Spatial modulation (SM) is a new paradigm of multiple-input multiple-output (MIMO) systems, in which only one antenna at the transmitter is activated during each symbol period. Recently, it is observed that SM training sequences derived from corss Z-complementary pairs (CZCPs) lead to optimal channel estimation performance over frequency-selective channels. CZCPs are special form of sequence pairs which have zero aperiodic autocorrelation zones and cross-correlation zone at certain time-shifts. Recent paper by Liu \textit{et al.} discussed only perfect CZCPs. In this paper, we focus on non-perfect CZCPs. We introduce the term cross Z-complementary ratio and re-categorise the CZCPs, both perfect and non-perfect, based on that. We propose a systematic construction of CZCPs based on generalised Boolean functions (GBFs). We further extend the lengths of the CZCPs by using the insertion method. The proposed CZCPs are all of new lengths of the form $2^\alpha10^\beta26^γ+2~(α\geq1)$, $10^β+2$, $26^γ+2$ and $10^β26^γ+2$. Finally we propose a construction of optimal binary CZCPs having parameters $(12,5)$ and $(24,11)$ from binary Barker sequences. These CZCPs are also extended to $(12N,5N)$- CZCPs and $(24N,11N)$- CZCPs, where $N$ is the length of a binary Golay complementary pair (GCP). During the proof, we also found a new structural property of binary CZCPs and concluded all binary GCPs are CZCPs too. Finally, we give some numerical simulations to confirm that depending on the number of multi-paths, the proposed CZCPs can be used to design SM training matrix which attains the minimum mean square error.
The sequence pairs of length $2^{m}$ projected from complementary array pairs of Type-II of size $\mathbf{2}^{(m)}$ and mixed Type-II/III and of size $\mathbf{2}^{(m-1)}\times2$ are complementary sequence pairs Type-II and Type-III respectively. An exhaustive search for binary Type-II and Type-III complementary sequence pairs of small lengths $2^{m}$ ($m=1,2,3,4$) shows that they are all projected from the aforementioned complementary array pairs, whose algebraic normal forms satisfy specified expressions. It's natural to ask whether the conclusion holds for all $m$. In this paper, we proved t…
We generalize the three-stage process for constructing and enumerating Golay array and sequence pairs given in 2008 by Frank Fiedler et al. [A multi-dimensional approach to the construction and enumeration of Golay complementary sequences, Journal of Combinatorial Theory, Series A 115 (2008) 753-776] to $4^{q}$-QAM constellation based on para-unitary matrix method, which partly solves their open questions. Our work not only includes the main part of known results of Golay complementary sequences over $4^{q}$-QAM based on Boolean functions and standard Golay sequence pairs over QPSK, but also g…
This letter presents a direct construction of cross Z-complementary sequence sets (CZCSSs), whose aperiodic correlation sums exhibit zero correlation zones at both the front-end and tail-end shifts. CZCSS can be regarded as an extension of the symmetrical Z-complementary code set (SZCCS). The available construction of SZCCS has a limitation on the set size, with a maximum set size of 8. The proposed generalized Boolean function based construction can generate CZCSS of length in the form of non-power-of-two with variable set size $2^{n+1}$, where each code has $2^{n+1}$ constituent sequences. T…
One objective of this paper is to propose a novel class of sequence pairs, called "Quasi-orthogonal Z-complementary pairs (QOZCPs)", each depicting Z-complementary property for their aperiodic auto-correlation sums and also have a zero correlation zone when their aperiodic cross-correlation is considered. Construction of QOZCPs based on Successively Distributed Algorithms under Majorization Minimization (SDAMM) is presented. Another objective of this paper is to apply the proposed QOZCPs in fully polarimetric radar systems and analyse the corresponding ambiguity functions. It turns out that QO…
The contributions of this paper are twofold: Firstly, we introduce a novel class of sequence pairs, called "cross Z-complementary pairs (CZCPs)", each displaying zero-correlation zone (ZCZ) properties for both their aperiodic autocorrelation sums and crosscorrelation sums. Systematic constructions of perfect CZCPs based on selected Golay complementary pairs (GCPs) are presented. Secondly, we point out that CZCPs can be utilized as a key component in designing training sequences for broadband spatial modulation (SM) systems. We show that our proposed SM training sequences derived from CZCPs lea…
A pair $(C, D)$ of group codes over group algebra $R[G]$ is called a linear complementary pair (LCP) if $C \oplus D =R[G]$, where $R$ is a finite principal ideal ring, and $G$ is a finite group. We provide a necessary and sufficient condition for a pair $(C, D)$ of group codes over group algebra $R[G]$ to be LCP. Then we prove that if $C$ and $D$ are both group codes over $R[G]$, then $C$ and $D^{\perp}$ are permutation equivalent.
This letter proposes a direct construction for cross Z-complementary sets (CZCSs) with flexible lengths and a large zero correlation zone (ZCZ). CZCS is an extension of the cross Z-complementary pair (CZCP). The maximum possible ZCZ width of a CZCP is half of its sequence length. In this letter, for the first time, a generalized Boolean function based construction of CZCSs with a large number of constituent sequences and a ZCZ ratio of $2/3$ is presented. For integers $m$ and $δ$, the proposed construction produces CZCS with length expressed as $2^{m-1}+2^δ$ ($0 \leq δ<m-1,m\geq 4$), where bot…
Linear complementary dual (LCD) codes and linear complementary pair (LCP) of codes over finite fields have been intensively studied recently due to their applications in cryptography, in the context of side-channel and fault injection attacks. The security parameter for an LCP of codes $(C,D)$ is defined as the minimum of the minimum distances $d(C)$ and $d(D^\bot)$. It has been recently shown that if $C$ and $D$ are both 2-sided group codes over a finite field, then $C$ and $D^\bot$ are permutation equivalent. Hence the security parameter for an LCP of 2-sided group codes $(C,D)$ is simply $d…
Cross Z-complementary pairs (CZCPs) are a special kind of Z-complementary pairs (ZCPs) having zero autocorrelation sums around the in-phase position and end-shift position, also having zero cross-correlation sums around the end-shift position. It can be utilized as a key component in designing optimal training sequences for broadband spatial modulation (SM) systems over frequency selective channels. In this paper, we focus on designing new CZCPs with large cross Z-complementary ratio $(\mathrm{CZC}_{\mathrm{ratio}})$ by exploring two promising approaches. The first one of CZCPs via properly ca…
We present a general algorithm for generating arbitrary standard complementary pairs of sequences (including binary, polyphase, M-PSK and QAM) of length 2^N using Boolean functions. The algorithm follows our earlier paraunitary algorithm, but does not require matrix multiplications. The algorithm can be easily and efficiently implemented in hardware. As a special case, it reduces to the non-recursive (direct) algorithm for generating binary sequences given by Golay, to the algorithm for generating M-PSK sequences given by Davis and Jedwab (and later improved by Paterson) and to all published a…
Zero correlation zone (ZCZ) sequences and Golay sequences are two kinds of sequences with different preferable correlation properties. It was shown by Gong \textit{et al.} and Chen \textit{et al.} that some Golay sequences also possess a large ZCZ and are good candidates for pilots in OFDM systems. Known Golay sequences with ZCZ reported in the literature have a limitation in the length which is the form of a power of 2. One objective of this paper is to propose a construction of Golay complementary pairs (GCPs) with new lengths whose periodic autocorrelation of each of the Golay sequences and…
Motivated by the problem of reducing the peak to average power ratio (PAPR) of transmitted signals, we consider a design of complementary set matrices whose column sequences satisfy a correlation constraint. The design algorithm recursively builds a collection of $2^{t+1}$ mutually orthogonal (MO) complementary set matrices starting from a companion pair of sequences. We relate correlation properties of column sequences to that of the companion pair and illustrate how to select an appropriate companion pair to ensure that a given column correlation constraint is satisfied. For $t=0$, companion…
This paper is devoted to sequences and focuses on designing new two-dimensional (2-D) Z-complementary array pairs (ZCAPs) by exploring two promising approaches. A ZCAP is a pair of 2-D arrays, whose 2-D autocorrelation sum gives zero value at all time shifts in a zone around the $(0,0)$ time shift, except the $(0,0)$ time shift. The first approach investigated in this paper uses a one-dimensional (1-D) Z-complementary pair (ZCP), which is an extension of the 1-D Golay complementary pair (GCP) where the autocorrelations of constituent sequences are complementary within a zero correlation zone (…
A pair of sequences is called a Z-complementary pair (ZCP) if it has zero aperiodic autocorrelation sums at each of the non-zero time-shifts within {a} certain region, called the zero correlation zone (ZCZ). ZCPs are categorised into two types{:} Type-I ZCPs and Type-II ZCPs. Type-I ZCPs have {the} ZCZ around the in-phase position and Type-II ZCPs have the ZCZ around the end-shift position. {Till now only a few} constructions of Type-II ZCPs are reported {in the literature}, and all {have} lengths of the form $2^m\pm1$ or $N+1$ where $N=2^a 10^b 26^c$ and $a,~b,~c$ are non-negative integers. I…
We give a short and elementary proof of the fact that for a linear complementary pair $(C,D)$, where $C$ and $D$ are $2$-sided ideals in a group algebra, $D$ is uniquely determined by $C$ and the dual code $D^\perp$ is permutation equivalent to $C$. This includes earlier results of Carlet et al. and Güneri et al. on nD cyclic codes which have been proved by subtle and lengthy calculations in the space of polynomials.
It is shown that replacing the sinusoidal chip in Golay complementary code pairs by special classes of waveforms that satisfy two conditions, symmetry/anti-symmetry and quazi-orthogonality in the convolution sense, renders the complementary codes immune to frequency selective fading and also allows for concatenating them in time using one frequency band/channel. This results in a zero-sidelobe region around the mainlobe and an adjacent region of small cross-correlation sidelobes. The symmetry/anti-symmetry property results in the zero-sidelobe region on either side of the mainlobe, while quasi…
A pair of sequences is called a Z-complementary pair (ZCP) if it has zero aperiodic autocorrelation sums (AACSs) for time-shifts within a certain region, called zero correlation zone (ZCZ). Optimal odd-length binary ZCPs (OB-ZCPs) display closest correlation properties to Golay complementary pairs (GCPs) in that each OB-ZCP achieves maximum ZCZ of width (N+1)/2 (where N is the sequence length) and every out-of-zone AACSs reaches the minimum magnitude value, i.e. 2. Till date, systematic constructions of optimal OB-ZCPs exist only for lengths $2^α \pm 1$, where $α$ is a positive integer. In thi…
We extend the paraunitary (PU) theory for complementary pairs to comple- mentary sets and complete complementary codes (CCC) by proposing a new PU construction. A special, but very important case of complementary sets (and CC- C), based on standard delays, is analyzed in details and a new 'Radix-M generator' (RM-G) is presented. The RM-G can be viewed as a generalization of the Boolean generator for complementary pairs. An efficient correlator for standard complemen- tary sets and CCC is also presented. Finally, examples of polyphase, QAM and hexagonal PU sets of three sequences are given.
The construction of complementary sets (CSs) of sequences with different set size and sequence length become important due to its practical application for OFDM systems. Most of the constructions of CSs, based on generalized Boolean functions (GBFs), are of length $2^α$ ($α$ is a natural number). Recently some works have been reported on construction of CSs having lengths non-power of two, i.e., in the form of $2^{m-1}+2^v$ ($m$ is natural number, $0\leq v <m $), $N+1$ and $N+2$, where $N$ is a length for which $q$-ary complementary pairs exist. In this paper, we propose a construction of CSs …
A new method to construct $q$-ary complementary sequence sets (CSSs) and complete complementary codes (CCCs) of size $N$ is proposed by using desired para-unitary (PU) matrices. The concept of seed PU matrices is introduced and a systematic approach on how to compute the explicit forms of the functions in constructed CSSs and CCCs from the seed PU matrices is given. A general form of these functions only depends on a basis of the functions from $\Z_N$ to $\Z_q$ and representatives in the equivalent class of Butson-type Hadamard (BH) matrices. Especially, the realization of Golay pairs from the…
We present a construction for complementary pairs of arrays that exploits a set of mutually-unbiased bases, and enumerate these arrays as well as the corresponding set of complementary sequences obtained from the arrays by projection. We also sketch an algorithm to uniquely generate these sequences. The pairwise squared inner-product of members of the sequence set is shown to be $\frac{1}{2}$. Moreover, a subset of the set can be viewed as a codebook that asymptotically achieves $\sqrt{\frac{3}{2}}$ times the Welch bound.
We present a general method for constructing radar transmit pulse trains and receive filters for which the radar point-spread function in delay and Doppler (radar cross-ambiguity function) is essentially free of range sidelobes inside a Doppler interval around the zero-Doppler axis. The transmit and receive pulse trains are constructed by coordinating the transmission of a pair of Golay complementary waveforms across time according to zeros and ones in a binary sequence $P$. In the receive pulse train filter, each waveform is weighted according to an element from another sequence $Q$. We show …
The previous constructions of quadrature amplitude modulation (QAM) Golay complementary sequences (GCSs) were generalized as $4^q $-QAM GCSs of length $2^{m}$ by Li \textsl{et al.} (the generalized cases I-III for $q\ge 2$) in 2010 and Liu \textsl{et al.} (the generalized cases IV-V for $q\ge 3$) in 2013 respectively. Those sequences are presented as the combination of the quaternary standard GCSs and compatible offsets. By providing new compatible offsets based on the factorization of the integer $q$, we proposed two new constructions of $4^q $-QAM GCSs, which have the generalized cases I-V a…
To find the non-standard binary Golay complementary sequences (GCSs) of length $2^{m}$ or theoretically prove the nonexistence of them are still open. Since it has been shown that all the standard $q$-ary (where $q$ is even) GCSs of length $2^m$ can be obtained by standard $q$-ary Golay complementary array pair (GAP) of dimension $m$ and size $2\times 2 \times \cdots \times 2$ (abbreviated to size $\mathbf{2}^{(m)}$), it's natural to ask whether all the $q$-ary GAP of size $\mathbf{2}^{(m)}$ are standard. We give a positive answer to this question.
In recent years, complementary sequence sets have found many important applications in multi-carrier code-division multiple-access (MC-CDMA) systems for their good correlation properties. In this paper, we propose a construction, which can generate multiple sets of complete complementary codes (CCCs) over $\mathbb{Z}_N$, where $N~(N\geq 3)$ is a positive integer of the form $N=p_0^{e^0}p_1^{e^1}\dots p_{n-1}^{e^{n-1}}$, $p_0<p_1<\cdots<p_{n-1}$ are prime factors of $N$ and $e_0,e_1,\cdots,e_{n-1}$ are non-negative integers. Interestingly, the maximum inter-set aperiodic cross-correlation magni…
A major drawback of orthogonal frequency division multiplexing (OFDM) systems is their high peak-to-mean envelope power ratio (PMEPR). The PMEPR problem can be solved by adopting large codebooks consisting of complementary sequences with low PMEPR. In this paper, we present a new construction of polyphase complementary sets (CSs) using generalized Boolean functions (GBFs), which generalizes Schmidt's construction in 2007, Paterson's construction in 2000 and Golay complementary pairs (GCPs) given by Davis and Jedwab in 1999. Compared with Schmidt's approach, our proposed CSs lead to lower PMEPR…
Quasi-complementary sequence sets (QCSSs) can be seen as a generalized version of complete complementary codes (CCCs), which enables multicarrier communication systems to support more users. The contribution of this work is two-fold. First, we propose a systematic construction of Florentine rectangles. Secondly, we propose several sets of CCCs and QCSS, using Florentine rectangles. The CCCs and QCSS are constructed over $\mathbb{Z}_N$, where $N\geq2$ is any integer. The cross-correlation magnitude of any two of the constructed CCCs is upper bounded by $N$. By combining the proposed CCCs, we pr…
A new method to construct $q$-ary complementary sequence (or array) sets (CSSs) and complete complementary codes (CCCs) of size $N$ is introduced in this paper. An algorithm on how to compute the explicit form of the functions in constructed CSS and CCC is also given. A general form of these functions only depends on a basis of functions from $\Z_N$ to $\Z_q$ and representatives in the equivalent class of Butson-type Hadamard matrices. Surprisingly, all the functions fill up a larger number of cosets of a linear code, compared with the existing constructions. From our general construction, its…
The one-dimensional (1-D) Golay complementary set (GCS) has many well-known properties and has been widely employed in engineering. The concept of 1-D GCS can be extended to the two-dimensional (2-D) Golay complementary array set (GCAS) where the 2-D aperiodic autocorrelation of constituent arrays sum to zero except for the 2-D zero shift. The 2-D GCAS includes the 2-D Golay complementary array pair (GCAP) as a special case when the set size is 2. In this paper, 2-D generalized Boolean functions are introduced and novel constructions of 2-D GCAPs, 2-D GCASs, and 2-D Golay complementary array m…
We present a general method for constructing radar transmit pulse trains and receive filters for which the radar point-spread function in delay and Doppler, given by the cross-ambiguity function of the transmit pulse train and the pulse train used in the receive filter, is essentially free of range sidelobes inside a Doppler interval around the zero-Doppler axis. The transmit pulse train is constructed by coordinating the transmission of a pair of Golay complementary waveforms across time according to zeros and ones in a binary sequence P. The pulse train used to filter the received signal is …
On $\ell$-rank additive intersection pairs (RAIP) of codes
Indian Institute of Technology Guwahati（印度理工学院古瓦哈提分校）；University of Paris VIII（巴黎第八大学）；University Sorbonne Paris Nord, CNRS, UMR 7539（索邦巴黎北大学，法国国家科学研究中心）；Telecom Paris, Polytechnic institute of Paris（巴黎综合理工学院电信学院）
Certain applications require the use of signals that combine both the capability to operate with low signal-to-noise ratios and the ability to support multiple users without interference. In the case where many users have very different signal-to-noise ratios, it is necessary to consider coding schemes that can be used in a multi-user environment but with different noise immunity levels. Traditional detection systems based on the correlation function and coding sequences have significant limitations in satisfying both objectives, since the cross-correlation between coded signals corresponding …
Borwein and Mossinghoff investigated the Rudin-Shapiro-like sequences, which are infinite families of binary sequences, usually represented as polynomials. Each family of Rudin-Shapiro-like sequences is obtained from a starting sequence (which we call the seed) by a recursive construction that doubles the length of the sequence at each step, and many sequences produced in this manner have exceptionally low aperiodic autocorrelation. Borwein and Mossinghoff showed that the asymptotic autocorrelation merit factor for any such family is at most $3$, and found the seeds of length $40$ or less that…
Pursley and Sarwate established a lower bound on a combined measure of autocorrelation and crosscorrelation for a pair $(f,g)$ of binary sequences (i.e., sequences with terms in $\{-1,1\}$). If $f$ is a nonzero sequence, then its autocorrelation demerit factor, $\text{ADF}(f)$, is the sum of the squared magnitudes of the aperiodic autocorrelation values over all nonzero shifts for the sequence obtained by normalizing $f$ to have unit Euclidean norm. If $(f,g)$ is a pair of nonzero sequences, then their crosscorrelation demerit factor, $\text{CDF}(f,g)$, is the sum of the squared magnitudes of …
