QPSK Block-Modulation Codes for Unequal Error Protection

Unequal error protection (UEP) codes find applications in broadcast channels, as well as in other digital communication systems, where messages have different degrees of importance. Binary linear UEP (LUEP) codes combined with a Gray mapped QPSK signal set are used to obtain new efficient QPSK block-modulation codes for unequal error protection. Several examples of QPSK modulation codes that have the same minimum squared Euclidean distance as the best QPSK modulation codes, of the same rate and length, are given. In the new constructions of QPSK block-modulation codes, even-length binary LUEP codes are used. Good even-length binary LUEP codes are obtained when shorter binary linear codes are combined using either the well-known |u¯|u¯+v¯|-construction or the so-called construction X. Both constructions have the advantage of resulting in optimal or near-optimal binary LUEP codes of short to moderate lengths, using very simple linear codes, and may be used as constituent codes in the new constructions. LUEP codes lend themselves quite naturally to multistage decoding up to their minimum distance, using the decoding of component subcodes. A new suboptimal two-stage soft-decision decoding of LUEP codes is presented and its application to QPSK block-modulation codes for UEP illustrated

Prime-phase sequences with periodic correlation properties better than binary sequences

For the case where p is an odd prime, n&#62;or=2 is an integer, and omega is a complex primitive pth root of unity, a construction is presented for a family of p<SUP>n</SUP> p-phase sequences (symbols of the form omega<SUP>i</SUP>), where each sequence has length p<SUP>n</SUP>-1, and where the maximum nontrivial correlation value C<SUB>max</SUB> does not exceed 1+ square root p<SUP>n</SUP>. A complete distribution of correlation values is provided. As a special case of this construction, a previous construction due to Sidelnikov (1971) is obtained. The family of sequences is asymptotically optimum with respect to its correlation properties, and, in comparison with many previous nonbinary designs, the present design has the additional advantage of not requiring an alphabet of size larger than three. The new sequences are suitable for achieving code-division multiple access and are easily implemented using shift registers. They wee discovered through an application of Deligne's bound (1974) on exponential sums of the Weil type in, several variables. The sequences are also shown to have strong identification with certain bent functions

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

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