New Constructions for Near-Optimal Sets of Frequency-Hopping Sequences via the Gaussian Periods in Finite Fields

Frequency-hopping sequences (FHSs) have been widely applied in frequency-hopping code-division multiple-access (FH-CDMA) systems, since they can be used for transmitting messages efficiently along with switching frequencies at set intervals by each sender. The performance of the FHSs has a great impact on the performance of FH-CDMA systems. The optimality achieving exactly the Peng-Fan bounds is an important performance measure. However, optimal sets of FHSs do not always exist for all lengths and alphabet sizes. Thus, it is meaningful to seek and design more near-optimal FHS sets whose parameters are near to achieving the Peng-Fan bounds. Let <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> be a power of a prime. In this paper, we present some classes of near-optimal sets of FHSs, whose parameters are <inline-formula> <tex-math notation="LaTeX">$\left({\frac {3(q+1)}{2},\frac {2(q-1)}{3},3;q}\right)$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$q\equiv 1 \pmod {12}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$\left({\frac {q+1}{k},{k(q-1)},2;q}\right)$ </tex-math></inline-formula> with even <inline-formula> <tex-math notation="LaTeX">$\frac {q+1}{k}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$\left({2(q^{2}+1),\frac {q^{2}-1}{2},2(q+1);q}\right)$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$q \equiv 3 \pmod {4}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$(19,18,4;7)$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$(7,9,3;4)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$(91,45,7;16)$ </tex-math></inline-formula> respectively. Most importantly, these classes of near-optimal sets of FHSs have new parameters which are not covered in the foregoing literature.


I. INTRODUCTION
Frequency-hopping multiple access (FHMA) spread systems play an important role in military radio communications, mobile communications, modern radar and sonar echolocation systems [1], [2]. In these systems, each user is represented by a sequence of hopping frequencies, which is called a frequency-hopping sequence (FHS). In general, we need to minimise the maximum of Hamming out-of-phase autocorrelation and Hamming crosscorrelation of the FHS sets, whose purpose is to discriminate their own signals from the others and reduce multiple access collisions by simultaneous transmission. To accommodate a great quantity of users, it is also preferred that the size of the FHS sets is as large as possible. All in all, it is imperative to find FHSs with long length, small available frequencies, large size of FHS set and low Hamming correlation simultaneously. As an inseparable whole, these parameters of the FHS sets are closely connected with each The associate editor coordinating the review of this manuscript and approving it for publication was Khmaies Ouahada.
other. As a matter of fact, they are subjected to some theoretic bounds, for example, the Lempel-Greenberger bound [3] and the Peng-Fan bounds [4]. These bounds become standards of evaluating the performance of the FHSs.
Because of the specification of a given system or environment, the required length and alphabet size of an FHS set vary in practical applications. Hence it is very important to choose some FHS sets with (near-)optimal Hamming correlation under the given constraint condition. Generally speaking, (near-)optimality of an FHS set is measured by the Peng-Fan bounds. More specifically, we call an FHS set to be optimal if its maximum periodic Hamming correlation achieves exactly the Peng-Fan bounds, and we call an FHS set to be near-optimal if its maximum periodic Hamming correlation is bigger than the Peng-Fan bounds by one.
So far, both algebraic and combinatorial constructions of optimal sets of FHSs with respect to the Peng-Fan bound were provided (see, for example, [3], [5]- [23]). Generally speaking, optimal sets of FHSs do not always exist for all lengths and alphabet sizes. However, it is a difficult problem to verify whether an optimal FHS set exists for a given length and a given alphabet size. Hence it is also valuable to construct more near-optimal sets of FHSs. In 2011, Chung and Yang [24] introduced a class of near-optimal sets of FHSs via using k-fold cyclotomy. In 2012, Chung and Yang [16] presented a class of near-optimal sets of FHSs by using non-linear functions over Z p , where p is an odd prime. In 2013, Chung and Yang [25] got a near-optimal set of FHSs by applying the new class of balanced near-perfect nonlinear mappings. Except these constructions, Xu et al. [6], [21], [26] obtained some new classes of near-optimal sets of FHSs by means of interleaving technology and generalized cyclotomy.
Our purpose in this paper is to design more near-optimal sets of FHSs for some cases which are not covered in the literature. By means of the trace function and known Gaussian periods in finite fields, some classes of near-optimal sets of FHSs are presented, see Theorems 7, 9 and 11. In Table 2, we summarize some known near-optimal sets of FHSs, including the parameters obtained from this paper.
The rest of this paper is organized as follows. In Section II, we give some preliminaries to frequency-hopping sequence, the cyclotomic classes and Gaussian periods in finite fields. In Section III, we propose some constructions of near-optimal sets of FHSs. Finally, Section IV concludes this paper.

II. PRELIMINARIES
For the constructions of near-optimal sets of FHSs in the sequel, we need to recall the notions of frequency-hopping sequence, the cyclotomic classes and Gaussian periods in finite fields.

A. FREQUENCY-HOPPING SEQUENCE
Let F = {f 0 , f 1 , . . . , f l−1 } be an alphabet of l available frequencies which are shared by numerous senders. A sequence X = {x t } n−1 t=0 is called a frequency-hopping sequence (FHS) of length n over F if x t ∈ F for all 0 ≤ t ≤ n − 1. Given any two FHSs X = {x t } n−1 t=0 and Y = {y t } n−1 t=0 of length n over F, their periodic Hamming correlation H X ,Y at time delay τ is defined by and 0 otherwise, and t + τ is performed modulo n.
The maximum periodic Hamming out-of-phase autocorrelation H (X ) of X and the maximum periodic Hamming crosscorrelation H (X , Y ) for two different FHSs X and Y are defined as Let S be a set of M FHSs of length n over an alphabet F. The maximum periodic Hamming correlation of S is defined by Henceforth, we use (n, M , λ; l) to denote a set S containing M FHSs of length n over an alphabet of size l with H (S) = λ. In this case, we also say that a set S of FHSs has parameters (n, M , λ; l). In and where z denotes the largest integer less than or equal to z, and z denotes the smallest integer not less than z.
In 2017, Chen et al. [5] proved that the two Peng-Fan bounds given above are actually identical. Furthermore, Xu et al. [6] in 2016 gave a simplified form of the Peng-Fan bounds as follows.
Lemma 2 ( [5], [6]): Let S be a set of M FHSs of length n over an alphabet of size l and a = n l . Then An FHS set S is called optimal with respect to the Peng-Fan bound if the parameters of S meet the equality in (1) or (2), and an FHS set S is called near-optimal with respect to the Peng-Fan bound if H (S) is bigger than the righthand side of (1) or (2) by one.

B. CYCLOTOMIC CLASSES AND GAUSSION PERIODS
Throughout this paper, let p be a prime and q = p s for a positive integer s. Denote by F q (or F r ) the finite field with q (or r) elements, where r = q m and m is a positive integer. Let F * r be the multiplicative group consisting of all nonzero elements in F r and θ be a fixed primitive element of F r such that F * r = θ . Let Tr r/q denote the trace function from F r to F q defined by Let r = nN + 1 for two positive integers n, N ≥ 2. The cyclotomic classes of order N in F r are defined by In the remainder parts of this section, we shall introduce basic results on Gaussian periods. For more details, the reader is advised to refer to the paper [27] and the book [28].
be the primitive p-th root of unity. The canonical additive character of F r can be defined by where Tr r/p denotes the absolute trace function from F r to F p . According to the orthogonal property of additive characters, we have x∈F r Definition 3: Let χ be the canonical additive character of F r and D In general, it is very hard to calculate the values of Gaussian periods. However, they can be computed for several particular cases. In this paper, the following lemmas are important for determining the periodic Hamming correlation distributions of an FHS set in Section III.
Lemma 4: Let r = q m , q = p s and N = 2. The Gaussian periods of order 2 in F r are given by To present further known results about Gaussian periods in F r , we need to introduce the period polynomial (N ,r) (X ) defined as Lemma 5: Let r = q m , q = p s and N = 3. If p ≡ 1 (mod 3) and ms ≡ 0 (mod 3), then the factorization of (3,r) (X ) is given by (3,r) (2) In all the other cases,

III. SOME CONSTRUCTIONS OF NEAR-OPTIMAL SETS OF FHSS
Let r = q m = nN + 1 and θ be defined as Section II. The set S = {s (0) , s (1) , · · · , s (N − 1) } of N FHSs of length n over F q is defined by Up to now, the parameters of the set S of FHSs being optimal with respect to the Peng-Fan bound are listed in Table 1, where q is a power of a prime p. Our aim in this section is to present new constructions for such set being near-optimal with respect to the Peng-Fan bound. Specifically, these parameters with near-optimal Hamming correlation property are different from those parameters in [10,Theorem 4.8], [20,Theorem 2] and [23, Theorem 1], because gcd(N , q m −1 q−1 ) = 1 and N (q − 1). Furthermore, we also use the interleaving techniques in [13] to construct new near-optimal sets of FHSs with longer length and flexible maximum periodic Hamming correlation from old ones.

A. THE FIRST CLASS OF NEAR-OPTIMAL SETS OF FHSS
In this subsection, let q be a power of an odd prime p and r = q 2 . The set A = {a (0) , a (1) , · · · , a ( 2q−5
By Matlab software, there are simulation results to validate Theorem 7 in Figures 1 and 2 as follows:

B. THE SECOND CLASS OF NEAR-OPTIMAL SETS OF FHSS
In this subsection, let p be an odd prime, q = p s with an odd integer s and r = q 2 . Let k be a positive integer satisfying that p ≡ −1 (mod 2k) and q+1 k is even. The set Theorem 9: The set B of FHSs has the parameters ( q+1 k , k(q − 1), 2; q) and is near-optimal with respect to the Peng-Fan bound, where q = p s with an odd integer s, p ≡ −1 (mod 2k) and q+1 k is even. Proof: Firstly, we have since q+1 k is even. Secondly, for any two FHSs b (i 1 ) , b (i 2 ) ∈ B with 0 ≤ i 1 , i 2 < k(q − 1), their periodic Hamming correlation at time delay τ is given as Tr r/q (θ i 1 +kt(q−1) ) = Tr r/q (θ i 2 +k(t+τ )(q−1) )}| where (7) holds since kt(q − 1) + i(q + 1) takes on each element of 2k · Z q 2 −1 2k exactly 2 times for gcd(k(q − 1), q + 1) = 2k, when t and i range over Z q+1 k and Z q−1 respectively. For Lemma 6, p is an odd prime, f = 1 and γ = s. Hence, we have by Lemma 6 and Furthermore, Then we have Therefore, B is near-optimal with respect to the Peng-Fan bound according to Lemma 1. Example 10: Let q = p = 83, k = 2 and r = q 2 . Let θ be the primitive element of F r with θ 2 − θ + 2 = 0. Then the set B = {b (0) , b (1) , · · · , b (163) } defined by (6)  In this subsection, let q = p s and r = q 4 , where p is an odd prime with p ≡ 3 (mod 4) and s is an odd integer. The set 2 FHSs of length 2(q 2 + 1) over F q is defined by Theorem 11: The set C of FHSs has the parameters (2(q 2 + 1), q 2 −1 2 , 2(q + 1); q) and is near-optimal with respect to the Peng-Fan bound, where q is a power of an odd prime with q ≡ 3 (mod 4).
By Matlab software, there are simulation results to validate Theorem 11 in Figures 5 and 6 as follows: Remark 13: Except these constructions given above, more near-optimal sets of FHSs can also be obtained from (3), Lemmas 5 and 6, whose parameters are (19, 18, 4; 7), (7, 9, 3; 4) and (91, 45, 7; 16). Now we compare our parameters with some known parameters of near-optimal sets of FHSs in Table 2, where p is a prime, q is a power of an odd prime and p 1 , p 2 , · · · , p h are h odd primes with 2 < p 1 < p 2 < · · · < p h .

IV. CONCLUDING REMARKS
By means of the trace function and known Gaussian periods in finite fields, some algebraic constructions for near-optimal sets of FHSs were presented. As a comparison, we listed some parameters of near-optimal sets of FHSs, including our parameters in this paper. Inspired by this idea, more near-optimal sets of FHSs with new parameters may be obtained by virtue of new Gaussian periods and exponential sums in finite fields. Here we welcome the scholars to solve this problem together.