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Construction of $p$-ary Sequence Families of Period $(p^n-1)/2$ and Cross-Correlation of $p$-ary m-Sequences and Their Decimated Sequences
DC Field | Value | Language |
---|---|---|
dc.contributor.advisor | 노종선 | - |
dc.contributor.author | 김지엽 | - |
dc.date.accessioned | 2017-07-13T07:06:52Z | - |
dc.date.available | 2017-07-13T07:06:52Z | - |
dc.date.issued | 2015-02 | - |
dc.identifier.other | 000000024766 | - |
dc.identifier.uri | https://hdl.handle.net/10371/119048 | - |
dc.description | 학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2015. 2. 노종선. | - |
dc.description.abstract | 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 $4k-2$. | - |
dc.description.tableofcontents | 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 109 | - |
dc.format | application/pdf | - |
dc.format.extent | 2024951 bytes | - |
dc.format.medium | application/pdf | - |
dc.language.iso | en | - |
dc.publisher | 서울대학교 대학원 | - |
dc.subject | 자기상관도 | - |
dc.subject | 상호상관도 | - |
dc.subject | 데시메이션 수열 | - |
dc.subject | 지수함수합 | - |
dc.subject | 클루스터만합 | - |
dc.subject | m-수열 | - |
dc.subject | 비이진수열 | - |
dc.subject | 의사난수수열 | - |
dc.subject | 이차형식 | - |
dc.subject | 수열군 | - |
dc.subject | Autocorrelation | - |
dc.subject | cross-correlation | - |
dc.subject | decimated sequence | - |
dc.subject | exponential sum | - |
dc.subject | Kloosterman sum | - |
dc.subject | m-sequence | - |
dc.subject | nonbinary sequence | - |
dc.subject | quadratic form | - |
dc.subject | sequence | - |
dc.subject | sequence family | - |
dc.subject.ddc | 621 | - |
dc.title | Construction of $p$-ary Sequence Families of Period $(p^n-1)/2$ and Cross-Correlation of $p$-ary m-Sequences and Their Decimated Sequences | - |
dc.type | Thesis | - |
dc.description.degree | Doctor | - |
dc.citation.pages | vii, 110 | - |
dc.contributor.affiliation | 공과대학 전기·컴퓨터공학부 | - |
dc.date.awarded | 2015-02 | - |
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