4 research outputs found
๋ p์ง ๋ฐ์๋ฉ์ด์ ์์ด ๊ฐ์ ์ํธ์๊ด๋
ํ์๋
ผ๋ฌธ (๋ฐ์ฌ)-- ์์ธ๋ํ๊ต ๋ํ์ : ์ ๊ธฐยท์ปดํจํฐ๊ณตํ๋ถ, 2017. 2. ๋
ธ์ข
์ .In this dissertation, the cross-correlation between two differently decimated sequences of a -ary m-sequence is considered. Two main contributions are as follows.
First, for an odd prime , , and a -ary m-sequence of period , the cross-correlation between two decimated sequences by and are investigated. Two cases of , with and with odd 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 -ary sequences of period with good correlation property are constructed.
Second, an upper bound on the magnitude of the cross-correlation function between two decimated sequences of a -ary m-sequence is derived. The two decimation factors are and , where is an odd prime, , and . In fact, these two sequences corresponds to the sequences used for the construction of -ary Kasami sequences decimated by . The upper bound is given as .
Also, using this result, an upper bound of the cross-correlation magnitude between a -ary m-sequence and its decimated sequence with the decimation factor 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 -ary Kasami Sequence Family 18
2.8 Previous Results on the Cross-Correlation for Decimations with 20
2.9 Cross-Correlation Between Two Decimated Sequences by and or 23
3 New -ary Sequence Families of Period with Good Correlation Property Using Two Decimated Sequences 26
3.1 Cross-Correlation for the Case of 27
3.2 Cross-Correlation for the Case of 37
3.3 Construction of New Sequence Families 43
4 Upper Bound on the Cross-Correlation Between Two Decimated -ary Sequences 52
4.1 Cross-Correlation Between and 53
4.2 Cross-Correlation Between and 66
5 Conclusions 69
Bibliography 72
Abstract (In Korean) 80Docto
Construction of -ary Sequence Families of Period and Cross-Correlation of -ary m-Sequences and Their Decimated Sequences
ํ์๋
ผ๋ฌธ (๋ฐ์ฌ)-- ์์ธ๋ํ๊ต ๋ํ์ : ์ ๊ธฐยท์ปดํจํฐ๊ณตํ๋ถ, 2015. 2. ๋
ธ์ข
์ .This dissertation includes three main contributions: a construction of a new family of -ary sequences of period with low correlation, a derivation of the cross-correlation values of decimated -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 and an odd integer , a new family of -ary sequences of period with low correlation is proposed. The family is constructed by shifts and additions of two decimated m-sequences with the decimation factors 2 and . The upper bound on the maximum value of the magnitude of the correlation of the family is shown to be by using the generalized Kloosterman sums. The family size is four times the period of sequences, .
Second, based on the work by Helleseth \cite{Helleseth1}, the cross-correlation values between two decimated m-sequences by 2 and are derived, where is an odd prime and is an integer. The cross-correlation is at most 4-valued and their values are . As a result, for , a new sequence family with the maximum correlation value and the family size is obtained, where 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 is investigated, where is an integer and the period of m-sequences is . The magnitude of the cross-correlation is upper bounded by . 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 .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^nโ1)/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}โ2 44
4.1. Introduction 44
4.2. Decimated Sequences of Period (p^nโ1)/2 49
4.3. Correlation Bound 53
4.4. Examples 59
4.5. A New Sequence Family of Period (p^nโ1)/2 60
4.6. Discussions 61
4.7. Conclusion 67
5. On the Cross-Correlation of Ternary m-Sequences of Period 3^{4k+2} โ 1 with Decimation (3^{4k+2}โ3^{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} โ 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
์ด๋ก 109Docto