1 Introduction
In 1951, Golay 1951Golay
first introduced a sequence pair whose aperiodic autocorrelation sums (AACSs) are zero at each outofphase time shift. Such a pair of sequences is called a Golay complementary sequence pair (GCP), and has been found many applications which include channel estimation
2001Spasojevic , radar2018Kumari , and peak power control in orthogonal frequency division multiplexing (OFDM) 1999GDJ ,2014Wang , etc. However, one drawback of GCPs is its limited availability. For example, binary GCPs are known to exist for lengths of the form where , and are nonnegative integers 20037Borwein . Due to its scarity, Fan et al. relaxed the autocorrelation constraint of GCPs, and introduced another type of sequences called Zcomplementary pair (ZCP) 20078Fan . ZCP consists of a pair of sequences whose aperiodic autocorrelation sums are zero within a zone around the zero shift position, called zero correlation zone (ZCZ). Compared with GCPs, ZCPs can have more flexible lengths. Actually, ZCPs are shown to available for all lengths 20111Li .In Liu2020 , Liu et al. investigated the training sequence design for the spatial modulation (SM) system and hence the cross Zcomplementary pair (CZCP) was proposed. In contrast to ZCPs, CZCPs deals with crosscorrelation sums in addition to calculating autocorrelation sums. They pointed out that CZCPs can be utilized as a key component in designing optimal training sequences for broadband spatial modulation (SM) systems over frequency selective channels. SM is a special class of MIMO techniques which trades multiplexing gain with complexity and performance. Unlike conventional MIMO, an SM system is equipped with multiple transmit antenna (TA) elements but only a single radiofrequency (RF) chain (see 2008Mesleh ,Yang2005 and Yang2016 for more details). CZCPs can be employed as training sequences in an SM system to mitigate the intersymbol interference (ISI) and the interchannel interference (ICI) resulting from multipath propagation Liu2020 . In Liu2020 , Liu et al. proposed binary using special types of binary GCPs, where is a Golay number. Besides, the authors also utilizes generalized Boolean functions (GBFs) to construct . It is easy to see that the and achieve the maximum ZCZ width, i.e., by Liu et al. discussed only perfect CZCPs. However, the length of perfect CZCPs is very limited. To check if the complementary sequences of other lengths also display the cross Zcomplementary property achieving the maximum ZACZ and ZCCZ width, Fan et al.Fan2020 proposed binary CZCPs of lengths , and having the ZACZ and ZCCZ width of , and , respectively. In Adhikary2020 , Adhikary et al. proposed binary and phase CZCPs of lengths , , and having the ZACZ and ZCCZ width of , and using insertion function, respectively. Besides, The authors proposed the construction of binary and using Barker sequences. These CZCPs are also extended to and , where is the length of a GCP. In Huang2020 , Huang et al. proposed binary using Boolean functions (BFs), where , . When , , the construction of with . In yang2021 , Yang et al. proposed binary CZCPs of lengths of the form , where evenlength binary ZCPs (EBZCPs) of length exists. They also utilised binary CZCPs to construct the quaternary CZCPs.
In this paper, we proposed two approaches to construct CZCPs. The first one of CZCPs via properly cascading sequences from a Golay complementary pair (GCP). The proposed construction leads to , and , where is the length of a binary GCP. Specially, we propose two optimal binary CZCPs with and . The second one of CZCPs based on BFs, and the construction of CZCPs have the largest CZC ratio, compared with known CZCPs but not perfect CZCPs in the literature.
This paper is organized as follows. In Section 2, we give some basic notation and definitions which will be needed in this paper. Section 3.1 and 3.2 are the core of the paper in which we present our two approaches for designing CZCPs. The first one (Section 3.1) uses the concatenation of Golay sequences of length while the second one (Section 3.2) employees Boolean functions. We also make a comparison of our results of CZCPs with known CZCPs in the literature. Section 4 concludes this paper.
2 Preliminaries
In this section, we give some basic notation and definitions which will be used throughout the paper:
2.1 Notations

denotes a binary sequence of length .

denotes the conjugation of .

denotes the reverse of .

denotes the negation of .

denotes the concatenation of and .

denotes the binary representations of .

is the binary complement of .

and are denoted by and , respectively.
2.2 ZComplementary Pair
Definition 1
The aperiodic crosscorrelation function (ACCF) of two complexvalued sequences and of length at shift is defined as
Note that . If , is called the aperiodic autocorrelation function (AACF) of , and refer to .
Definition 2
A pair of sequences and of length is called a complementary sequence pair (ZCP) with ZCZ width , denoted by ZCP, if they satisfy the following:
, for ,
where . If , then is called a Golay sequence pair (GCP).
2.3 Cross ZComplementary Pair
In contrast to ZCPs, CZCPs deals with crosscorrelation sums in addition to calculating autocorrelation sums.
Definition 3
(Liu2020 ) Let be a pair of sequences of identical length . For positive integers and with , define two intervals as and . Then is called an if it possesses symmetric zero (outofphase) AACF sums for timeshifts over and zero ACCF sums for timeshifts over . In short, it needs to satisfy the following two conditions:
(1) 
From two conditions given in (1), a CZCP has two zero autocorrelation zones (ZACZs) and one zero crosscorrelation zone (ZCCZ).
Definition 4
(Liu2020 ) Let be a binary . is called perfect if ( even). In this case, a perfect reduces to a sequence pair, called strengthened GCP.
Definition 5
(Adhikary2020 ) For an , the cross Zcomplementary ratio is defined as
where denotes the maximum value of for a given length . Note that since . When which implies that is achieved, such is called an optimal .
Note that for strengthened GCPs, , otherwise it is since of (1).
2.4 Boolean Function
A Boolean function is a function consisting of variables , where for . For a Boolean function with variables, we define the associated sequence and let where is the binary representation vector of the integer .
Obviously, the length of a sequence constructed by the Boolean function with variable is . But, sometimes, we need sequence lengths that are not limited to be powers of . Therefore, we define the truncated sequence , where is the result by removing the last elements from , that is, . Note that , where .
3 A Construction of CZCPs From GCPs
The ultimate objective of this section is to design CZCPs. We shall follow two directions as presented in the next subsections 3.1 and 3.2.
3.1 Our first approach for constructing CZCPs
In this section, we will propose two constructions of binary CZCPs from binary GCPs. Before then, we introduce some lemmas which will be used in our construction.
Lemma 1
(Huang2020 ) Let and be two sequences of length over . If and for all , then the pair satisfies the following
Lemma 2
(1974Turyn ) Let and be the first and the second binary GCPs of lengths and , respectively. Then is a GCP of length , where
Lemma 3
Now we are ready to present the main result.
Theorem 1
Let be a binary of length . Define the sequence pair with
where
Then, the sequence pair is a binary .
Proof
Case 1: Firstly, we compute the value of as follows:

For . When , we can obtain
When , we can obtain
Then, we have , .

For . When , we can obtain
When , we have
Then, we have , .

Similarly, for and , .

For ,
Then .
Based on the above discussion, we can obtain
Case 2: We compute the value of .
Obviously, we can see that and are sequences of length , , , . Then, by Lemma 1, we have
Remark 1
When in Theorem 1, the sequence pair is a binary . Moreover, .
Let and . By Theorem 1, we have
Then, we list the sums of their AACF and ACCF, respectively, as follows:
Hence, is a . Moreover, .
Next, we present one example to illustrate our above theorem.
Example 1
Let and . By Theorem 1, we have
where
Then, we list the sums of their AACF and ACCF, respectively, as follows:
Hence, is a . Moreover, .
Corollary 1
If is a Golay pair and in Theorem 1. Moreover, , , . Then is , and . Specially, when , .
Proof
It is easy to see that and are sequences of length , , , . Then, by Lemma 1, we have
Example 2
Let and . By Theorem 1, we can obtain
where
Then, we list the sums of their AACF and ACCF, respectively, as follows:
Hence, is a . Moreover, .
Theorem 2
Let be a GCP of length and be an . Also, let be an , where not is a Golay number. Moreover, , and . Define the sequence pair with
Then is an and .
Remark 2
If is a GCP of length and be an (strengthened GCP) in Theorem 2, we can obtain
Theorem 3
Let be a binary of length . Define the sequence pair with
where
Then, the sequence pair is a binary .
Proof
The proof process is similar to that of Theorem 1, so we omit it here.
Remark 3
When in Theorem 3, the sequence pair is a binary .
Example 3
Let and . By Theorem 3, we have
where
Then, we list the sums of their AACF and ACCF, respectively, as follows:
Hence, is a . Moreover, .
A summary of known CZCPs till date is given in Table 1. Compared with known CZCPs (but the length not Golay number) in Adhikary2020 , Fan2020 and yang2021 , we proposed the construction of CZCPs have the largest .
Table 1 Summary of CZCPs
ref.  Constraint  Method  

Adhikary2020  Insertion function  
Adhikary2020  Insertion function  
Adhikary2020  Insertion function  
Adhikary2020  Barker sequence  
Adhikary2020  Kronecker product  
Fan2020  GCP and Kronecker product  
Fan2020  GCP and Kronecker product  
Fan2020  GCP and Kronecker product  
Fan2020  GCP and Kronecker product  
yang2021  ZCP and concatenation  
yang2021  ZCP and concatenation  
yang2021  ZCP and concatenation  
yang2021  ZCP and concatenation  
yang2021  ZCP and concatenation  
Remark 1    GCP and concatenation  
Corollary 1  ,  GCP and concatenation  
Theorem 1  GCP and concatenation  
Corollary 1  ,  GCP and concatenation  
Theorem 3  GCP and concatenation 
3.2 Constructions of CZCPs from BFs
In this section, two constructions of CZCPs based on BFs are given and the CZC ratios of the CZCPs are also discussed. Compared with known CZCPs in the literature, we proposed CZCPs have the largest CZC. Before then, we introduce the following lemma which is useful for our construction.
Lemma 4
(1999GDJ ) Let be a permutation of the set . Define
where and . Let be the binary representations of the integer , then the two pairs of associated binary sequences
all form binary GCPs of length .
Theorem 4
Let be integer and be a permutation of . Define two binary BFs with variables as:
(3) 
where
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