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 our general form exactly coincides with the standard Golay pairs. The realization of ternary complementary sequences of size $3$ is first reported here. For the realization of the quaternary complementary sequences of size 4, almost all the sequences derived here are never reported before. Generalized seed PU matrices and the recursive constructions of the desired PU matrices are also studied, and a large number of new constructions of CSSs and CCCs are given accordingly. From the perspective of this paper, all the known results of CSSs and CCCs with explicit GBF form in the literature (except non-standard Golay pairs) are constructed from the Walsh matrices of order 2. This suggests that the proposed method with the BH matrices of higher orders will yield a large number of new CSSs and CCCs with the exponentially increasing number of the sequences of low peak-to-mean envelope power ratio.
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…
Previously, we have presented a framework to use the para-unitary (PU) matrix-based approach for constructing new complementary sequence set (CSS), complete complementary code (CCC), complementary sequence array (CSA), and complete complementary array (CCA). In this paper, we introduce a new class of delay matrices for the PU construction. In this way, generalized Boolean functions (GBF) derived from PU matrix can be represented by an array of size $2\times 2 \times \cdots \times 2$. In addition, we introduce a new method to construct PU matrices using block matrices. With these two new ingred…
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…
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.
This paper presents a coding approach for achieving omnidirectional transmission of certain common signals in massive multi-input multi-output (MIMO) networks such that the received power at any direction in a cell remains constant for any given distance. Specifically, two-dimensional (2D) Golay complementary array set (GCAS) can be used to design optimal massive MIMO precoding matrix so as to achieve omnidirectional transmission due to its complementary autocorrelation property. In this paper, novel constructions of new 2D GCASs with arbitrary array lengths are proposed. Our key idea is to ca…
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…
Due to the zero nontrivial aperiodic correlation of complete complementary code (CCCs), it is used in asynchronous multi carrier code division multiple access (MC-CDMA) communication to provide zero interference performance. However, there is no CCC for some lengths. In this case, if the system pays more attention to the correlation, it can use mutually orthogonal complementary sequence sets (MOCSSs), and if the system pays more attention to the set size, it can use Z-complementary code sets (ZCCSs). In this paper, we propose a direct construction of $q$-ary $(q^{v+1},q,q^m,q^{m-v})$-ZCCS. In …
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…
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.
We generalize the construction of Doppler-tolerant Golay complementary waveforms by Pezeshki-Calderbank-Moran-Howard to complementary code sets having more than two codes. This is accomplished by exploiting number-theoretic results involving the sum-of-digits function, equal sums of like powers, and a generalization to more than two symbols of the classical two-symbol Prouhet-Thue-Morse sequence.
In this paper, a recent method to construct complementary sequence sets and complete complementary codes by Hadamard matrices is deeply studied. By taking the algebraic structure of Hadamard matrices into consideration, our main result determine the so-called $δ$-linear terms and $δ$-quadratic terms. As a first consequence, a powerful theory linking Golay complementary sets of $p$-ary ($p$ prime) sequences and the generalized Reed-Muller codes by Kasami et al. is developed. These codes enjoy good error-correcting capability, tightly controlled PMEPR, and significantly extend the range of codin…
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…
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…
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 (…
In this work, we consider complementary lattice arrays in order to enable a broader range of designs for coded aperture imaging systems. We provide a general framework and methods that generate richer and more flexible designs than existing ones. Besides this, we review and interpret the state-of-the-art uniformly redundant arrays (URA) designs, broaden the related concepts, and further propose some new design methods.
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…
This paper introduces a novel class of code sets, called "symmetrical Z-complementary code sets (SZCCSs)" , whose aperiodic auto- and cross- correlation sums exhibit zero-correlation zones (ZCZs) at both the front-end and tail-end of the entire correlation window. Three constructions of (optimal) SZCCSs based on general Boolean functions are presented. As a second major contribution, we apply SZCCSs to design optimal training sequences for broadband generalized spatial modulation (GSM) systems over frequency-selective channels. Key words: Complementary code set, channel estimation, training se…
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.
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…
In this paper, we propose a direct construction of optimal two-dimensional Z-complementary array code sets (2D-ZCACS) using multivariable functions (MVFs). In contrast to earlier works, the proposed construction allows for a flexible array size and a large set size. Additionally, the proposed design can be transformed into a one-dimensional Z-complementary code set (1D-ZCCS). Many of the 1D-ZCCS described in the literature appeared to be special cases of this proposed construction. At last, we compare our work with the current state of the art and then draw our conclusions.
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…
Mutually orthogonal complementary sets (MOCSs) have received significant research attention in recent years due to their wide applications in communications and radar. Existing MOCSs which are constructed based on generalized Boolean functions (GBFs) mostly have lengths of power-of-two. How to construct MOCSs with non-power-of-two lengths whilst having large set sizes is a largely open problem. With the aid of GBFs, in this paper, we present new constructions of such MOCSs and show that the maximal achievable set size is $1/2$ of the flock size of an MOCS.
Constructions of Polyphase Golay Complementary Arrays
Key Laboratory for Information Science of Electromagnetic Waves (MoE), Department of Communication Science and Engineering, School of Information Science and Technology, Fudan University（复旦大学信息科学与技术学院通信科学与工程系电磁波信息科学教育部重点实验室）
Z-complementary code sets (ZCCSs) are used in multicarrier code-division multiple access (MC-CDMA) systems, for interference-free communication over multiuser and quasi-asynchronous environments. In this letter, we propose three new constructions of optimal binary $\left(R2^{k+1},2^{k+1}, Rγ,γ\right)$-ZCCS, $\left(R2^{k+1},2^{k+1}, R2^{m_{2}},2^{m_{2}}\right)$-ZCCS and $\left(2^{k+1},2^{k+1},3γ,2γ\right)$-ZCCS based on generalized Boolean functions (GBFs), where $γ=2^{m_{1}-1}+2^{m_{1}-3}, m_{1}\geq 5, k\geq 1,m_{2}\geq 1$ and $R$ is any even number. The proposed ZCCSs cover many unreported le…
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 …
Compared with the perfect complementary sequence sets, quasi-complementary sequence sets (QCSSs) can support more users to work in multicarrier CDMA communications. A near-optimal periodic QCSS is constructed in this paper by using an optimal quaternary sequence set and an almost difference set. With the change of the values of parameters in the almost difference set, the near-optimal QCSS can become asymptotically optimal and the number of users supported by the subcarrier channels in CDMA system has an exponential growth.
Z-complementary code set (ZCCS) are well known to be used in multicarrier code-division multiple access (MCCDMA) system to provide a interference free environment. Based on the existing literature, the direct construction of optimal ZCCSs are limited to its length. In this paper, we are interested in constructing optimal ZCCSs of all possible even lengths using Pseudo-Boolean functions. The maximum column sequence peakto-man envelop power ratio (PMEPR) of the proposed ZCCSs is upper-bounded by two, which may give an extra benefit in managing PMEPR in an ZCCS based MC-CDMA system, as well as th…
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…
Five Constructions of Asymptotically Optimal Aperiodic Doppler Resilient Complementary Sequence Sets with New Parameters
School of Mathematics and Statistics, Fujian Normal University（福建师范大学数学与统计学院）；Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), Fujian Normal University（福建师范大学分析数学与应用教育部重点实验室）；School of Mathematics and Statistics, Zhengzhou University（郑州大学数学与统计学院）
The use of error-correcting codes for tight control of the peak-to-mean envelope power ratio (PMEPR) in orthogonal frequency-division multiplexing (OFDM) transmission is considered in this correspondence. By generalizing a result by Paterson, it is shown that each q-phase (q is even) sequence of length 2^m lies in a complementary set of size 2^{k+1}, where k is a nonnegative integer that can be easily determined from the generalized Boolean function associated with the sequence. For small k this result provides a reasonably tight bound for the PMEPR of q-phase sequences of length 2^m. A new 2^…
Complementary set sequences (CSSs) are useful for dealing with the high peak-to-average power ratio (PAPR) problem in orthogonal frequency division multiplexing (OFDM) systems. In practical OFDM transmission, however, certain sub-carriers maybe reserved and/or prohibited to transmit signals, leading to the so-called \emph{spectral null constraint} (SNC) design problem. For example, the DC sub-carrier is reserved to avoid the offsets in D/A and A/D converter in the LTE systems. While most of the current research focus on the design of low PAPR CSSs to improve the code-rate, few works address th…
The concept of paraunitary (PU) matrices arose in the early 1990s in the study of multi-rate filter banks. So far, these matrices have found wide applications in cryptography, digital signal processing, and wireless communications. Existing PU matrices are subject to certain constraints on their existence and hence their availability is not guaranteed in practice. Motivated by this, for the first time, we introduce a novel concept, called $Z$-paraunitary (ZPU) matrix, whose orthogonality is defined over a matrix of polynomials with identical degree not necessarily taking the maximum value. We …
The correlation properties of sequences form a focal point in the design of multiple access systems of communications. Such a system must be able to serve a number of simultaneous users while keeping interference low. A popular choice for the set of sequences to deploy is the quasi-complementary sequence set (QCSS). Its large set size enables the system to accommodate a lot of users. The set has low nontrivial correlation magnitudes within a zone around the origin. This keeps undue interference among users under control. A QCSS performs better than the perfect complementary sequence set (PCSS)…
This paper aims to construct optimal Z-complementary code set (ZCCS) with non-power-of-two (NPT) lengths to enable interference-free multicarrier code-division multiple access (MC-CDMA) systems. The existing ZCCSs with NPT lengths, which are constructed from generalized Boolean functions (GBFs), are sub-optimal only with respect to the set size upper bound. For the first time in the literature, we advocate the use of pseudo-Boolean functions (PBFs) (each of which transforms a number of binary variables to a real number as a natural generalization of GBF) for direct constructions of optimal ZCC…
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…
Let PCS_p^N denote a set of p binary sequences of length N such that the sum of their periodic auto-correlation functions is a delta-function. In the 1990, Boemer and Antweiler addressed the problem of constructing such sequences. They presented a table covering the range p <= 12, N <= 50 and showing in which cases it was known at that time whether such sequences exist, do not exist, or the question of existence is undecided. The number of undecided cases was rather large. Subsequently the number of undecided cases was reduced to 26 by the author. In the present note, several cyclic difference…
This paper presents an efficient construction of two-dimensional (2D) complete complementary codes (CCCs) for their modern application as omnidirectional precoding matrices in massive MIMO systems to attain enhanced cell coverage. Unlike the traditional 1D CCCs, little progress has been made on efficient and systematic constructions of the 2D counterpart. In contrast to the existing recursive constructions with the aid of various sequence operations, certain 1D seed sequences or 2D arrays, we propose to use 2D generalized Boolean functions for direct synthesis of 2D CCCs. Simulation results sh…
A functional PIR array code is a coding scheme which encodes some $s$ information bits into a $t\times m$ array such that every linear combination of the $s$ information bits has $k$ mutually disjoint recovering sets. Every recovering set consists of some of the array's columns while it is allowed to read at most $\ell$ encoded bits from every column in order to receive the requested linear combination of the information bits. Functional batch array codes impose a stronger property where every multiset request of $k$ linear combinations has $k$ mutually disjoint recovering sets. Locality funct…
A quasi-complementary sequence set (QCSS) refers to a set of two-dimensional matrices with low non-trivial aperiodic auto- and cross- correlation sums. For multicarrier code-division multiple-access applications, the availability of large QCSSs with low correlation sums is desirable. The generalized Levenshtein bound (GLB) is a lower bound on the maximum aperiodic correlation sum of QCSSs. The bounding expression of GLB is a fractional quadratic function of a weight vector $\mathbf{w}$ and is expressed in terms of three additional parameters associated with QCSS: the set size $K$, the number o…
Zero correlation zone (ZCZ) sequences and Golay complementary sequences are two kinds of sequences with different preferable correlation properties. Golay-ZCZ sequences are special kinds of complementary sequences which also possess a large ZCZ and are good candidates for pilots in OFDM systems. Known Golay-ZCZ sequences reported in the literature have a limitation in the length which is the form of a power of 2. In this paper, we propose two constructions of Golay-ZCZ sequence sets with new parameters which generalize the constructions of Gong et al. (IEEE Transaction on Communications 61(9),…
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…
Linear nested codes, where two or more sub-codes are nested in a global code, have been proposed as candidates for reliable multi-terminal communication. In this paper, we consider nested array-based spatially coupled low-density parity-check (SC-LDPC) codes and propose a line-counting based optimization scheme for minimizing the number of dominant absorbing sets in order to improve its performance in the high signal-to-noise ratio regime. Since the parity-check matrices of different nested sub-codes partially overlap, the optimization of one nested sub-code imposes constraints on the optimiza…
If the number of runs in a (mixed-level) orthogonal array of strength 2 is specified, what numbers of levels and factors are possible? The collection of possible sets of parameters for orthogonal arrays with N runs has a natural lattice structure, induced by the ``expansive replacement'' construction method. In particular the dual atoms in this lattice are the most important parameter sets, since any other parameter set for an N-run orthogonal array can be constructed from them. To get a sense for the number of dual atoms, and to begin to understand the lattice as a function of N, we investiga…
Golay complementary pairs (GCPs) and complete complementary codes (CCCs) have found a wide range of practical applications in coding, signal processing and wireless communication due to their ideal correlation properties. In fact, binary CCCs have special advantages in spread spectrum communication due to their simple modulo-2 arithmetic operation, modulation and correlation simplicity, but they are limited in length. In this paper, we present a direct construction of GCPs, mutually orthogonal complementary sets (MOCSs) and binary CCCs of non-power of two lengths to widen their application in …
