence is considered. Two main contributions are as follows. First, for an odd prime $p$, $n=2m$, and a $p$-ary m-sequence of period $p^n -1$, the cross-correlation between two decimated sequences by $2$ and $d$ are investigated. Two cases of $d$, $d=\frac{(p^m +1)^2}{2}$ with $p^m \equiv 1 \pmod4$ and $d=\frac{(p^m +1)^2}{p^e +1}$ with odd $m/e$ are considered. The value distribution of the cross-correlation function for each case is completely deterimined. Also, by using these decimated sequences, two new families of $p$-ary sequences of period $\frac{p^n -1}{2}$ with good correlation property are constructed. Second, an upper bound on the magnitude of the cross-correlation function between two decimated sequences of a $p$-ary m-sequence is derived. The two decimation factors are $2$ and $2(p^m +1)$, where $p$ is an odd prime, $n=2m$, and $p^m \equiv 1 \pmod4$. In fact, these two sequences corresponds to the sequences used for the construction of $p$-ary Kasami sequences decimated by $2$. The upper bound is given as $\frac{3}{2}p^m + \frac{1}{2}$. Also, using this result, an upper bound of the cross-correlation magnitude between a $p$-ary m-sequence and its decimated sequence with the decimation factor $d=\frac{(p^m +1)^2}{2}$ is derived.1 Introduction 1 1.1 Background 1 1.2 Overview of This Dissertation 7 2 Preliminaries 9 2.1 Finite Fields 9 2.2 Trace Functions and Sequences 11 2.3 Cross-Correlation Between Two Sequences 13 2.4 Characters and Weils Bound 15 2.5 Trace-Orthogonal Basis 16 2.6 Known Exponential Sums 17 2.7 Cross-Correlation of $p$-ary Kasami Sequence Family 18 2.8 Previous Results on the Cross-Correlation for Decimations with $\gcd(p^n -1, d)=\frac{p^{n/2}+1}{2}$ 20 2.9 Cross-Correlation Between Two Decimated Sequences by $2$ and $d=4$ or $\frac{p^n +1}{2}$ 23 3 New $p$-ary Sequence Families of Period $\frac{p^n -1}{2}$ with Good Correlation Property Using Two Decimated Sequences 26 3.1 Cross-Correlation for the Case of $d=\frac{(p^m +1)^2}{2}$ 27 3.2 Cross-Correlation for the Case of $d=\frac{(p^m +1)^2}{p^e +1}$ 37 3.3 Construction of New Sequence Families 43 4 Upper Bound on the Cross-Correlation Between Two Decimated $p$-ary Sequences 52 4.1 Cross-Correlation Between $s(2t+i)$ and $s(2(p^m +1)t +j)$ 53 4.2 Cross-Correlation Between $s(t)$ and $s(\frac{(p^m +1)^2}{2} t)$ 66 5 Conclusions 69 Bibliography 72 Abstract (In Korean) 80Docto