1 Introduction
A pair of sequences is called a Golay complementary pair (GCP) [11] if their aperiodic autocorrelation sums for any nonzero shifts are all equal to zero. Each sequence in the GCP is called a Golay complementary sequence (GCS). The concept of binary GCP was extended later to the polyphase case [22] and complementary sequence sets [23]. These sequences have found numerous applications in various fields of science and engineering, especially in orthogonal frequencydivision multiplexing (OFDM) systems. One of the major impediments to deploying OFDM is the high peaktomean envelope power ratio (PMEPR) of uncoded OFDM signals. PMEPR reduction in OFDM transmission can be implemented [1][19] by using codes constructed from the sequences in the complementary sequence sets, especially GCSs.
GCPs were initially constructed by the recursive methods [11][3]. An extensive study on this topic was made by Davis and Jedwab in [7] by a direct construction of polyphase complementary sequences based on generalized Boolean functions (GBFs), which have been referred to as the standard GCSs subsequently. Nonstandard GCPs were studied in [16][8][9] and complementary sequence sets were constructed in [18][21] based on GBFs later on.
All the aforementioned sequences are constructed over the phaseshift keying (PSK) constellations. Since quadrature amplitude modulation (QAM) are widely employed in high rate OFDM transmissions, QAM sequences based on weighted quaternary PSK (QPSK) GCSs were studied by Rößing and Tarokh [20] in 2001. Chong et al. [5] then proposed a construction of 16QAM GCSs based on standard GCSs over QPSK and firstorder offsets in three cases. It was pointed out in [5] that an OFDM system with 16QAM GCSs has a higher code rate than that with binary or quaternary standard GCSs, given the same PMEPR constraint. In 2006, Lee and Golomb [13] proposed a construction of QAM GCSs with the weightedsum of three standard GCSs over QPSK and firstorder offsets in five cases. Further improvements of constructions of GCSs over 16QAM and 64QAM were given by Li et al. [14] and Chang et al. [6] later on. These results were extended to the general construction GCSs over QAM by Li et al. [15] in 2010 and Liu et al. [17] in 2013, respectively. All these GCSs over QAM are constructed based on standard QPSK GCSs and compatible offsets. Depending on the algebraic structure of the compatible offsets, the GCSs proposed in [15] and [17] are referred to as the generalized cases IIII and the generalized cases IVV, respectively.
In 2018, Budišin and Spasojevi [2] introduced a new recursive algorithm in multiplicative form to generate GCPs over QAM by paraunitary (PU) matrices, where any element of a sequence can be generated by indexing the entries of unitary matrices with the binary representation of the discrete time index. Sequences derived from unitary matrix over QAM constellation are referred to as the Qum case. It is shown that the 1Qum case and 2Qum case can generate the sequences in the generalized cases IIII [15] and cases IVV [17], respectively. Moreover, a large number of new GCSs over QAM are produced from the Qum case when . For instance, the numerical results show that the overall increase in the total number of GCSs including 1Qum and 2Qum cases of length 1024 is up to and , for 64, 256, and 1024QAM, respectively. However, for given and sequence length , the acceptable unitary matrices over QAM can be obtained only by exhaustive search, and the lack of explicit algebraic expression of GCSs leads to unexpected duplication for Qum case when .
In this paper, we propose two new general constructions of GCSs over QAM of length by explicit GBFs based on standard GCSs over QPSK and compatible offsets. The known generalized cases IV in [15, 17] are special cases of our new constructions. Moreover, if is a composite number, the proposed constructions significantly increase the number of the GCSs given in [15, 17]. An example for shows that the number of new GCSs from our constructions is seven times more than that in generalized cases IVV [17]. Another example for shows that the ratio of the number of new GCSs and the generalized cases IVV is greater than six and will increase in proportion with . Note that the number of GCSs in generalized cases IVV is much larger than that in cases IIII when is large.
Although the GCSs over QAM proposed in our construction are represented by explicit algebraic expression of weightedsum of standard GCSs over QPSK, the ideas here are inspired by the paraunitary algorithm [2] and the Golay array pairs (GAPs) [10]. GAPs and their relationship with GCPs over PSK by a threestage construction process were introduced by F. Fiedler et al. [10] in 2008. We extend this idea from PSK modulation to QAM modulation, and propose a mapping from a GAP of size to a large number of GCPs over QAM of length instead of previous threestage construction process. We also make a connection of the construction of the GAPs and the specified PU matrices with multivariables over QAM. Finally, by the technique to derive GBFs from PU matrices introduced in our recent work [24], two new general constructions of GCSs over QAM are obtained.
For the general cases IIII [15], general cases IVV [17], and the constructions in this paper, the GCSs over QAM of length are all expressed by the standard GCSs over QPSK and compatible offsets. If is a prime, the proposed constructions are identical to the general case IV. However, if is a composite number, the proposed constructions comprise of not only the general case IV, but also a great number of new GCSs. The numbers of the GCSs proposed in the general cases IV and this paper are all equal to the product of the number of the standard GCSs over QPSK (which is determined) and the number of the compatible offsets. It has been shown that the numbers of the compatible offsets in general cases IIII and cases IVV are linear polynomials of and quadratic polynomial of , respectively. We show that the numbers of the new compatible offsets in this paper is seven times more than that in general cases IVV for , and is at least a cubic polynomial of for .
On the other hand, from the proof of the proposed constructions, the GCSs proposed in this paper give the explicit algebraic expressions of GCSs from the PU construction in [2] for special Qum cases when . Moreover, our proof implies all the mentioned GCSs over QAM in this paper can be regarded as projections of Golay complementary arrays of size , so the results in this paper provide a partial solution to an open problem from [10] for GAPs of size :
How can the threestage construction process be used to simplify or extend known results on the construction of Golay sequences in QAM modulation?
The rest of this paper is organized as follows. In the next section, we introduce the definitions of GCP and GCS, and the known constructions of the GCPs over QAM. In Section 3, we present two new general constructions of QAM GCSs including the generalized cases IV as special cases in Theorem 1 and 2. Enumerations of the new GCSs other than the generalized cases IV for and are given in Section 4. The proofs of our main results are presented in the following two sections. In Section 5, we explain our main theory on GAPs, PU matrices and corresponding GBFs. What’s more, we propose the array form of our result in Theorem 5 and 6, namely QAM GAPs based on PU matrices, from which the main result on GCSs can be derived directly. The proofs of Theorem 5 and 6 are shown in Section 6. We conclude the paper in Section 7.
2 Preliminary
The following notations will be used throughout the paper.

are all positive integers, where .

For any positive integer , is the residue class ring modulo .

For positive integers and , define as a set and .

, where each is a indeterminate for .

is the complex field, and is the real number field.

For any , is the conjugation of .

is a permutation of symbols , and is a permutation of symbols .
2.1 Golay Complementary Pair
A complexvalued sequence of length can be expressed by a function , i.e.,
The aperiodic autocorrelation of at shift () is defined by
where if or is not defined.
A pair of sequences of length is said to be Golay complementary pair (GCP) if
(1) 
And either sequence in a GCP is called a Golay complementary sequence (GCS) [11].
2.2 GCPs over QPSK
A generalized Boolean function (GBF) (or over is a function from to . Such a function can be uniquely expressed as a linear combination over of the monomials
where the coefficient of each monomial belongs to .
For , can be written uniquely in a binary expansion as where . Then a sequence of length over QPSK can be associated with a GBF over by
(2) 
where is a fourth primitive root of unity.
There are several constructions of GCPs over QPSK based on GBFs, such as [7], [16], [8] and [9]. The most typical GCPs are so called standard GCPs given in [7], which are associated with GBFs over given below.
Fact 1
2.3 GCPs over QAM
In this subsection, we show some of the known results about sequence and GCPs over QAM based on the weighted sums of sequences over QPSK.
A vectorial GBF (VGBF) is a function from to , denoted by
where each component function is a GBF over .
A sequence over QAM can be viewed as the weighted sums of sequences over QPSK, with weights in the ratio of . Then a sequence over QAM of length can be associated with a VGBF over by
(4) 
where and . Obviously, the sequence over QPSK can be seen as a special case of QAM sequence when .
The GCPs of length over QAM were well studied by their associated VGBFs in the literature. Such VGBFs are usually given by

standard GCSs in form (3),

offset VGBF ,

paring difference VGBF ,
or more clearly,
(5) 
where denotes the dimension vector . The offset VGBFs and paring difference VGBFs proposed in the generalized cases IIII given by Li et al. [15] and the generalised cases IVV given by Liu et al. [17] are shown below respectively.
Fact 2
(The generalized cases IIII [15]) form a QAM GCP of length if the offset VGBF and paring difference VGBF in their associated VGBFs satisfy one of the following cases:

The generalized case I:

The generalized case II:

The generalized case III:
with or , , and
Definition 1
([17]) A complex number is called a Gaussian integer if its real and imaginary part are both integers. Define
(6) 
Then a onetoone mapping from to , which is a set consisting of Gaussian integers.
For and , a pair of distinct Gaussian integers with identical magnitude, and which are not conjugate with each other, namely:
(7) 
is called a nonsymmetrical Gaussian integer pair (NSGIP).
Fact 3
(The generalized cases IVV [17]) Given a nonsymmetrical Gaussian integer pair , forms a QAM GCP of length if the offset VGBFs and paring difference VGBFs in their associated VGBFs satisfy one of the following cases:

The generalized case IV:
with or , .

The generalized case V:
with or , , .
3 Main Results
In this section, we will propose two new constructions of GCPs over QAM. The generalized cases IIII [15] and generalized cases IVV [17] are special cases of our first and second constructions, respectively. Different from the offset VGBFs and paring difference VGBFs shown in Fact 2 and 3, the new proposed and relate to the factorization of the integer .
Definition 2
Let be a factorization of , where are positive integers. For , define as sets of integers by
(8) 
For the case , the above definition is extended to . For any given permutation of symbols , we can define mappings for such that .
The rationale of the definition of is guaranteed by the following remark.
Remark 1
Any can be uniquely decomposed as , where . In fact, () can be uniquely determined by the following recursive formula:
It is easy to verify .
Example 1
For and the factorization , we have

Suppose that the permutation satisfies and . For , the decomposition of () is given below:

Definition 3
For , define such that . In particular, we always define .
For any , There are possible values of , which are given as follows
3.1 The First Construction
Our first construction includes the results in the generalized cases IIII in [15] as a special case.
Theorem 1
For any factorization and permutation , let () be mappings given in Definition 2, and () elements over given in Definition 3. Denote the vectors and respectively by
(9) 
Sequences over QAM of length associated with VGBFs in (5) form a GCP if the offset VGBFs and paring difference VGBFs satisfy the following conditions:
(10) 
The detailed proof of Theorem 1 will be given in Sections 5 and 6. We first explain the sequences in Theorem 1 by the following examples.
Let the factorization of be trivial, i.e., such that for and for . Let be the identity permutation. We have
From Definition 2, we obtain equals if , and equals otherwise. Then the offset and pairing difference set VGBFs for different can be obtained immediately by Theorem 1.

If , i.e., , then and We have
(11) or
(12) 
If , i.e., , then and We have
(13) or
(14) 
If , i.e., , then We have
(15)
It is obvious that the offset VGBFs and paring difference VGBFs shown in (11), (13), (15) agree with the generalized cases IIII in [15]. New paring difference VGBFs shown in (12), (14) lead to new GCPs over QAM, but they do not produce new GCSs.
Moreover, new GCPs and GCSs over QAM can be constructed from Theorem 1 if is a composite number and is a nontrivial factorization. Two examples are given below to illustrate it.
Example 2
For , suppose that the factorization and is the identity permutation. Then we have

From Theorem 1, the QAM sequence pair associated with the VGBFs
form a GCP of length , where is given in (3). Considering for , the offset and vectors are given in the table respectively:

where for and .
Example 3
For and , suppose that the factorization is given by and permutation satisfies and . Then and the decomposition of () are shown in Example 1. Notice that for . For , the offset in (10) is simplified to
(16) 
where for and , i.e.,

Thus the sequence pair over QAM associated with the VGBFs
form a GCP of length , where is given in (3) and vector is given in the table above.
To the best of our knowledge, the expressions of offset VGBFs shown in the above examples have never been reported before.
3.2 The Second Construction
In this subsection, we slightly modify the conditions in the first construction, and obtain our second construction which include the generalized cases IVV in [17].
Definition 4
Let be a factorization of , where and are positive integers. For , define as sets of integers by
(17) 
For the case , we extend the definition to . For any given permutation of symbols , define mappings and , such that .
Similar to the mappings in the first construction, can be uniquely decomposed as
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