New families of Strictly optimal Frequency hopping sequence sets

Frequency hopping sequences (FHSs) with favorable partial Hamming correlation properties have important applications in many synchronization and multiple-access systems. In this paper, we investigate constructions of FHS sets with optimal partial Hamming correlation. We present several direct constructions for balanced nested cyclic difference packings (BNCDPs) and balanced nested cyclic relative difference packings (BNCRDPs) such that both of them have a special property by using trace functions and discrete logarithm. We also show three recursive constructions for FHS sets with partial Hamming correlation, which are based on cyclic difference matrices and discrete logarithm. Combing these BNCDPs, BNCRDPs and three recursive constructions, we obtain infinitely many new strictly optimal FHS sets with respect to the Peng-Fan bounds.


I. INTRODUCTION
Frequency hopping (FH) multiple-access is widely used in the modern communication systems such as ultrawideband (UWB), military communications, Bluetooth and so on, for example, [3], [17], [24].In FH multiple-access communication systems, frequency hopping sequences are employed to specify the frequency on which each sender transmits a message at any given time.An important component of FH spread-spectrum systems is a family of sequences having good correlation properties for sequence length over suitable number of available frequencies.The optimality of correlation properties is usually measured according to the well-known Lempel-Greenberger bound [21] and Peng-Fan bounds [22].During these decades, many algebraic or combinatorial constructions for FHSs or FHS sets meeting these bounds have been proposed, see [2], [6]- [10], [12]- [14], [18], [19]- [20], [25]- [28], and the references therein.
Compared with the traditional periodic Hamming correlation, the partial Hamming correlation of FHSs is much less well studied.Nevertheless, in many application scenarios where the synchronization time is limited or the hardware is complex [15], the length of a correlation window should be much shorter than the period of the chosen FHSs [15].Therefore, the partial Hamming correlation, rather than the periodic Hamming correlation, will paly a major role in determining the performance.
In recent years, a little progress on the study of the partial Hamming correlation of FHSs has been made.In 2004, Eun et al. [15] generalized the Lempel-Greenberger bound on the periodic Hamming autocorrelation to the case of partial Hamming autocorrelation, and obtained a class of FHSs with optimal partial autocorrelation [23].In 2012, Zhou et al. [29] extended the Peng-Fan bounds on the periodic Hamming correlation of FHS sets to the case of partial Hamming correlation.Based on the so-called array structure, Zhou et al. [29] constructed both individual FHSs and FHS sets with optimal partial Hamming correlation.In 2014, Cai et al. [4] improved lower bounds on the partial Hamming correlation of FHSs and FHS sets, and based on generalized cyclotomy, they constructed FHS sets with optimal partial Hamming correlation.Very recently, Cai et al. [5] derived upper bounds on the family sizes of FHS sets with respect to partial Hamming correlation from some classical bounds on error-correcting codes, and they presented strictly optimal FHS sets having optimal family sizes with respect to one of the new bounds.Fan et al. [16] established a generic connection between strictly optimal FHSs and disjoint cyclic perfect Mendelsohn difference families.Bao et al. [1] established a correspondence between FHS sets with optimal partial Hamming correlation and multiple partition-type balanced nested cyclic difference packings with a special property.By virtue of this correspondence, they obtained some classes of strictly optimal FHSs and FHS sets by cyclotomic classes, they presented two recursive constructions for strictly optimal FHS sets and they also yielded some classes of strictly optimal FHS sets by these recursive constructions.
In this paper, we present some constructions for FHS sets with optimal partial Hamming correlation.First of all, We present several direct constructions for BNCDPs and BNCRDPs such that both of them have a special property by using trace functions and discrete logarithm.Next, we present three recursive constructions for FHS sets with partial Hamming correlation.Combing these BNCDPs, BNCRDPs and three recursive constructions, we yield infinitely many strictly optimal FHS sets with new and flexible parameters not covered in the literature.The parameters of FHS sets with optimal partial Hamming correlation from the known results and the new ones are listed in the Table I and Table II, respectively.
The outline of the paper is as follows.Section II introduces the known bounds on the partial Hamming correlation of FHSs and FHS sets.Section III presents several direct constructions for BNCDPs and BNCRDPs such that both of them have a special property by using trace functions and discrete logarithm.Section IV presents three recursive constructions of FHS sets with partial Hamming correlation.Section V concludes this paper with some remarks.

II. LOWER BOUNDS ON THE PARTIAL HAMMING CORRELATION OF FHSS AND FHS SETS
In this section, we introduce some known lower bounds on the partial Hamming correlation of FHSs and FHS sets.
For any positive integer l ≥ 2, let F = {f 0 , f 1 , . . ., f l−1 } be a set of l available frequencies, also called an alphabet.A sequence X = {x(t)} n−1 t=0 is called a frequency hopping sequence (FHS) of length n over F if v ≥ p1e r and gcd(w, e) = 1 q is a prime power and p is a prime; v is an integer with prime factor decomposition e, f are integers such that e > 1 and e|gcd(p 1 − 1, p 2 − 1, . . ., p s − 1), and f = p1−1 e ; w is an integer with prime factor decomposition w = q n1 1 q n2 2 • • • q nt t with q 1 < q 2 < • • • < q t ; r is an integer such that r > 1 and r|gcd(e, q 1 − 1, q 2 − 1, . . ., q t − 1); d, m are positive integers.
x(t) ∈ F for all 0 ≤ t ≤ n − 1.For any two FHSs X = {x(t)} n−1 t=0 and Y = {y(t)} n−1 t=0 of length n over F , the partial Hamming correlation function of X and Y for a correlation window length L starting at j is defined by where τ, L, j are integers with and q1 > p m , or v is a prime with f ≥ e, Corollary 4.5 Corollary 4.11 u1, u2, . . .us are positive integers such that u1 ≤ u2 ≤ . . .≤ us ; q, q ′ are prime powers; v is an integer with prime factor decomposition e is an integer such that e|gcd(p1 − 1, p2 − 1, . . ., ps − 1), and f = p 1 −1 e ; w is any an integer with prime factor decomposition w and For any FHS of length n over an alphabet of size l and each window length L with 1 ≤ L ≤ n, Eun et al. [15] derived a lower bound, which is a generalization of the Lempel-Greenberger bound [21].Recently, such a lower bound was improved by Cai et al. [4].
Let S be a set of M FHSs of length n over an alphabet F of size l.For any given correlation window length L, the maximum nontrivial partial Hamming correlation H(S; L) of the sequence set S is defined by Throughout this paper, we use (n, M, λ; l) to denote a set S of M FHSs of length n over an alphabet F of size l, where λ = H(S; n), and we use (n, λ; l) to denote an FHS X of length n over an alphabet F of size l, where λ = H(X; n).
When L = n, Peng and Fan [22]   Remark: By the Lemma 2.2, we have the two Peng-Fan bounds are identical.
Recall that the correlation window length may change from case to case according to the channel conditions in practical systems.Hence, it is very desirable that the involved FHS sets have optimal partial Hamming correlation for any window length.Cai et al. gave the following definition of strictly optimal FHS sets in [4].
Definition 2.3: An FHS set S is said to be strictly optimal or an FHS set with optimal partial Hamming correlation if the bounds in Lemma 2.1 is met for an arbitrary correlation window length L with 1 ≤ L ≤ n.
When L = n, the bounds in Lemma 2.1 are exactly the Peng-Fan bounds.It turns out that each strictly optimal FHS set is also optimal with respect to the Peng-Fan bounds, but not vice versa.

A. A combinatorial characterization of Strictly optimal FHS sets
In 2009, Ge et al. [20] revealed a connection between FHS sets and families of partition-type balanced nested cyclic difference packings. Set A, where is the multiset union, i.e., the multiset m • A contains m copies of each element of A. Let A, D be subsets of Z v , we define ∆(D) ≤ m • A, i.e., the multiset ∆(D) contains each element of A at most m times and no elements of Z v \ A occur.Let Tr q m /q (x) be the trace function from F q m to F q .
Let A, B be two subsets of Z v .The list of external difference of ordered pair (A, B) is the multiset Note that the list of external difference ∆ E (A, B) may contain zero.For any residue τ ∈ Z v , the number of Let B j , 0 ≤ j ≤ M − 1, be a collection of l subsets B j 0 , . . ., B j l−1 of Z n , respectively.The list of external difference of ordered pair (B j , B j ′ ), 0 ≤ j = j ′ < M , is the union of multisets If each B j is an (n, K j , λ)-CDP of size l, and ∆ E (B j , B j ′ ) contains each residue of Z n at most λ times for 0 ≤ j = j ′ < M , then the set {B 0 , . . ., B M−1 } of CDPs is said to be balanced nested with index λ and denoted by (n, {K 0 , . . ., K M−1 }, λ)-BNCDP.If each B j is a partition-type CDP for 0 ≤ j < M , then the (n, {K 0 , . . ., K M−1 }, λ)-BNCDP is called partition-type.For convenience, the number l of the base blocks in B j is also said to be the size of the BNCDP.
In 2009, Ge et al. [20] revealed a connection between FHS sets and partition-type BNCDPs as follows.
For a u-tuple T = (a 0 , a 1 , . . ., a u−1 ) over Z n , the multiset ∆ i (T ) = {a j+i − a j : 0 ≤ j ≤ u − 1} is called i-apart difference list of the tuple, where 1 ≤ i ≤ u, j + i is reduced modulo u, and a j+i − a j is taken as the least positive residue modulo n.
Let B = {B X : X ∈ S} be a family of M partition-type CDPs of size l over Z n , where In 2016, Bao et al. [1] revealed a connection between strictly optimal FHS sets and partition-type BNCDPs with a special property as follows.

Theorem 3.2: ([1]
) There is a strictly optimal (n, M, λ; l)-FHS set S with respect to the Peng-Fan bounds if and only if there exists a partition-type (n, {K X : Let B j be an (mg, g, K j , 1)-CRDP over Z mg for 0 ≤ j < M , where contains each element of Z mg \ mZ mg at most once and no elements of mZ g occur for 0 ≤ j = j ′ < M .For convenience, the number u of the base blocks in B j is also said to be the size of the BNCRDP.
One importance of a BNCRDP is that we can put an appropriate BNCRDP on its subgroup to derive a new BNCRDP.
Lemma 3.3: Suppose that there exists an (mg such that all elements of base blocks of B j , together with 0, g 2 , . . ., ( mg1 s − 1)g 2 , form a complete system of representatives for the cosets of mg1g2 s Z mg1g2 in Z mg1g2 for 0 ≤ j < M , where s|m.If there exists an , such that all elements of base blocks of B ′ j , together with 0, g 1 , . . ., ( m s − 1)g 1 , form a complete system of representatives for the cosets of mg1 s Z mg1 in Z mg1 for 0 ≤ j < M .Then there exists an (mg such that all elements of base blocks of A j , together with 0, g 1 g 2 , . . ., ( m s − 1)g 1 g 2 , form a complete system of representatives for the cosets of mg1g2 s Z mg1g2 in Z mg1g2 for 0 ≤ j < M .

Proof:
that all elements of base blocks of A i , together with 0, g 1 g 2 , . . ., ( m s − 1)g 1 g 2 , form a complete system of representatives for the cosets of mg1g2 s we have that It follows that all elements of base blocks of A i , together with 0, g 1 g 2 , . . ., ( m s − 1)g 1 g 2 , form a complete system of representatives for the cosets of mg1g2 s Then, Hence, at most once.This completes the proof.

Lemma 3.4: ([1]
) Suppose that there exists an (mg, g, {K 0 , . . ., K M−1 }, 1)-BNCRDP of size u, {B 0 , . . ., B M−1 }, such that all elements of base blocks of B j , together with 0, m, . . ., (s − 1)m, form a complete system of representatives for the cosets of smZ mg in Z mg for 0 ≤ j < M , where s|g.If there exists a partition-type Then there exists a partition-type When we replace the BNCRDP in Lemma 3.4 with a (g, {K 0 , K 1 , . . ., K M−1 }, 1)-BNCDP such that all elements of base blocks of each CDP form a complete system of representatives for the cosets of sZ g in Z g where s|g, the same procedure yields a new partition-type BNCDP.This proof is similar to that of Lemma 4.4 in [1].Lemma 3.5: Suppose that there exists a (g, {K 0 , . . ., K M−1 }, 1)-BNCDP of size u, B, such that all elements of base blocks of each CDP form a complete system of representatives for the cosets of sZ g in Z g where s|g.Then there exists a partition-type (g, {K 0 , . . ., Since all elements of base blocks of B i form a complete system of representatives for the cosets of sZ g in Z g , we have that D i is a partition of Z g . of the form (a 0 , a 0 +s, a 0 +2s, . . ., a 0 +g−s); otherwise Similarly, it is readily checked that d

B. Some direct constructions of BNCDPs and BNCRDPs
In this subsection, we give several direct constructions for BNCDPs and BNCRDPs such that both of them have a special property.
Let d, m be positive integers such that m ≥ 2 and gcd(m, d) = 1, q a prime power such that d|q − 1.Let α be a primitive element of F q m , set g = α d and β = α q m −1 q−1 .Clearly, F * q = {β i : 0 ≤ i < q − 1}.Identifying F q m with the m-dimensional F q -vector space F m q , then each element in F q m can be viewed as a vector over F q .Let a 1 , a 2 , . . ., a m−1 ∈ F q m be m − 1 linearly independent elements over F q .Define a sequence where 0 ≤ t < q m −1 d and 0 ≤ i < d.
In 2012, Zhou et al. [29] obtained the following result based on this construction.Then there is a strictly optimal ( q m −1 d , d, q−1 d ; q m−1 )-FHS set S, and H(S; L) = L(q−1) By Theorem 3.2 and Theorem 3.6, where and 0 ≤ j < q−1 d .From the proof of Theorem 2 in [29], we have that A i − → 0 = ∅ for each i, 0 ≤ i < d.Hence, there exist an element , where G − → x = {g ′− → x : g ′ ∈ G}.Denote by R a system of distinct representatives for the equivalence } and For , we have |{t : Tr q m /q (α i a1g t ) = b1, Tr q m /q (α i a2g t ) = b2, . . ., Tr q m /q (α i am−1g t ) = bm−1, 0 ≤ t < q m −1 d }| = |{t : Tr q m /q (a1α t ) = b1, Tr q m /q (a2α t ) = b2, . . ., Tr q m /q (am−1α t ) = bm−1, 0 ≤ t < q m − 1}|. Then, q − 1 otherwise. ( Theorem 3.7: Let q be a prime power, d, m positive integers such that m ≥ 2, d|q − 1 and gcd(m, d) = 1.Let q−1 such that all elements of base blocks of each CRDP, together with 0, form a complete system of representatives for the cosets of q m −1 In view of equality (4), it holds that for 0 ≤ i < d.Then, It follows that all elements of base blocks of A ′ i , together with 0, form a complete system of representatives for the cosets of q m −1 In view of equality (4), we get where G = {1, β d , . . ., β q−1−d }.By Theorem 3.2, we have

This completes the proof
Starting with a ( q m −1 d , q−1 d , {K 0 , . . ., K d−1 }, 1)-BNCRDP in Theorem 3.8.By adding a block {0} to each CRDP, we obtain the following ( q m −1 d , {K 0 , . . ., K d−1 }, 1)-BNCDP.Corollary 3.8: Let q be a prime power, d, m positive integers such that d|q − 1, m ≥ 2 and gcd(m, d) = 1.Then there exists a ( q m −1 d , {K ′ 0 , . . ., K ′ d−1 }, 1)-BNCDP of size d(q m−1 −1) q−1 + 1 such that all elements of base blocks of each CDP form a complete system of representatives for the cosets of q m −1 Construction B Let q be a prime power and let α be a primitive element.Using the discrete logarithm in F * q , define a function from F * q to Z q−1 as ǫ(x) = log α (x).
Let p be a prime and let m be an integer with m > 1.Let α be a primitive element of F p m and denote Lemma 3.9: Let p be a prime and let m be an integer with m > 1.Let A x be defined in Construction B. Then Then, According to the values of x, x ′ and (c, d), we distinguish six cases.
Case 1: x ′ = x, c = 0 and d = 0. Since |{z + i : 0 ≤ i < p} ∩ R| = 1 for each z ∈ F p m , it holds that Case 2: x ′ = x, c = 0 and d = 0.In this case α c − 1 = 0. Then Case 3: x ′ = x, c = 0 and d = 0.In this case α c = 1.Then Case 4: x ′ = x, c = 0 and d = 0.In this case α c = 1, there exists a unique element a c ∈ Z p m −1 such that Case 5: x ′ = x and c = 0.In this case α c = 1.Since x, x ′ ∈ R, we have d = x − x ′ .Then Case 6: x ′ = x and c = 0.In this case α c = 1.Since x, x ′ ∈ R, we have d = x − x ′ .Then there exists a unique In summary, the discussion in the six cases above shows that Theorem 3.10: Let p be a prime and let m be an integer with m > 1.Let A x be defined in Construction A.
Then {A x : x ∈ R} is a (p(p m − 1), {K 0 , . . ., K p m−1 −1 }, 1)-BNCDP of size p m−1 such that all elements of base blocks of each CDP form a complete system of representatives for the cosets of where Proof: By Lemma 3.9, A x is a (p(p m − 1), {K 0 , . . ., K p m−1 −1 }, 1)-BNCDP.It is left to show that all elements of base blocks of each CDP form a complete system of representatives for the cosets of (p m − 1)Z p(p m −1) in Z p(p m −1) .
Since |{z + i : 0 ≤ i < p} ∩ R| = 1 for each z ∈ F p m , it holds that This completes the proof.
Applying Lemma 3. Comparing with their proof, ours seems simpler.
Let A x be block sets defined in construction B, set B x = {A x x \ {(0, p − 1)} : A x x ∈ A x } for each x ∈ R. Since gcd(p, p m − 1) = 1, we have that Z p(p m −1) is isomorphic to Z p × Z p m −1 .By using Lemma 3.9 and Theorem 3.10, we obtain the following (p(p m − 1), p, {K 0 , . . ., K p m−1 −1 }, 1)-NBCRDP.Corollary 3.12: Let p be a prime and let m be an integer with m > 1.Then there exists a (p(p m −1), p, {K 0 , . . ., K p m−1 −1 }, 1)-NBCRDP of size p m−1 such that all elements of base blocks of each CRDP, together with 0, form a complete system of representatives for the cosets of (p m − 1)Z p(p m −1) in Z p(p m −1) where K 0 = . . .such that all elements of base blocks of each CRDP, together with 0, form a complete system of representatives for the cosets of vZ uv in Z uv where Construction C Starting with a (uv, u, {K 0 , . . ., K f −1 }, 1)-BNCRDP in Lemma 3.13 where By adding a block {0} to each CRDP, we obtain the following corollary.
Corollary 3.14: Let v be a positive integer of the form v = p m1 1 p m2 2 • • • p ms s for s positive integers m 1 , m 2 , . . ., m s and s distinct primes p 1 , p 2 , . . ., p s .Let u be a positive integer such that gcd(u, v) = 1.Let e be a common factor of u, p 1 − 1, p 2 − 1, . . ., p s − 1 and e > 1, and let f = min{ pi−1 e : 1 ≤ i ≤ s}.Then there exists a (uv, {K 0 , . . ., K f −1 }, 1)-BNCDP of size v−1 e + 1 such that all elements of base blocks of each CDP form a complete system of representatives for the cosets of vZ uv in Z uv where

A. Construction based on difference matrices
In this subsection, three recursive constructions are used to construct strictly optimal individual FHSs and FHS sets.The first recursive construction is based on the cyclic difference matrix (CDM).
A (w, t, 1)-CDM is a t × w matrix D = (d ij ) (0 ≤ i ≤ t − 1, 0 ≤ j ≤ w − 1) with entries from Z w such that, for any two distinct rows R r and R h , the vector difference R h − R r contains every residue of Z w exactly once.It is easy to see that the property of a difference matrix is preserved even if we add any element of Z w to all entries in any row or column of the difference matrix.Then, without loss of generality, we can assume that all entries in the first row are zero.Such a difference matrix is said to be normalized.The (w, t − 1, 1)-CDM obtained from a normalized (w, t, 1)-CDM by deleting the first row is said to be homogeneous.The existence of a homogeneous (w, t − 1, 1)-CDM is equivalent to that of a (w, t, 1)-CDM.Observe that difference matrices have been extensively studied.A large number of known (w, t, 1)-CDMs are well documented in [11].In particular, the multiplication table of the prime field Z p is a (p, p, 1)-CDM.By using the usual product construction of CDMs, we have the following existence result.Lemma 4.1: ( [11]) Let w and t be integers with w ≥ t ≥ 3.If w is odd and the least prime factor of w is not less than t, then there exists a (w, t, 1)-CDM.
When we replace the BNCRDP in Theorem 5.2 in [1] with a (g, {K 0 , K 1 , . . ., K M−1 }, 1)-BNCDP such that all elements of base blocks of each CDP form a complete system of representatives for the cosets of sZ g in Z g where s|g, the same procedure yields a new partition-type BNCDP.This proof is similar to that of Theorem 5.2 in [1].Theorem 4.2: Assume that {B 0 , . . ., B M−1 } is a (g, {K 0 , K 1 , . . ., K M−1 }, 1)-BNCDP of size u such that all elements of base blocks of B j form a complete system of representatives for the cosets of sZ g in Z g for 0 ≤ j < M , where s|g and B j = {B j 0 , B j 1 , . . ., B j u−1 }.If there exists a homogeneous (w, t, 1)-CDM over Z w with t = max k |} and gcd(w, g s ) = 1, then there also exists a (gw, {K 0 , K 1 , . . ., K M−1 }, 1)-BNCDP of size uw, B ′ = {B ′ 0 , . . ., B ′ M−1 } such that all elements of base blocks of B ′ j form a complete system of representatives for the cosets of swZ gw in Z gw for 0 ≤ j < M .
Since all elements of base blocks of B j , form a complete system of representatives for the cosets of sZ g in Z g , we have that 0≤i<u B j i ≡ {0, 1, 2, . . ., s − 1} (mod s) and as desired.
Secondly, we show that each B ′ j is a (gw, K j , 1)-CDP.Since B j is a (g, K j , 1)-CRDP, we have ∆(B j ) ⊂ Z g \{0}.Simple computation shows that Therefore, B ′ is the required BNCDP.This completes the proof.
Combining Theorem 4.2, Lemma 3.5 and Corollary 3.8 together, we obtain the following corollary.Let w be an odd integer whose the least prime factor is greater than q.Then there exists a strictly optimal ( w(q m −1) d , d, q−1 d ; (q m−1 − 1 + q−1 d )w)-FHS set with respect to the Peng-Fan bounds.Proof: By using Corollary 3.8, there exists a ( q m −1 d , {K 0 , . . ., K d−1 }, 1)-BNCDP of size d(q m−1 −1) q−1 + 1 such that all elements of base blocks of each CDP form a complete system of representatives for the cosets of q m −1 q−1 Z q m −1 . Since w is an odd integer whose the least prime factor is greater than q, there exists a homogeneous (w, q, 1)-CDM over Z w .Since the least prime factor of w is greater than q > q−1 d , we have gcd(w, q−1 d ) = 1.In view of equation ( 5), we have max |B j k |} = q.By Theorem 4.2 with g = q m −1 d and s = q m −1 q−1 yields a ( w(q m −1) d , {K 0 , . . ., K d−1 }, 1)-BNCDP of size ( d(q m−1 −1) q−1 + 1)w such that all elements of base blocks of each CDP form a complete system of representatives for the cosets of w(q m −1)
Combining Theorem 4.2, Lemma 3.5 and Theorem 3.10 together, we get the following corollary.
Corollary 4.4: Let p be a prime and let m be an integer with m > 1.Let w be an odd integer whose the least prime factor is greater than p m .Then there exists a strictly optimal (wp(p m − 1), p m−1 , p; p m w)-FHS set with respect to the Peng-Fan bounds.
Proof: By using Theorem 3.10, there exists a (p(p m − 1), {K 0 , . . ., K p m−1 −1 }, 1)-BNCDP of size p m−1 such that all elements of base blocks of each CDP form a complete system of representatives for the cosets of Since w is an odd integer whose the least prime factor is greater than p m , there exists a homogeneous (w, p m , 1)-CDM over Z w .Since the least prime factor of w is greater than p m , we have gcd(w, p) = 1.Since e : 1 ≤ i ≤ s}.Let w be an odd integer whose the least prime factor is greater than p 1 − 1.If v is not a prime with f > 1 or v is a prime with f ≥ e, then there exists a strictly optimal (ewv, f, e; (v − 1 + e)w)-FHS set with respect to the Peng-Fan bounds.
Proof: By Corollary 3.14 with u = e, there exists an (ev, {K 0 , . . ., K f −1 }, 1)-BNCDP of size v−1 e + 1 such that all elements of base blocks of each CDP form a complete system of representatives for the cosets of vZ ev in Since w is an odd integer whose the least prime factor is greater than p 1 − 1, there exists a homogeneous (w, p 1 − 1, 1)-CDM over Z w .Since the least prime factor of w is greater than If v is not a prime, we have v ≥ p 2  1 > e 2 f 2 and Then, it holds that If v is a prime with f ≥ e, we have v = ef + 1 and Since e > 1, f ≥ e and w ≥ p 1 − 1 = ef , we have By Theorem 3.2, D is a strictly optimal (ewv, f, e; (v − 1 + e)w)-FHS set with respect to the Peng-Fan bounds.
This completes the proof.

B. Constructions based on discrete logarithm
Now, we present two recursive constructions for strictly optimal frequency hopping seuqences based on discrete logarithm.
ǫ(x) is defined in Section III-B.
Lemma 4.6: [] Let q be a prime power.Let a, b ∈ F q be two distinct elements, then Theorem 4.7: Assume that {B 0 , . . ., B M−1 } is an (mg, g, {K 0 , K 1 , . . ., K M−1 }, 1)-BNCRDP of size u such that all elements of base blocks of B j , together with 0, m, . . ., (s − 1)m, form a complete system of representatives for the cosets of smZ mg in Z mg for 0 ≤ j < M , where s|g and B j = {B j 0 , B j 1 , . . ., B j u−1 }.Let q be a prime power such that q ≥ max 0≤k<u { M−1 j=0 |B j k |} and gcd(q − 1, g s ) = 1, then there also exists an (mg(q − 1), g(q − 1), such that all elements of base blocks of B ′ j , together with 0, m, . . ., (s(q − 1) − 1)m, form a complete system of representatives for the cosets of sm(q − 1)Z mg(q−1) in Z mg(q−1) for 0 ≤ j < M where K |B j i | distinct elements x i,j,c ∈ F q (0 ≤ j < M and 1 ≤ c ≤ |B j i |) and set η i (a i,j,c ) = x i,j,c .
Clearly, η i (a i,b,c ) = η i (a i,b ′ ,c ′ ) if and only if (b, c) = (b ′ , c ′ ) for each i.
By Theorem 3.2, D is a strictly optimal ((q − 1)p(p m − 1), p m−1 , p; p m q)-FHS set with respect to the Peng-Fan bounds.This completes the proof.

Dj a = as for 1
≤ a ≤ g s .Therefore, d D a = min{ min 0≤i =j<M {d (Di,Dj ) a }, min 0≤j<M {d Dj a }} = as for 1 ≤ i ≤ g s , and D is the required BNCDP.This completes the proof.
5 and Theorem 3.2, we obtain the following corollary.In 2016, Cai et al. [5] obtained a (p(p m − 1), p m−1 , p; p m )-FHS set.It is easy to check that the (p(p m − 1), p m−1 , p; p m )-FHS set are strictly optimal with respect to the Peng-Fan bounds.Corollary 3.11: ([5]) Let p be a prime and let m be an integer with m > 1.Then there exists a strictly optimal (p(p m − 1), p m−1 , p; p m )-FHS set with respect to the Peng-Fan bounds over the alphabet F p m .Remark: In 2016, Cai et al. have constructed such a strictly optimal (p(p m − 1), p m−1 , p; p m )-FHS set [5].