THREE CLASSES OF PARTITIONED DIFFERENCE FAMILIES AND THEIR OPTIMAL CONSTANT COMPOSITION CODES

. Cyclotomy, ﬁrstly introduced by Gauss, is an important topic in Mathematics since it has a number of applications in number theory, combi- natorics, coding theory and cryptography. Depending on v prime or com-posite, cyclotomy on a residue class ring Z v can be divided into classical cyclotomy or generalized cyclotomy. Inspired by a foregoing work of Zeng et al. [40], we introduce a generalized cyclotomy of order e on the ring GF( q 1 ) × GF( q 2 ) × ··· × GF( q k ), where q i and q j ( i (cid:54) = j ) may not be co- prime, which includes classical cyclotomy as a special case. Here, q 1 , q 2 , ··· , q k are powers of primes with an integer e | ( q i − 1) for any 1 ≤ i ≤ k . Then we obtain some basic properties of the corresponding generalized cyclotomic numbers. Furthermore, we propose three classes of partitioned diﬀerence fam- ilies by means of the generalized cyclotomy above and d -form functions with diﬀerence balanced property. Afterwards, three families of optimal constant composition codes from these partitioned diﬀerence families are obtained, and their parameters are also summarized.


Introduction
For a positive integer v, let Z v be the residue class ring module v and Z * v be the set of all invertible elements of Z v . A partition {D 0 , D 1 , · · · , D d−1 } of Z * v is a collection of some subsets with D i ∩ D j = ∅ for any i = j, and If D 0 is a multiplicative subgroup of Z v , and there exist elements a 1 , · · · , a d−1 in Z * v such that D i = a i D 0 for any 1 ≤ i < d, then these cosets D i are called generalized cyclotomic classes of order d when v is composite, and classical cyclotomic classes of order d when v is prime. The (generalized) cyclotomic numbers of order d are defined by (i, j) = |(D i + 1) ∩ D j | with 0 ≤ i, j < d.
The theory of cyclotomy can date back to Gauss and has some important applications in number theory. In the beginning, cyclotomy means "circle-division" and refers to the problem of dividing the circumference of the unit circle into a given number, v, of arcs of equal lengths. We refer the reader to the classical book by Storer [32] for an exposition. Note that different subgroups D 0 of the ring Z v could give different generalized cyclotomies and cyclotomic numbers. Nowadays, to the knowledge of ours, there are mainly five classes of cyclotomy: Classical cyclotomy [19], Whiteman generalized cyclotomy [35], Ding-Helleseth generalized cyclotomy [15], Zeng-Cai-Tang-Yang generalized cyclotomy [40] and Fan-Ge generalized cyclotomy [17]. Specifically speaking, classical cyclotomy was firstly dealt to a good extent by Gauss in his book "Disquisitiones Arithmeticae" [19], where he introduced the so-called Gaussian periods and cyclotomic numbers. Later, it was extended to finite fields as well and the properties of cyclotomic numbers were extensively investigated in [32]. For searching for residue difference sets, a generalized cyclotomy of order d with respect to pq was introduced by Whiteman [35] and was applied to design of sequences with good autocorrelation, where p and q are distinct odd primes with gcd(p − 1, q − 1) = d. Obviously, this generalized cyclotomy is not consistent with classical cyclotomy. Later, Ding and Helleseth proposed a new generalized cyclotomy of order 2 with respect to v = p m1 1 p m2 2 · · · p m k k [15] and gave error-correcting codes derived from this generalized cyclotomy [16], where odd primes p i and p j satisfy gcd(p i − 1, p j − 1) = 2 for any 1 ≤ i = j ≤ k.
It includes classical cyclotomy of order 2 and Whiteman generalized cyclotomy of order 2 as special cases. In 2013, Zeng et al. [40] introduced the generalized cyclotomy of order φ(v)/e with respect to v = p m1 1 p m2 2 · · · p m k k and presented a construction of optimal frequency-hopping sequences sets and two constructions of optimal frequency-hopping sequences, where φ(x) denotes the Euler function and e > 1 is a common factor of p 1 − 1, p 2 − 1, · · · , p k − 1. In 2015, Zha and Hu [41] introduced cyclotomic cosets of the set Z v , which may help the reader to have a better understanding of generalized cyclotomy proposed by Zeng et al. [40]. Moreover, Fan and Ge [17] introduced the generalized cyclotomy order e with respect to v, where they constructed an infinite series of near-optimal codebooks over Z pq and asymptotically optimal difference systems of sets over Z v . Especially, Fan-Ge generalized cyclotomy includes Whiteman and Ding-Helleseth generalized cyclotomies as special cases. More recently, the cyclotomy has proved to be valuable in other applied fields such as sequences [40,15,8,22,9], coding theory [15,17,16,10,25] and cryptography [15]. The combinatorics has also benefited from the use of cyclotomy, which can be applied for constructing difference sets, difference families, and so on [32,35,17,41,36,7,39,34,37,6,29].
Let A be an additive group of order n and P = {B i : 0 ≤ i < m} be a collection of nonempty subsets (blocks) of A. P is called a difference family (DF) in A, if every nonzero element of A occurs exactly λ times in the multiset In brief, one says that P is an (A, K, λ) DF. In particular, if A is a cyclic group of order n, we also denote P as an (n, K, λ) DF. A difference family is called disjoint (DDF) if its blocks are pairwise disjoint. Let P be an (A, K, λ) DF, if P forms a partition of A, then it is called a partitioned difference family (PDF) and denoted as an (A, K, λ) PDF. In the sequel, we sometimes use a more informative notation to describe the multiset K: an (A, [k u1 1 k u2 2 . . . k us s ] , λ) DF is a difference family in which there are u i blocks of size k i for 1 ≤ i ≤ s.
It is meaningful to point out that ZDB functions and PDFs are two equivalent objects as follows.
Lemma 1.1. [42] Let A be an additive group with |A| = n and B be a set with Then f is an (n, m, λ) ZDB function if and only if P is an (A, K, λ) PDF, where the multiset K = {|B i | : 0 ≤ i < m}.
In 2019, Buratti and Jungnickel [4] made a comparison between the equivalent notions of a PDF and a ZDB function, raising four reasons for which they prefer to use the terminology and notation of PDFs. It was pointed out that some papers published in the last decade on ZDB functions may be repetitive and of little value since they rediscovered simple results on DFs which were known since the 90s or even earlier. This problem deserves our full attention. Hence, in the present paper, we use the terminology of PDFs.
The main contribution of the present paper consists of the following three parts: (1) This paper builds a general bridge between the generalized cyclotomy over the ring Z q1 × Z q2 × · · · × Z q k and the generalized cyclotomy over the product ring R = GF(q 1 )×GF(q 2 )×· · ·×GF(q k ) of finite fields, see Section 2.1. More specifically, we present the generalized cyclotomy of order k i=1 (q i − 1)/e on the ring R, which is a generalization of the classical cyclotomy. Here, q 1 , q 2 , · · · , q k are powers of primes with an integer e|(q i − 1) for any 1 ≤ i ≤ k, where q i and q j (i = j) may not be co-prime. Whereafter, some basic properties of the corresponding generalized cyclotomic numbers are derived. Compared to the Zeng-Cai-Tang-Yang generalized cyclotomy, we have essentially replaced the requirement e|(p i − 1) by e|(p mi i − 1), where p i , 1 ≤ i ≤ k, are primes. In 2011, Chung and Yang [9] introduced the k-fold cyclotomy of order (v − 1)/e over the ring R, where v = q 1 q 2 · · · q k and the set of its cyclotomic classes is a partition of R \ { 0}. In comparison, our method is quite neat and more clear to understand. And above all, by virtue of the generalized cyclotomy in this paper, we can construct more PDFs and optimal constant composition codes. It seems that these constructions can not be obtained by the other cyclotomies or generalized cyclotomies, such as the classical cyclotomy, the Zeng-Cai-Tang-Yang generalized cyclotomy and so on.
(2) In this paper, we present three constructions of PDFs based on the generalized cyclotomy above and d-form functions with difference balanced property, see Theorems 3.4, 3.6 and 3.9. Firstly, compared with Construction 1 in [6] and Construction 1 in [29] respectively, Construction 1 and Construction 2 in this paper provide new parameters since the requirement e|(p mi i − 1) gives more flexibility. Furthermore, compared with the recursive constructions of PDFs in [27,Theorem 18] and [4,Chapter 3], our constructions are direct. Secondly, Construction 3 in this paper not only includes [37,Theorem 13] as a special case, but also gives flexible parameters due to the free choice of k and e. In particular, Construction 3 works for every d-form function with difference-balanced property and the parameters of Theorem 3.9 are new when k = 1 or e = q − 1.
(3) According to [14,Construction 6], every PDF leads to an optimal constant composition code. Employing our newly constructed PDFs, three classes of optimal constant composition codes with new parameters are obtained.
The outline of the paper is organized as follows. In Section 2, we introduce the generalized cyclotomy on product ring of finite fields and difference balanced functions. In Section 3, we construct three classes of PDFs by means of the generalized cyclotomy and d-form functions with difference balanced property of Section 2. In Section 4, we give an application of such PDFs. Finally, Section 5 concludes this paper.

Preliminaries
In this section we introduce some properties of the generalized cyclotomy on product ring of finite fields and difference balanced functions. These will be used to construct more PDFs in Section 3.
2.1. The generalized cyclotomy on product ring of finite fields and its properties. Let an integer v = q 1 q 2 · · · q k , where q 1 , q 2 , · · · , q k are powers of primes. For each i, let GF(q i ) be a finite field of order q i and g i be a generator of the multiplicative group GF( For any element x = (x 1 , x 2 , · · · , x k ) and y = (y 1 , y 2 , · · · , y k ) in R, we define an addition and a multiplication in R as follows: x + y = (x 1 + y 1 , x 2 + y 2 , · · · , x k + y k ) x · y = (x 1 y 1 , x 2 y 2 , · · · , x k y k ) where x i + y i and x i y i are operated in GF(q i ). Further, we have where R * denotes the set of all invertible elements in (R, ·). Obviously, For a subset H of R and an element x in R, define x + H and x · H as Let e > 1 be a common factor of q 1 − 1, q 2 − 1, · · · , q k − 1, i.e.
Since the order of g i is q i − 1 for any i, the order of g (K,e) in R is e and thus the set In the sequel, we will employ the subgroup D (K,e) and its cosets to give a partition of R * . Let Ω for any K . In particular, if I K = (0, 0, · · · , 0) ∈ Ω (e) K , then D (K,e) I K = D (K,e) . Furthermore, for any two k-dimensional vectors I K = (i 1 , i 2 , · · · , i k ) and J K = (j 1 , j 2 , · · · , j k ) in Ω (e) K , the addition is defined as the following where the operation i 1 +j 1 is performed in the ring Z f1 and the operations i r +j r (2 ≤ r ≤ k) are performed in the ring Z qr−1 , respectively. By a similar analysis as in [40], it can be proved that {D K } is a partition of R * . In accordance with the notation of [15], we call D the corresponding generalized cyclotomic numbers of order Remark 1. In 2013, Zeng et al. introduced the Zeng-Cai-Tang-Yang cyclotomy [40] for constructing frequency-hopping sequences with optimal Hamming correlation. On one hand, the group Z v in [40] is cyclic and the group (R, +) in our paper is not cyclic in general; on the other hand, compared with [40], the parameter e is more flexible, since we replace e|( In the next section, we will employ this generalized cyclotomy to construct more PDFs. To this end, we study some necessary properties of the generalized cyclotomic classes and generalized cyclotomic numbers above. From now on, we always assume that v is an odd integer unless particularly stated. The following lemma can be easily verified, we omit its proof.
Proof. See Appendix A.
Similar to Proposition 1 in [40], the following proposition can be proved by virtue of the definitions of generalized cyclotomic classes and generalized cyclotomic numbers.
K , the generalized cyclotomic numbers defined in equality (2) have the following properties: Then (4) Corollary 2.4. Let e ≥ 2 be an integer. Then (1) (2) For any element a ∈ R * , Proof. See Appendix B.
Corollary 2.5. Let e ≥ 2 be an odd integer and Λ K be any fixed element with 0 ≤ Λ K < f 1 . Then (1) (2) For each J K ∈ Ω (e) K , K and a ∈ R * , Proof. See Appendix C.
and Λ 1 is any fixed element x q ri , x ∈ GF(q m ), be the trace mapping from GF(q m ) to its subfield GF(q r ). Definition 2.6. A function f (x) from GF(q m ) onto GF(q r ) is said to be balanced if any element of GF(q r ) appears q m−r times with x ranging over GF(q m ). It is said to be difference balanced, if for any δ ∈ GF(q m ) \ {0, 1}, the difference function As you know, d-form functions give a rich source of functions with difference balanced property, which were first defined in [26].
for any y ∈ GF(q r ) and x ∈ GF(q m ).
In the literature, there are only a few constructions of difference balanced functions from GF(q m ) onto GF(q r ): (1) The single trace function f (x) = Tr q m /q r (x d ) taken from m-sequence, where d is a positive integer with gcd(d, q m − 1) = 1.
It is observed that all currently known difference balanced functions listed above are d-form functions for some d. In 2004, No derived the following lemma:

Three classes of PDFs
In this section, we will construct three classes of PDFs based on the generalized cyclotomy and d-form functions with difference balanced property introduced in Section 2. Before proposing our constructions, we give some necessary definitions and notation.
For each non-empty subset K ⊆ K, let Without confusion we may identify R K with i∈K GF(q i ) * . Assume that K = {t 1 , t 2 , · · · , t |K | } ⊆ K and define g (K ,e) = (g Thus the set D (K ,e) = {( g (K ,e) ) s : 0 ≤ s < e} is a cyclic subgroup of R K of order e. Let Ω (e) for each I K = (i 1 , i 2 , · · · , i |K | ) ∈ Ω (e) K . The assertion in the following lemma is obvious.
Lemma 3.1. Let an integer v = q 1 q 2 · · · q k and R = GF(q 1 )×GF(q 2 )×· · ·×GF(q k ), where q 1 , q 2 , · · · , q k are powers of primes. Then we have The following lemma can be used to analyze the difference property of the PDFs generated by Construction 1 and Construction 2.
Then we have Proof. Firstly, if K = K , then where the last equality is from Lemma 2.1. Secondly, if K = K , we need to show that |(D (K ,e) I K + a) ∩ D (K ,e) I K | = 0. Now we prove it by contradiction. In this case, the discussion is divided into two cases.
Case 1: There exists an integer 1 ≤ t d ≤ k such that t d ∈ K and t d / where s 0 is a fixed integer with 0 ≤ s 0 < e. Hence, 0 + g which is a contradiction.
Based on the previous discussion, we know that for different choice of the number e, one can get different partitions of R. In what follows, we consider two choices of e and use the corresponding generalized cyclotomies to construct PDFs. In Construction 1, we always assume that e and e−1 are divisors of (q i −1), i = 1, ..., k.
K . Consider the commutative ring R × Z e , where we define an addition and a multiplication in R × Z e as follows: for any element ( x, s) and ( y, t) in R × Z e , ( x, s) + ( y, t) = ( x + y, s + t e ) ( x, s) · ( y, t) = ( x · y, st e ) where x + y and x · y are operated in R. The following result on the function η will be useful in the sequel. Lemma 3.3. Let the function η be defined as above. Then for any given ( a 1 , a 2 ) ∈ R × Z e with a 1 = 0 and a 2 = 0, there exists a unique x 1 in A∈Ae\{ 0} { x ∈ A : x + a 1 ∈ A} such that Proof. Suppose there exist K = {t 1 , t 2 , · · · , t |K | } ⊆ K and (i 1 , i 2 , · · · , i |K | ) ∈ Ω (e) K such that Hence, the assertion is proved.
We are now in a position to propose our constructions of PDFs. The first construction of PDFs is presented as follows.
Construction 1: Let the notation be defined as above and c ∈ Z * e . Generate a function f over R × Z e as For the above construction, we have the first main result. Proof. For any given nonzero ( a 1 , a 2 ) ∈ R × Z e , we have , then x 1 = 0, x 2 = 0, x 1 + a 1 = 0 and x 2 + a 2 e = 0. Therefore, ∆ 1 ( a 1 , a 2  When a 2 = 0, it follows from the definition of the function f that ∆ 2 ( a 1 , a 2 ) = 0. When a 1 = 0 and a 2 = 0, it follows from the definitions of the functions f and ψ that where the last identity followed from Corollary 2.4-(2). Hence, we have and ca 2 + η( x 1 + a 1 ) ≡ η( x 1 ) (mod e), which lead to K = K and I K = I K . Hence, ∆ 3 ( a 1 , a 2 ) is equal to (e − 2)|T |, where Now we distinguish the following three cases to discuss |T |.
With a simple calculation by using a computer, we can check that . This is consistent with the result of Theorem 3.4.
By equality (13), we need to prove that the number of x ∈ R such that x and x + a belong to some same set in S is always e−1 2 . Hence, we have where the second and last equalities follow from Lemma 3.2 and Corollary 2.5- (3) respectively. This completes the proof.
Here we employ the following example to illustrate Construction 2 and Theorem 3.6. where α is a generator of the multiplicative group GF(5 2 ) * . We can check that the A is a (GF(5 2 )×GF (7), [3 29 1 88 ], 1) PDF by using a computer, which is consistent with the result of Theorem 3.6.
The following result is useful for the construction of the third class of PDFs. : The third construction of PDFs is presented as the following. Proof. Firstly, we have by Lemma 3.8 Secondly, for any 1 ≤ a < k q m −1 e , there exists only one (t, j) such that a = tk + j with t ∈ Z q m −1 e and j ∈ Z k . Let δ = α tk+j . Hence, we have by Lemma 3.8 This completes the proof of Theorem 3.9.
Remark 3. In this paper, we propose three constructions of PDFs by virtue of the generalized cyclotomy and difference balanced functions in Section 2. It can be easily checked that Theorem 3.4 still holds for even v. We summarize these PDFs in Table 1, where q 1 , q 2 , · · · , q k are powers of primes and R = GF(q 1 ) × GF(q 2 ) × · · · × GF(q k ).
(1) In [6] and [29], two classes of PDFs are given using the terminology of ZDB functions. Compared with [6] and [29], Construction 1 and Construction 2 provide many new and more flexible parameters, since the requirement e|(p mi i − 1) or e(e − 1)|(p mi i − 1) gives more flexibility in our constructions. Nevertheless, these parameters are identical to those of [27,Theorem 18] and [4, Chapter 3] respectively via the recursive constructions of PDFs. Compared with [27] and [4], our constructions are direct.
(2) Construction 3 is generic in the sense that it works for every d-form function with difference balanced property. When k = 1 and e = q − 1, the parameters of Theorem 3.9 are equivalent to those given by [37,Theorem 13]. When k = 1 or e = q − 1, the parameters of Theorem 3.9 are new. Table 1. (A, K, λ) PDF constructed in this paper In the following, an example of PDF generated by Construction 3 is given. x + x q + x q 2 + x q 3 in Construction 3. If k = 1, then the set J is equivalent to the function h in [37,Example 15]. If k = 2, then the set D in Construction 3 is given as In this case, we can check that the set J = {D} {{x} : x ∈ (Z 85 × Z 2 )\D} is a (Z 85 × Z 2 , [42 1 1 128 ], 10) PDF by using a computer.

An application of PDFs
Once PDFs are constructed, many interesting objects can be obtained. In this section, we will construct optimal constant composition codes by virtue of Constructions 1, 2 and 3 in Section 3.
By virtue of the method in the proof of Theorem 4.2, every PDF leads to an optimal CCC. In Table 2, we obtain new optimal CCCs using three classes of PDFs in Section 3, where q 1 , q 2 , · · · , q k are powers of primes. Table  2. Some optimal CCCs with parameters (n, M, d, [ω 0 , ω 1 , · · · , ω m−1 ]) m from our PDFs

Concluding remarks
In this paper, we give a unified treatment for the Zeng-Cai-Tang-Yang generalized cyclotomy over the ring Z q1 × Z q2 × · · · × Z q k and the generalized cyclotomy over the ring GF(q 1 ) × GF(q 2 ) × · · · × GF(q k ). By virtue of the generalized cyclotomy on product ring of finite fields and d-form functions with difference balanced property, we presented three classes of PDFs. These PDFs can be used to construct optimal constant composition codes. In the future work, we are expected to propose more PDFs or ZDB functions, by which we can construct more optimal cryptographic objects. E-mail address: sdxzx11@163.com E-mail address: ljqu happy@hotmail.com E-mail address: xwcao@nuaa.edu.cn