243 research outputs found
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
On the List-Decodability of Random Linear Codes
For every fixed finite field \F_q, and , we
prove that with high probability a random subspace of \F_q^n of dimension
has the property that every Hamming ball of radius
has at most codewords.
This answers a basic open question concerning the list-decodability of linear
codes, showing that a list size of suffices to have rate within
of the "capacity" . Our result matches up to constant
factors the list-size achieved by general random codes, and gives an
exponential improvement over the best previously known list-size bound of
.
The main technical ingredient in our proof is a strong upper bound on the
probability that random vectors chosen from a Hamming ball centered at
the origin have too many (more than ) vectors from their linear
span also belong to the ball.Comment: 15 page
๋ 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
Secondary constructions of vectorial -ary weakly regular bent functions
In \cite{Bapic, Tang, Zheng} a new method for the secondary construction of
vectorial/Boolean bent functions via the so-called property was
introduced. In 2018, Qi et al. generalized the methods in \cite{Tang} for the
construction of -ary weakly regular bent functions. The objective of this
paper is to further generalize these constructions, following the ideas in
\cite{Bapic, Zheng}, for secondary constructions of vectorial -ary weakly
regular bent and plateaued functions. We also present some infinite families of
such functions via the -ary Maiorana-McFarland class. Additionally, we give
another characterization of the property for the -ary case via
second-order derivatives, as it was done for the Boolean case in \cite{Zheng}
Replacing the Soft FEC Limit Paradigm in the Design of Optical Communication Systems
The FEC limit paradigm is the prevalent practice for designing optical
communication systems to attain a certain bit-error rate (BER) without forward
error correction (FEC). This practice assumes that there is an FEC code that
will reduce the BER after decoding to the desired level. In this paper, we
challenge this practice and show that the concept of a channel-independent FEC
limit is invalid for soft-decision bit-wise decoding. It is shown that for low
code rates and high order modulation formats, the use of the soft FEC limit
paradigm can underestimate the spectral efficiencies by up to 20%. A better
predictor for the BER after decoding is the generalized mutual information,
which is shown to give consistent post-FEC BER predictions across different
channel conditions and modulation formats. Extensive optical full-field
simulations and experiments are carried out in both the linear and nonlinear
transmission regimes to confirm the theoretical analysis
Applications of finite geometries to designs and codes
This dissertation concerns the intersection of three areas of discrete mathematics: finite geometries, design theory, and coding theory. The central theme is the power of finite geometry designs, which are constructed from the points and t-dimensional subspaces of a projective or affine geometry. We use these designs to construct and analyze combinatorial objects which inherit their best properties from these geometric structures.
A central question in the study of finite geometry designs is Hamadaโs conjecture, which proposes that finite geometry designs are the unique designs with minimum p-rank among all designs with the same parameters. In this dissertation, we will examine several questions related to Hamadaโs conjecture, including the existence of counterexamples. We will also study the applicability of certain decoding methods to known counterexamples.
We begin by constructing an infinite family of counterexamples to Hamadaโs conjecture. These designs are the first infinite class of counterexamples for the affine case of Hamadaโs conjecture. We further demonstrate how these designs, along with the projective polarity designs of Jungnickel and Tonchev, admit majority-logic decoding schemes. The codes obtained from these polarity designs attain error-correcting performance which is, in certain cases, equal to that of the finite geometry designs from which they are derived. This further demonstrates the highly geometric structure maintained by these designs.
Finite geometries also help us construct several types of quantum error-correcting codes. We use relatives of finite geometry designs to construct infinite families of q-ary quantum stabilizer codes. We also construct entanglement-assisted quantum error-correcting codes (EAQECCs) which admit a particularly efficient and effective error-correcting scheme, while also providing the first general method for constructing these quantum codes with known parameters and desirable properties. Finite geometry designs are used to give exceptional examples of these codes
Two-tuple balance of non-binary sequences with ideal two-level autocorrelation
AbstractLet p be a prime, q=pm and Fq be the finite field with q elements. In this paper, we will consider q-ary sequences of period qn-1 for q>2 and study their various balance properties: symbol-balance, difference-balance, and two-tuple-balance properties. The array structure of the sequences is introduced, and various implications between these balance properties and the array structure are proved. Specifically, we prove that if a q-ary sequence of period qn-1 is difference-balanced and has the โcyclicโ array structure then it is two-tuple-balanced. We conjecture that a difference-balanced q-ary sequence of period qn-1 must have the cyclic array structure. The conjecture is confirmed with respect to all of the known q-ary sequences which are difference-balanced, in particular, which have the ideal two-level autocorrelation function when q=p
Quantum error control codes
It is conjectured that quantum computers are able to solve certain problems more
quickly than any deterministic or probabilistic computer. For instance, Shor's algorithm
is able to factor large integers in polynomial time on a quantum computer.
A quantum computer exploits the rules of quantum mechanics to speed up computations.
However, it is a formidable task to build a quantum computer, since the
quantum mechanical systems storing the information unavoidably interact with their
environment. Therefore, one has to mitigate the resulting noise and decoherence
effects to avoid computational errors.
In this dissertation, I study various aspects of quantum error control codes - the key component of fault-tolerant quantum information processing. I present the
fundamental theory and necessary background of quantum codes and construct many
families of quantum block and convolutional codes over finite fields, in addition to
families of subsystem codes. This dissertation is organized into three parts:
Quantum Block Codes. After introducing the theory of quantum block codes, I
establish conditions when BCH codes are self-orthogonal (or dual-containing)
with respect to Euclidean and Hermitian inner products. In particular, I derive
two families of nonbinary quantum BCH codes using the stabilizer formalism. I study duadic codes and establish the existence of families of degenerate quantum
codes, as well as families of quantum codes derived from projective geometries.
Subsystem Codes. Subsystem codes form a new class of quantum codes in which
the underlying classical codes do not need to be self-orthogonal. I give an
introduction to subsystem codes and present several methods for subsystem
code constructions. I derive families of subsystem codes from classical BCH and
RS codes and establish a family of optimal MDS subsystem codes. I establish
propagation rules of subsystem codes and construct tables of upper and lower
bounds on subsystem code parameters.
Quantum Convolutional Codes. Quantum convolutional codes are particularly
well-suited for communication applications. I develop the theory of quantum
convolutional codes and give families of quantum convolutional codes based
on RS codes. Furthermore, I establish a bound on the code parameters of
quantum convolutional codes - the generalized Singleton bound. I develop a
general framework for deriving convolutional codes from block codes and use it
to derive families of non-catastrophic quantum convolutional codes from BCH
codes.
The dissertation concludes with a discussion of some open problems
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