A pair of sequences is called a Golay complementary pair (GCP)  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  and complementary sequence sets . These sequences have found numerous applications in various fields of science and engineering, especially in orthogonal frequency-division multiplexing (OFDM) systems. One of the major impediments to deploying OFDM is the high peak-to-mean envelope power ratio (PMEPR) of uncoded OFDM signals. PMEPR reduction in OFDM transmission can be implemented  by using codes constructed from the sequences in the complementary sequence sets, especially GCSs.
GCPs were initially constructed by the recursive methods . An extensive study on this topic was made by Davis and Jedwab in  by a direct construction of polyphase complementary sequences based on generalized Boolean functions (GBFs), which have been referred to as the standard GCSs subsequently. Non-standard GCPs were studied in  and complementary sequence sets were constructed in  based on GBFs later on.
All the aforementioned sequences are constructed over the phase-shift 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  in 2001. Chong et al.  then proposed a construction of 16-QAM GCSs based on standard GCSs over QPSK and first-order offsets in three cases. It was pointed out in  that an OFDM system with 16-QAM GCSs has a higher code rate than that with binary or quaternary standard GCSs, given the same PMEPR constraint. In 2006, Lee and Golomb  proposed a construction of -QAM GCSs with the weighted-sum of three standard GCSs over QPSK and first-order offsets in five cases. Further improvements of constructions of GCSs over 16-QAM and 64-QAM were given by Li et al.  and Chang et al.  later on. These results were extended to the general construction GCSs over -QAM by Li et al.  in 2010 and Liu et al.  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  and  are referred to as the generalized cases I-III and the generalized cases IV-V, respectively.
In 2018, Budišin and Spasojevi  introduced a new recursive algorithm in multiplicative form to generate GCPs over QAM by para-unitary (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 1-Qum case and 2-Qum case can generate the sequences in the generalized cases I-III  and cases IV-V , 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 1-Qum and 2-Qum cases of length 1024 is up to and , for 64-, 256-, and 1024-QAM, 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 I-V 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 IV-V . Another example for shows that the ratio of the number of new GCSs and the generalized cases IV-V is greater than six and will increase in proportion with . Note that the number of GCSs in generalized cases IV-V is much larger than that in cases I-III when is large.
Although the GCSs over QAM proposed in our construction are represented by explicit algebraic expression of weighted-sum of standard GCSs over QPSK, the ideas here are inspired by the para-unitary algorithm  and the Golay array pairs (GAPs) . GAPs and their relationship with GCPs over PSK by a three-stage construction process were introduced by F. Fiedler et al.  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 three-stage construction process. We also make a connection of the construction of the GAPs and the specified PU matrices with multi-variables over -QAM. Finally, by the technique to derive GBFs from PU matrices introduced in our recent work , two new general constructions of GCSs over -QAM are obtained.
For the general cases I-III , general cases IV-V , 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 I-V. However, if is a composite number, the proposed constructions comprise of not only the general case I-V, but also a great number of new GCSs. The numbers of the GCSs proposed in the general cases I-V 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 I-III and cases IV-V 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 IV-V 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  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  for GAPs of size :
How can the three-stage 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 I-V as special cases in Theorem 1 and 2. Enumerations of the new GCSs other than the generalized cases I-V 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.
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 .
is the finite field with two elements, and
is m-dimension vector space over.
, 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 complex-valued sequence of length can be expressed by a function , i.e.,
The aperiodic auto-correlation 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
And either sequence in a GCP is called a Golay complementary sequence (GCS) .
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
where is a fourth primitive root of unity.
There are several constructions of GCPs over QPSK based on GBFs, such as , ,  and . The most typical GCPs are so called standard GCPs given in , which are associated with GBFs over given below.
 For GBF
where , and , the sequence pair associated with the GBFs over
form a GCP over QPSK.
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 (V-GBF) 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 V-GBF over by
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 V-GBFs in the literature. Such V-GBFs are usually given by
standard GCSs in form (3),
offset V-GBF ,
paring difference V-GBF ,
or more clearly,
where denotes the -dimension vector . The offset V-GBFs and paring difference V-GBFs proposed in the generalized cases I-III given by Li et al.  and the generalised cases IV-V given by Liu et al.  are shown below respectively.
(The generalized cases I-III ) form a -QAM GCP of length if the offset V-GBF and paring difference V-GBF in their associated V-GBFs satisfy one of the following cases:
The generalized case I:
The generalized case II:
The generalized case III:
with or , , and
() A complex number is called a Gaussian integer if its real and imaginary part are both integers. Define
Then a one-to-one 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:
is called a non-symmetrical Gaussian integer pair (NSGIP).
(The generalized cases IV-V ) Given a non-symmetrical Gaussian integer pair , forms a -QAM GCP of length if the offset V-GBFs and paring difference V-GBFs in their associated V-GBFs 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 I-III  and generalized cases IV-V  are special cases of our first and second constructions, respectively. Different from the offset V-GBFs and paring difference V-GBFs shown in Fact 2 and 3, the new proposed and relate to the factorization of the integer .
Let be a factorization of , where are positive integers. For , define as sets of integers by
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.
Any can be uniquely decomposed as , where . In fact, () can be uniquely determined by the following recursive formula:
It is easy to verify .
For and the factorization , we have
Suppose that the permutation satisfies and . For , the decomposition of () is given below:
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 I-III in  as a special case.
Sequences over -QAM of length associated with V-GBFs in (5) form a GCP if the offset V-GBFs and paring difference V-GBFs satisfy the following conditions:
Let the factorization of be trivial, i.e., such that for and for . Let be the identity permutation. We have
If , i.e., , then and We have
If , i.e., , then and We have
If , i.e., , then We have
It is obvious that the offset V-GBFs and paring difference V-GBFs shown in (11), (13), (15) agree with the generalized cases I-III in . New paring difference V-GBFs 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 non-trivial factorization. Two examples are given below to illustrate it.
For , suppose that the factorization and is the identity permutation. Then we have
From Theorem 1, the -QAM sequence pair associated with the V-GBFs
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 .
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
where for and , i.e.,
Thus the sequence pair over -QAM associated with the V-GBFs
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 V-GBFs 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 IV-V in .
Let be a factorization of , where and are positive integers. For , define as sets of integers by
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