Construction and Assignment of Orthogonal Sequences and Zero Correlation Zone Sequences Over GF(p)

Orthogonal sequences can be assigned to a regular tessellation of hexagonal cells, typical for synchronised code-division multiple-access (S-CDMA) systems. The sequences within any cell should both be orthogonal in the adjacent cells and require a sufficient number of users in each cell. However, for binary sequences, the capacity of communication system is limited. So, a new family of non-binary orthogonal sequences is constructed, which has more sequences per cell than binary sequences in the network. Next, an efficient assignment of these orthogonal sequence sets to a regular tessellation of hexagonal cells is given, and also the sequence sets have large re-use distance. Finally, based on above the construction, we construct a family of orthogonal sequences with zero correlation zone (ZCZ) property which can reduce the interference among users in a multi-path environment. The constructed ZCZ sequences can be applied to the quasi-synchronous CDMA (QS-CMDA) spread spectrum systems. And the assignment of sequences has the same re-use distance.


I. INTRODUCTION
Orthogonal sequences are used in many applications, such as synchronous code division multiple access (S-CDMA) spread spectrum systems, detection signals in satellite communications and mobile devices in cloud computing [1], [2]. To prevent interference from the neighboring cells, the regular tessellation of hexagonal cells be used as a model [3], [4], [5]. The sequences within any cell should both be orthogonal in the adjacent cells and require a sufficient number of users in each cell [6]. Employing correlation-constrained sets of Hadamard matrices to construct spreading codes in these systems are a usual way such that the maximal cross-correlation The associate editor coordinating the review of this manuscript and approving it for publication was R. K. Tripathy . of sequences lies in the range [2 m/2 , 2 (m+2)/2 ] [7], [8]. Akansu et al. presented a class of Walsh-like nonlinear binary orthogonal sequence sets which can be used in asynchronous direct sequence CDMA communications system [9]. However, the number of binary sequences in each cell is 2 m−3 or 2 m−2 and it is difficult to increase in S-CDMA systems. This raises a challenge to how to increase the capacity of each cell in S-CDMA systems. In order to increase the number of users in each cell, we propose a class of sequences over GF (p) such that each cell has more sequences. To more sufficient using the sequence sets, the re-use distance was proposed. The so-called re-use distance D reflects the ability to use the same codewords in non-adjacent cells that are far away from the cell where these codewords have originally been placed. At the same time, to prevent interference from the users in neighbouring cells, a standard requirement for the assignment of orthogonal sequences in the network is that the sequences within any cell should be orthogonal to the sequences in the neighbouring cells.
In S-CDMA systems, orthogonal sequences, such as well-known Walsh sequences, can guarantee that the correlation of any pairwise sequence is zero without shifting. However, whenever there is a relative shift between the sequences which will loss their orthogonality. To reduce the interference among users in a multipath environment or in a qusai-synchronised CDMA (QS-CDMA) system, Fan et al. proposed the concept of so-called zero correlation zone (ZCZ) property [10]. Tang and Fan obtained the lower bound of the sequence set of ZCZ [11]. Tang and Mow presented a systematic construction of new families of sequences with zero correlation zone [12]. In these systems it is of importance that two sequences have zero cross-correlation within a certain range time-shifts. Such the the time-shifts are called ZCZ width of the sequences. Tang and Wai proposed a new systematic construction of zero correlation zone sequences based on interleaved perfect sequences [13]. Tang et al. obtained a class of ZCZ sequences based on complete complementary codes [14]. Hu and Gong provided a construction ZCZ sequence sets based on interleaving technique [15]. Li and Xu presented a construction of ZCZ sequence sets over Gaussian integers [16], [17]. Zhou et al. proposed a class of optimal ZCZ sequence sets based on interleaving technique and perfect nonlinear functions [18], [19]. Gu et al. constructed a new family of polyphase sequences with low correlation [20]. Wang et al. obtained a class of M-sequences, which have good auto-correlation properties [21], [22].
The main contribution this paper is an efficient method to construct a large set of p-phase orthogonal sequences, using (vectorial) semi-bent functions such that there are p m−1 sequences (users) in each cell. The problem of allocating these cells of orthogonal sequences to a regular tessellation of hexagonal cells is also addressed, thus allowing an efficient practical implementation of our method. Based on the results above, a family of p-phase ZCZ sequence sets is constructed based on p-phase complementary triads and p-phase complete complementary codes. We also compare the results with those of previous results. Our constructed ZCZ sequence sets have new parameters. And the assignment of sequences has the same re-use distance.
This article is organized as follows. In Section II, the useful notations are given. In Section III, we construct a new class of p-ary orthogonal sequences, and a family of orthogonal sequences with ZCZ is presented in Section IV. Finally, the conclusion is presented in Section V.

II. PRELIMINARIES
For a prime p, let F p m and F m p denote the finite field GF(p m ) and the corresponding m-dimensional vector space, respectively. Let B m denote the set of all functions in m-variables. The set of all linear functions in m variables is defined by The Fourier transform of function f at λ is denoted by W f (λ) and it is computed as where ω = e 2πi/p .
Let f 1 , f 2 , · · · , f p be p functions with length p m . The concatenation of f 1 , f 2 , · · · , f p , denoted by f = f 1 f 2 · · · f p , is a function with length p m+1 . The true table of the function f is divided into p equipartitions such that the first part of function f corresponds to f 1 , the second part to f 2 and the last part to f p , the rest can be done in the same manner.
The sequence of a function f ∈ B m is defined as where the period of sequence is p m .
where f * denotes the conjugate of f .
Noticing that then the following important characterization of orthogonal sequences is obtained. Lemma 1: Let f 1 , f 2 be the functions over the vector space F m p , then f 1 ⊥f 2 if and only if W f 1 −f 2 (0 m ) = 0. According to the Lemma 1, it is easily deducing that the set of linear functions L m over the vector space F m p is a set of orthogonal sequences.
Definition 2: Let a = (a 0 , a 1 , · · · , a N −1 ) and b = (b 0 , b 1 , · · · , b N −1 ) be two complex sequences, the aperiodic correlation function of a and b at shift τ is given as follows: R a,b (τ ) is called aperiodic cross-correlation function (ACCF) if a = b; otherwise, it is called the aperiodic auto-correlation function (AACF). For simplicity, the AACF of a will be written as R a (τ ). Definition 3: Let A = {A 1 , A 2 , · · · , A K } be a sequences set, let the length of each sequence of A is L,

VOLUME 10, 2022
A is said to be a zero correlation zone (ZCZ) sequence set with ZCZ width Z if and only if the set A satisfies the following two conditions: i) 1) R A i (τ ) = 0 holds for any 1 ≤ i ≤ K and 1 ≤ |τ | ≤ Z ; 2) R A i ,A j (τ ) = 0 holds for any i = j and 0 ≤ |τ | ≤ Z .

III. CONSTRUCTION OF A LARGE FAMILY OF p-ARY ORTHOGONAL SEQUENCES
In this section we obtain the orthogonal sequences such that the number of users equals p m−1 per cell based on linear functions. Construction 1: Let m, s and t be three positive integers where [y] denotes the integer representation of y, and c ∈ {1, 2, · · · , p t−1 } is an integer. For y ∈ F s p , x ∈ F t p , the semi-bent functions can be obtained as follows, }, p t disjoint sequence sets are constructed as follows: According to (7), note that the f c (y, x) is the concatenation of p s t-variable functions, the length of the function f c (y, x) is p s · p t = p m . These sequences can be easily assigned to hexagonal cells in such a way that the sequences assigned to adjacent cells is orthogonal, while the correlation between sequences assigned to non-adjacent cells is small.
Theorem 1: Let the sequence set S c,i be defined by (9) as in Construction 1. Then, we always have where #S denotes the number of a sequence set S; 2) All the sequences in S c,i are semi-bent sequences; S c,i ⊥S c ,i holds. Proof: 1) Note that #L α = p m−1 , which implies that 1) hold. 2) For the sequence set S c,i , without loss of generality, we only consider the case of i = 0. Then the sequence set f c , and l is the sequence of a linear function l. For a nonzero c ∈ {1, 2, · · · , p s } and for y ∈ F s p , x ∈ F t p , According to (7), note that the f c (y, x) is the concatenation of p s t-variable linear functions, the length of the function An m-variable linear function β · y + (α 1 , α) · x can be regarded as the concatenation of a t-variable linear function (α 1 , α) · x and the corresponding affine functions (α 1 , α) · x + c, where c ∈ F p .
In order to get the magnitude of the fourier transform of the function f c (y, x), we just need to know if the linear function (α 1 , α) · x appears in the function f c (y, x). When Then all the sequences in S c,i are semi-bent sequences.
3) Let f c + l ∈ S c,i and f c + l ∈ S c ,i , where l = β · y + (α 1 , α) · x ∈ T i and l = β · y + (α 1 , α ) · x ∈ T i . To analyze the orthogonality between f c + l and f c + l , we consider From the analysis of the 2), when i = i , we can obtain α = α , then W h (0 m ) = 0. From Lemma 1, we have S c,i ⊥S c ,i . Remark 1: The assignment of the sequences is consistent with the above assignment such that the correlation between sequences assigned to adjacent cells is zero.
Example 1: Let m = 4, p = 3 and γ ∈ F 3 2 be a root of the primitive polynomial z 2 + z + 2. Define the isomorphism π : For y ∈ F 1 3 , x ∈ F 3 3 , then the function f c (y, x) can be denoted as follows, where c ∈ {1, 2, · · · , 9}. We can get 9 semi-bent functions. For simplicity, we only list two functions f 1 and f 2 as follows, 3 3 . For c ∈ {1, 2, · · · , 9} and i ∈ F 3 , a set of orthogonal sequences can be defined as follows, where From (11), then we have The orthogonality between f c and H i is shown in the following Table 1. Then we construct 27 sets of orthogonal sequences {S c,i | c ∈ {1, 2, · · · , 9}, i ∈ F 3 }. All the 27 sets of orthogonal sequences can be used to get an assignment with the re-use distance D = √ 27 as depicted in Fig. 1.

Construction 2:
Let m, k ≥ 2 be two positive integers with m = 2k + 2. Let γ be a primitive element of F p k , and {1, γ , · · · , γ k−1 } be a polynomial basis of F p k over F p . Define the isomorphism π: F p k → F k p by where [y] denotes the integer representation of y. Let y ∈ F k p , x ∈ F k+2 p . For i = 1, . . . , k, let For any fixed α ∈ F 2 p , linear function can be defined as follows, We construct disjoint sequence sets as follows: Theorem 2: Let the sequence set S c,i be defined by (17) as in Construction 2. Then, we always have 1) For i ∈ F k p , #S i,j = p m−1 , where j ∈ {0, 1, · · · , p − 1}; 2) All the sequences in S i,j are semi-bent sequences, where c is nonzero; and only if j = j . Proof: 1) Note that #L α = p m−2 , and #S i,j = p m−2 · p = p m−1 , which implies that 1) holds.
3) Let f i + l ∈ S i,j and f i + l ∈ S i ,j , where the Boolean function corresponding to the sequence l is l = β · y + (α 1 , α 2 ) · x ∈ H j and the Boolean function corresponding to the sequence l is l = β · y + (α 1 , α 2 ) · x ∈ H j . To analyze the orthogonality between f i + l and f i + l , we consider When j = j , then W h (0 m ) = 0 holds. From Lemma 1, we have S i,j ⊥S i ,j . The orthogonal sequences can be applied to the S-CDMA systems successfully. But orthogonal sequences can not be applied to the QS-CDMA systems directly. This problem is solved in the next section by constructing large sets of orthogonal sequences with ZCZ property.

IV. A FAMILY OF ORTHOGONAL SEQUENCES WITH ZCZ
Based on the orthogonal sequences obtained in Section III, we next construct a family of sequences satisfying ZCZ properties, which implies that the sequences can be applied to QS-CDMA systems. Furthermore, the re-use distance D is the same with the orthogonal sequences obtained in Section III.
is the Kronecker product operation.
For τ = 0, according to (22), we have if T c,α and T c ,α are in two adjacent cells.
If T c,α and T c ,α are in two non-adjacent cells, then When τ = 0, we consider the following case, Since the set {A 1 , B 1 , C 1 } is a three 3-phase Golay triad, for 0 < τ < N , and 0 < N −τ < N , R A 1 (τ )+R B 1 (τ )+R C 1 (τ ) = 0 holds. Then for 0 < τ < N , we can deduce We can determine that T c,α is a ZCZ sequence set with Z cz = N .
Let m = 4, p = 3 and γ ∈ F 3 2 be a root of the primitive polynomial z 2 + z + 2. We can obtain 27 sets of orthogonal sequences {S c,α | c ∈ {1, 2, · · · , 9}, α ∈ F 3 } as in Example 1. Then a ZCZ sequence set T c,α is obtained with length N = 3 5 · 6 + 10 = 1468 as follows, The constructed sequence set T cα has ZCZ width N , and also can be arranged as Fig. 1 in a similar way with the re-use distance D = √ 27, where we just replace S c,α with T c,α . Corollary 1: Let S c,α be the sequence set generated by the section III, where p = 3. Let {A 1 , B 1 , C 1 } be a GCT with length N , and {A 2 , B 2 , C 2 } be a mate of {A 1 , B 1 , C 1 }. We define a new sequence triads as follows: We construct 3 k+1 disjoint sequence sets of (3 m+1 N + 4N − 2)-length with ZCZ width 2N as follows: Corollary 1 has a similar proof to Theorem 3 and is therefore omitted here. Remark 3: Corollary 1 obtain a class of ZCZ sequence sets with Z cz = 2N , which has larger ZCZ width compared with Theorem 3.
Definition 6: ( [26], [27]) Let the sequence set Corollary 2: Let S c,α be the sequence set generated by the section III. Let A = {A 0 , A 1 , · · · , A p−1 } be a (p, p, p)-CCC, and A j = {a j,0 , a j,1 , · · · , a j,p−1 }, where a j,l is a sequence with length p, and 0 ≤ j, l ≤ p − 1. Defining A j,l = (a j,l , a j,l , · · · , a j,l ) with p m -tuples, we construct p k+2 disjoint sequence sets of (p m+2 +(p−1) 2 )-length with ZCZ as follows: In each ZCZ sequence set, we have p m constituent sequence. Corollary 2 has a similar proof to Theorem 3 and is therefore omitted here. Remark 4: Corollary 2 obtains a class of ZCZ sequence sets with Z cz = N , and the number of constituent sequences is p m in each sequence set, which has a larger number of sequences compared with Theorem 3.
Remark 5: In Table 3, we compare the main parameters of ZCZ sequence. Comparison with the literature [12] and [23]. A class of p-phase ZCZ sequence sets is constructed, which the length of sequence is 3 m+1 ·N +2N −2 in Theorem 3. This is the first time we obtain a ZCZ sequence with new length. In Corollary 1, a class of ZCZ sequence sets is presented with ZCZ width 2N of 3 m+1 ·N +4N −2 length. The ZCZ sequence sets in Corollary 1 have larger ZCZ width than sequence sets in Theorem 3. In Corollary 2, a family of ZCZ sequence sets is proposed based on p-phase CCC, which has p m+2 + (p − 1) 2 length.
In regular tessellation of hexagonal cells, the constructed ZCZ sequence sets have the same reuse distance as orthogonal sequence sets constructed in Theorem 1.

V. CONCLUSION
In this article, we construct a class of orthogonal sequences over GF(p), and assign them to a tessellation of hexagonal cells. The method increases the number of sequences to be p m−1 per cell in the network. Compared with binary sequences, non-binary sequences have more sequences in each sequence set. Next we extended the orthogonal sequences to ZCZ sequences which can be applied to QS-CDMA system, and analyzed the properties of the new sequence sets. For the sequence set in each cell, the ZCZ sequence sets not only have the same re-use distance as the orthogonal sequences, but also have large zero correlation zone which can be applied to QS-CDMA system.