MORE CYCLOTOMIC CONSTRUCTIONS OF OPTIMAL FREQUENCY-HOPPING SEQUENCES

. In this paper, some general properties of the Zeng-Cai-Tang-Yang cyclotomy are studied. As its applications, two constructions of frequency- hopping sequences (FHSs) and two constructions of FHS sets are presented, where the length of sequences can be any odd integer larger than 3. The FHSs and FHS sets generated by our construction are (near-) optimal with respect to the Lempel–Greenberger bound and Peng–Fan bound, respectively. By choosing appropriate indexes and index sets, a lot of (near-) optimal FHSs and FHS sets can be obtained by our construction. Furthermore, some of them have new parameters which are not covered in the literature.


Introduction
For convenience, we introduce the following notation in this paper: • x y : the least nonnegative residue of x modulo y for two positive integers x and y; • z : the least integer greater than or equal to z; • z : the largest integer less than or equal to z; • Z v : the residue class ring modulo v for a positive integer v; • Z * v : the set consisting of all elements in Z v relatively prime to v; • (Z v ) k : the k-dimensional space over Z v ; • a|b : the integer a divides the integer b. Let F ={f 0 , f 1 , ..., f l−1 } be an alphabet of l available frequencies. A sequence X = {x 0 , x 1 , ..., x n−1 } is called a frequency-hopping sequence (FHS) of length n over F if x t ∈ F for 0 ≤ t < n. Given any two sequences X = {x 0 , x 1 , ..., x n−1 } and Y = {y 0 , y 1 , ..., y n−1 } of length n over F , the periodic Hamming correlation H X,Y is defined by h[x t , y t+τ ], 0 ≤ τ < n (1) where h[a, b] = 1 if a = b and 0 otherwise, and the addition operation in the subscript is performed modulo n.
The maximum Hamming out-of-phase autocorrelation H(X) of X and the maximum Hamming crosscorrelation H(X, Y ) for two distinct FHSs X and Y are defined, respectively, by Throughout this paper, let (n, l, λ) denote an FHS X of length n over an alphabet F of size l with λ = H(X). In this case, we say that the sequence X has parameters (n, l, λ).
To estimate the measurement of a single FHS X, Lempel and Greenberger in 1974 established the first bound of H(X) as follows.  [20] For any FHS X of length n over an alphabet F of size l, we have (2) H(X) ≥ (n − n l )(n + n l − l) l(n − 1) .
The Lempel-Greenberger bound can also be rewritten by the following lemma.
Lemma 1.2. [15] For any FHS X of length n over an alphabet F of size l, H(X) ≥ 0, if n = l n/l , otherwise.
An FHS X is called optimal with respect to the Lempel-Greenberger bound if the Lempel-Greenberger bound in Lemma 1.1 or Lemma 1.2 is met with equality. Furthermore, an (n, l, λ)-FHS X is said to be near-optimal with respect to the Lempel-Greenberger bound if H(X) is bigger than the right-hand side of (2) or (3) by one.
Let S be the set of M FHSs of length n over an alphabet F of size l, the maximum Hamming correlation of S is defined by Henceforth, we use (n, M, λ; l) to denote an FHS set S containing M FHSs of length n over an alphabet F of size l with λ = H(S) , and we also say that the set S has parameters (n, M, λ; l).
In 2004, Peng and Fan developed the following bound on H(S) by taking into account the parameter M .  Recently, Xu et al. [23] provided a simplified form of the Peng-Fan bound as follows.  where a = n/l .
An FHS set S is called optimal with respect to the Peng-Fan bound if one of the Peng-Fan bounds in Lemma 1.3 or Lemma 1.4 is met with equality. Roughly speaking, an FHS set S is called near-optimal with respect to the Peng-Fan bound if H(S) is bigger than the right-hand side of (4) or (5) by one.
In general, optimal FHSs with respect to the Lempel-Greenberger bound and optimal FHS sets with respect to the Peng-Fan bound do not always exist for all lengths and alphabet sizes. However, it is a difficult problem to verify whether an optimal FHS with respect to the Lempel-Greenberger bound or an optimal FHS set with respect to the Peng-Fan bound exists for a given length and a given alphabet size. Under the circumstances, it is also valuable to construct more near-optimal FHSs or FHS sets. So far both algebraic and combinatorial constructions of such sequences were provided (see, for example, [19,15,4,16,12,13,17,14,18,5,6,9,7,28,25,26,27,22,24,3]). Based on cyclotomies, many optimal FHSs and FHS sets have been constructed. More specifically, the classical cyclotomy over a prime field F p was employed to construct optimal FHSs and FHS sets in [4] and [6]. Chung et al. [5] made a slight modification of the construction in [4] and obtained more optimal FHSs. By introducing the discrete logarithm function, Ding and Yin [13] generalized the construction in [4] to a general finite filed F p r . They used the cyclotomy over F p r to construct optimal FHSs with very flexible parameters. Later, Han and Yang [18] improved Ding and Yin's construction from the viewpoint of Sidel'nikov sequences. In 2011, Chung and Yang [7] applied k-fold cyclotomy to construct optimal FHSs and FHS sets. Later, Zeng et al. [27] presented a construction of optimal FHS sets and two constructions of optimal FHSs based on the Zeng-Cai-Tang-Yang cyclotomy. Ren et al. [22] proposed a construction of optimal FHS sets employing the technique of cyclotomy over F p r and the Chinese Remainder Theorem. Recently, Xu et al. [24] presented a construction of a class of optimal FHS sets by means of the cyclotomy on Z p 2 .
Our purpose in this paper is to design more (near-) optimal FHSs and FHS sets for some cases which are not covered in the literature based on the Zeng-Cai-Tang-Yang cyclotomy. By means of the above cyclotomy and choosing appropriate indexes and index sets, two classes of (near-) optimal FHSs with respect to the Lempel-Greenberger bound, see Constructions A and B, and two classes of (near-) optimal FHS sets with respect to the Peng-Fan bound, see Constructions C and D, are presented, where the length of FHSs can be any odd integer larger than 3. Some parameters of (near-) optimal FHSs and FHS sets constructed in [6], [7], [27] and [8] can be obtained by our construction, see Remark 2 and Remark 3. Most importantly, our constructions can generate some FHSs and FHS sets with new parameters.
The rest of this paper is organized as follows. In Section 2, we introduce the Zeng-Cai-Tang-Yang cyclotomy and discuss its general properties. In Section 3, we give two constructions of FHSs and two constructions of FHS sets. Finally, Section 4 concludes this paper.

Zeng-Cai-Tang-Yang cyclotomy and its properties
In this section we introduce the definition and some properties of the Zeng-Cai-Tang-Yang cyclotomy [27]. This cyclotomy will be used to construct more (near-) optimal FHSs and FHS sets later.
For a positive integer v, an integer a in Z * v is called a primitive root modulo v if the multiplicative order of a modulo v is equal to φ(v), where φ(v) is the Euler function. It is well known that there exists g 0 such that g 0 is a primitive root modulo p j for all j ≥ 1 [1], where p is an odd prime. For a subset H of Z v and an element a in Z v , define a + H and aH as Let v be an odd positive integer. Then v can be written as v = p m1 1 p m2 2 · · · p m k k for k odd primes p 1 , p 2 , · · · , p k with 2 < p 1 < p 2 < · · · < p k and k positive integers m 1 , m 2 , · · · , m k . For any i with 1 ≤ i ≤ k, let g i be a primitive root modulo p t i for all t ≥ 1. And let e > 1 be a common factor of p 1 − 1, p 2 − 1, · · · and p k − 1, i.e.
for k positive integers f i with 1 ≤ i ≤ k. Obviously, f 1 < f 2 < · · · < f k . By the Chinese Remainder Theorem [10], there exists a unique integer g (v) ∈ Z * v satisfying It turns out in [27] that the multiplicative order of g (v) modulo v is e and thus the set where H (v) = (h 1 , h 2 , · · · , h k ) and H I [27]. In accordance with the notation of [11], we call D In 2013, the Zeng-Cai-Tang-Yang cyclotomy was used to construct FHSs with optimal Hamming correlation. In the next section, we will further employ it to construct more (near-) optimal FHSs. To this end, we recall some necessary properties of the cyclotomic classes and cyclotomic numbers which were first introduced in [27].
v , then there exists an integer s with 0 ≤ s < e such that (9) H . By the Chinese Remainder Theorem, (9) is equivalent to By (10) and I ∈ Ψ v , the cyclotomic numbers defined in (8) have the following properties: Hence, the assertion is proved.
The following results will be needed in the sequel. Then where the second equality and the last equality are from Lemmas 2.1 and 2.4, respectively.
Proof. The following proof is basically the same as Lemma 11 in [27].
where the last equality is from Lemma 2.1.
Otherwise, there exist two integers s 1 and s 2 with 0 ≤ s 1 , s 2 < e such that (16) v Now we will distinguish the following two cases to discuss the solutions of (16).
Case 1. s 1 = s 2 . In this case, (16) is equivalent to Case 2. s 1 = s 2 . In that case, the discussion is divided into two subcases.
Case 2-1. There exist two integers r 1 and c r1 such that p according to (6) and p Hence, j l = 0, which is in contradiction with 1 ≤ j l < f 1 .
Case 2-2. There exist two integers r 2 and c r2 such that p Remark 1. Lemma 2.8 can be viewed as a generalization of Lemma 11 in [27]. In other words, if we choose , where 1 ≤ a 1 , a 2 < f 1 with a 1 = a 2 and I 0 is the π(v 1 )−dimensional all-ones vector, then Lemma 2.8 is in accord with Lemma 11 in [27]. Note that Lemma 2.8 is very important to construct more optimal FHS sets in Section 3, see the proofs of Constructions C and D for details.
The following result follows directly from Lemmas 2.7 and 2.8, which will be useful in this paper. Lemma 2.9. For any element a ∈ Z v \ {0} and J = (j 1 , j 2 , · · · , j k )

More optimal FHSs and FHS sets based on the Zeng-Cai-Tang-Yang cyclotomy
In this section, we will propose two constructions of FHSs and two constructions of FHS sets based on the Zeng-Cai-Tang-Yang cyclotomy in Section 2. Before presenting our construction, we will give some necessary notation below.
Define a set A as Obviously, |A| = v−1 e + 1 since |D Define a set U as and a set V as It is easy to prove that |U | = f 1 /2 (f 1 −1) k−1 and there exist (f 1 !) k−1 V s satisfying the above definitions. Furthermore, there is the following special case: where I 0 is the k−dimensional all-ones vector. From now on, denote the first component of a vector I by Λ(I).
Construction A : Let the set U defined as above and I ∈ U . And let Theorem 3.1. If e is odd, p i ≡ 3 (mod 4) for 1 ≤ i ≤ k and I ∈ U , then the FHS X (I) generated by Construction A is optimal when v is composite or v = ef + 1 is a prime with v ≡ 3 (mod 4) and 4e ≤ f + 4.
Proof. The Hamming out-of-phase autocorrelation H X (I) (τ ) of X (I) at shift τ = 0 is given by where the last equality is from Lemma 2.6.
v2 . By Lemmas 2.1, 2.3 and 2.8, we have where the last equality is from Lemma 2.4.
The first element ϕ({0}) of X (I) in Construction A may be replaced by ϕ(D (v) ). Then, the alphabet size and the maximum Hamming out-of-phase autocorrelation are slightly changed.
Construction B : Let the set U defined as above and I ∈ U . And let Y (I) = {y defined by Theorem 3.3. If e is odd, p i ≡ 3 (mod 4) for 1 ≤ i ≤ k and I ∈ U , then the FHS Y (I) generated by Construction B is optimal when v is composite or v = ef + 1 is a prime with v ≡ 3 (mod 4) and I = f 2 , otherwise near-optimal. Proof. The Hamming out-of-phase autocorrelation H Y (I) (τ ) of Y (I) at shift τ = 0 is easily calculated by adding ∆ to H X (I) (τ ) in the proof of Theorem 3.1, where If v = ef + 1 is an odd prime with v ≡ 3 (mod 4), then we have for I = f 2 . Therefore, Construction B gives optimal (v, v−1 2e , 2e)-FHSs with respect to the Lempel-Greenberger bound if I = f 2 , while Y (I) is a near-optimal (v, v−1 2e , 2e+ 1)-FHS with respect to the Lempel-Greenberger bound.
If v is composite, then (2) If v is an odd prime with v ≡ 3 (mod 4) and I = 2a + 1, then Construction A and Construction B are exactly the same as the second construction and first construction in [6], respectively.
(3) If v = p 1 p 2 · · · p k for k different odd primes p i with p i ≡ 3 (mod 4), then the FHSs generated by Construction B share the same parameters with those generated in Construction B of [7] by applying k-fold cyclotomy. (4) If v = p m1 1 p m2 2 · · · p m k k for k different odd primes p i with p i ≡ 3 (mod 4) and I = yI 0 , where I 0 is the k−dimensional all-ones vector, then Construction B is exactly the same as Construction C in [27]. In other words, compared with Construction C in [27], Construction B in our paper derives more optimal FHSs by choosing appropriate index I from the set U , which can be seen from Example 3.2. (5) Compared with Construction C in [27], Construction A slightly changes the alphabet size and derives more optimal FHSs with respect to the Lempel-Greenberger bound which are not covered in the literature.
v1 . Theorem 3.4. Let the set V defined as above and f 1 > 1. Then the FHS set S has the following properties: (1) The family size is M = f 1 , the sequence length n = v, and |F | = v−1 e + 1; (2) The Hamming out-of-phase autocorrelation of S (I) ∈ S for any I ∈ V is given by H S (I) (τ ) = e − 1; (3) The Hamming crosscorrelation between any two distinct FHSs S (I) , S (J) ∈ S with I, J ∈ V is given by Proof.
(2) The Hamming out-of-phase autocorrelation of S (I) at shift τ with τ = 0 is where the last equality comes from Lemma 2.6.
(3) For any FHSs S (I) , S (J) ∈ S with I = J and I, J ∈ V , their Hamming crosscorrelation at shift τ = 0 is given by where the last equality comes from the definition of V and Lemma 2.9. If τ = 0, then it is obvious that H S (I) ,S (J) (τ ) = 1.
Corollary 3.5. If f 1 > 1, then the FHS set S generated by Construction C is optimal with respect to the Peng-Fan bound when v is composite or v = ef + 1 is prime with f ≥ e > 1.
Proof. If f 1 > 1 and v is composite, by Theorem 3.4, then the set S has parameters (v, f 1 , e; v−1 e + 1). Note that a = n l = v v−1 e +1 = e − 1 and Considering the facts p 1 < p 2 < · · · < p k , p r = ef r + 1 for 1 ≤ r ≤ k and v is composite, we have v ≥ p 2 1 , which results in Therefore, by Lemma 1.4, the FHS set S with parameters (v, f 1 , e; v−1 e + 1) is optimal with respect to the Peng-Fan bound when v is composite.
With a slight modification of the above proof, we can get the proof when v = ef + 1 is prime with f ≥ e > 1. The proof is complete. Corollary 3.6. Each FHS in the S generated by Construction C is optimal with respect to the Lempel-Greenberger bound when v is composite or v = ef + 1 is a prime with f ≥ e > 1.
Proof. We only give the proof when v is a prime with f ≥ e > 1 since the other is completely parallel. When v is a prime, each FHS generated by our construction has parameters (v, f +1, e−1) from the proof of Theorem 3.4. According to Lemma 1.2, the fact v = (e − 1)(f + 1) + f + 2 − e implies that each FHS is optimal with respect to the Lempel-Greenberger bound.
The first element ϕ({0}) of S (I) in Construction C may be replaced by ϕ(D (v) ). Then, the alphabet size and the maximum Hamming correlation value are slightly changed. Construction Theorem 3.8. Let E be defined as Lemma 2.1 and the set V defined as above.
Then the FHS set Z has the following properties: (1) The family size is M = f 1 , the sequence length n = v, and |F | = v−1 e ; (2) The Hamming out-of-phase autocorrelation of Z (I) ∈ Z for any I ∈ V is given by (2.1) When e is odd, (2.2) When e is even, (3) The Hamming crosscorrelation between any two distinct FHSs Z (I) , Z (J) ∈ Z with I, J ∈ V is given by (3.1) When v = ef + 1 is prime and e is odd, (3.2) When v is composite and e is odd, otherwise.
(2) The Hamming out-of-phase autocorrelation of Z (I) at shift τ with τ = 0 is I | where the last equality comes from Lemma 2.6. Then by Lemma 2.2, the results follow.
(3) For any FHSs Z (I) , Z (J) ∈ Z with I = J and I, J ∈ V , their Hamming crosscorrelation at shift τ with τ = 0 is given by where the last equality comes from the definition of V and Lemma 2.9. Combining Lemma 2.2, the statements follow. Corollary 3.9. If f 1 > 1, then the FHS set Z generated by Construction D is near-optimal with respect to the Peng-Fan bound when v is composite or v = ef + 1 is prime with 2 e, and optimal with respect to the Peng-Fan bound when v = ef + 1 is prime with 2 | e.
Proof. If f 1 > 1 and v is composite, by Theorem 3.8, then the set Z has parameters (v, f 1 , e + 1; v−1 e ). Note that a = n l = e and Considering the facts p 1 < p 2 < · · · < p k , p r = ef r + 1 for 1 ≤ r ≤ k and v is composite, we have v ≥ p 2 1 , which results in Therefore, by Lemma 1.4, the FHS set Z with parameters (v, f 1 , e + 1; v−1 e ) is near-optimal with respect to the Peng-Fan bound when v is composite.
With a slight modification of the above proof, we can get the proof when v = ef + 1 is prime. The proof is complete. Corollary 3.10. Each FHS in the Z generated by Construction D is optimal with respect to the Lempel-Greenberger bound when e is odd, and near-optimal with respect to the Lempel-Greenberger bound when e is even.
Proof. We only give the proof when e is odd since the other is completely parallel.
When e is odd, each FHS generated by our construction has parameters (v, v−1 e , e) from the proof of Theorem 3.8. According to Lemma 1.2, the fact v = e v−1 e + 1 implies that each FHS is optimal with respect to the Lempel-Greenberger bound. Therefore, Z is a near-optimal FHS set with parameters (175, 2, 3; 87) and each FHS in Z is a near-optimal FHS with parameters (175, 87, 3). (2) When v is an odd prime, the FHS sets generated by Construction D share the same parameters with those generated in Construction A of [8] by applying the classical cyclotomy. Furthermore, if v = p 1 p 2 · · · p k for k different odd primes p i , then the FHS sets generated by Construction D share the same parameters with those generated in Construction A 1 of [7] by applying k-fold cyclotomy. Yet, the length of the FHSs obtained by our construction can be any odd integer larger than 3.
(3) Based on the Zeng-Cai-Tang-Yang cyclotomy, Zeng et al. [27] obtained a class of optimal FHSs with parameters (v, v−1 e , e). Obviously, the FHS S ( 0) in our construction is equivalent to the FHS W constructed in Construction B of [27], where 0 is the k−dimensional all-zeros vector. (4) If v = p m1 1 p m2 2 · · · p m k k for k different odd primes p i and V = {iI 0 : 0 ≤ i < f 1 }, where I 0 is the k−dimensional all-ones vector, then Construction C is exactly the same as Construction A in [27]. In other words, compared with Construction A in [27], Construction C in our paper derives more optimal FHS sets by choosing appropriate index set V . (5) Compared with Construction A in [27], Construction D slightly changes the alphabet size and obtained a lot of (near-) optimal FHSs and FHS sets which are not covered in the literature by choosing the appropriate set V .

Concluding remarks
Based on the Zeng-Cai-Tang-Yang cyclotomy, we present two constructions of FHSs and two constructions of FHS sets, where the length of sequences can be any odd integer larger than 3. The results show that the FHSs and FHS sets generated by our construction are optimal and (near-) optimal with respect to the Lempel-Greenberger bound and Peng-Fan bound, respectively. By choosing appropriate indexes and index sets, many new (near-) optimal FHSs and FHS sets can be obtained by our construction.