New constructions for disjoint partial difference families and external partial difference families

Recently, new combinatorial structures called disjoint partial difference families (DPDFs) and external partial difference families (EPDFs) were introduced, which simultaneously generalize partial difference sets, disjoint difference families and external difference families, and have applications in information security. So far, all known construction methods have used cyclotomy in finite fields. We present the first noncyclotomic infinite families of DPDFs which are also EPDFs, in structures other than finite fields (in particular cyclic groups and nonabelian groups). As well as direct constructions, we present an approach to constructing DPDFs/EPDFs using relative difference sets (RDSs); as part of this, we demonstrate how the well‐known RDS result of Bose extends to a very natural construction for DPDFs and EPDFs.


Introduction
Difference sets and difference families are well-studied combinatorial objects dating back to the 1930s; difference families are useful for constructing balanced incomplete block designs (BIBDs) ( [6], [25]).Disjoint difference families (DDFs) have recently received attention [4], with applications to design theory and information security.In the early 2000s, motivated by applications in cryptography, external difference families (EDFs) were introduced ( [22], [23]).In a recent paper [14], partial analogues of DDFs and EDFs were introduced; these are called Disjoint Partial Difference Families (DPDFs) and External Partial Difference Families (EPDFs).A (v, s, k, λ, µ)-DPDF (respectively, EPDF) is a set S of s disjoint k-subsets of an order-v group, such that the multiset union of internal (respectively, external) differences of the sets in S comprises λ copies of each non-identity element in S, and µ copies of each non-identity element not in S.These also generalize the concept of a partial difference set (PDS) (see [19], [20]) and have applications in information security.In [14], construction methods were given for DPDFs and EPDFs in GF (q) (where q is a prime power) using cyclotomic techniques.Cyclotomy has long been used to produce traditional difference families, beginning with the work of [25].This paper takes the first step in going beyond cyclotomy to present a range of other construction methods in structures other than finite fields.
It is of particular interest to construct families of sets which are simultaneously DPDFs and EPDFs.It is shown in [14] that such families must partition a partial difference set, which is regular if it is proper.As well as the natural theoretical appeal of such examples, DPDFs which partition a regular PDS correspond to a two-class association scheme which means they can be used to obtain partially balanced incomplete block designs (PBIBDs) ( [16]).
In [14], DPDFs/EPDFs are obtained by partitioning cyclotomic PDSs in finite fields; for fields of prime order, these partition the quadratic residues (or non-residues) modulo p.In this paper, we address the following questions.
• Can DPDF/EPDF constructions be obtained in abelian groups other than (GF (q), +); particularly for cyclic groups Z v where v is not prime?
• Can DPDF/EPDF constructions be obtained in non-abelian groups?
We provide a range of constructions answering both of these questions in the affirmative.
It is known ( [19], [20]) that if D is a regular PDS in Z v , then there are just two possibilities: D or its complement is the set of non-zero squares (equivalently, non-squares) in GF (v) with v ≡ 1 mod 4 prime, or else D ∪ {0} or G \ D is a subgroup of G. Constructions of DPDFs and EPDFs partitioning the former type of PDS was addressed in [14]; in this paper we address the latter situation (although not limited to the cyclic group setting).In non-abelian groups, while the definitions of these difference family-type structures remain valid, very little is known.There are just a few non-abelian EDFs in the literature (see for example [13]), and prior to this paper there were no known constructions for non-abelian DPDFs or EPDFs, so the non-abelian constructions presented here are significant.
We first present explicit constructions in cyclic groups.We next develop constructions in general finite groups based on relative difference sets (RDSs), in which the subgroup not present in the union of the sets of the DPDF/EPDF is precisely the forbidden subgroup for the RDS.In particular, we show how the classic result of Bose [1] which originally constructed relative difference sets using finite geometry, very naturally extends to a DPDF/EPDF construction in cyclic groups.We obtain a framework for using RDSs for DPDF/EPDF constructions which encompasses this example and generates many others.Finally, we briefly present DPDFs and EPDFs in cyclic groups which demonstrate that not all DPDFs must be EPDFs, and vice versa.
Any (v, s, k, λ, 0)-DPDF which partitions G \ H (for some subgroup H of G) gives an instance of a relative difference family.Relative difference families were introduced in [2] and subsequently explored in various further papers (eg [3], [21]).These have mostly been studied in abelian groups, and are closely related to the concept of group divisible designs.EPDFs also give examples of bounded external difference families (see [23]).
We begin with a summary of relevant definitions (see [6] and [14]).
Definition 2.1.Let G be a group of order v. (iii) A (v, s, k, λ, µ)-disjoint partial difference family (DPDF) is a collection of disjoint k-subsets S ′ = {A 1 , . . ., A s } of G * with the property that Int(S ′ ) comprises each non-identity element of S = ∪ s i=1 A i precisely λ times, and each non-identity element of with the property that Ext(S ′ ) comprises each non-identity element of G precisely λ times.An EDF which partitions G * is called near-complete.
(iii) A (v, s, k, λ, µ)-external partial difference family (EPDF) is a collection of disjoint k-subsets S ′ = {A 1 , . . ., A s } of G * with the property that Ext(S ′ ) comprises each non-identity element of S = ∪ s i=1 A i precisely λ times, and each non-identity element of G \ S µ times.
Proof.This is immediate upon double-counting the elements of Int(S ′ ) and Ext(S ′ ).
We will also need the definition of a relative difference set, and its generalization, the divisible difference set.For a comprehensive survey article on these structures, see [24].Definition 2.3.Let G be a group of order mn and let H be a normal subgroup of G of order n.A k-subset R of G is an (m, n, k, λ)-relative difference set (RDS) in G relative to H if the multiset ∆(R) comprises each element in G \ H exactly λ times, and each non-identity element in H exactly 0 times.If n = 1 then R is a difference set.
A counting argument shows that, for an (m, n, k, λ)-RDS, we have the relation k(k − 1) = (mn − n)λ.Definition 2.4.Let G be a group of order mn and let H be a normal subgroup of G of order n.A k-subset D of G is an (m, n, k, λ, µ)-divisible difference set (DDS) relative to H if the multiset ∆(D) comprises each non-identity element of H exactly λ times, and each element of G \ H exactly µ times.If µ = 0 then D is a relative difference set.
In general, more is known about RDSs than about DDSs.Remark 2.5.In Definition 2.3, it is possible to relax the condition that H is a normal subgroup.An example of an RDS in A 5 relative to a subgroup H of order 2 is presented in [5], which satisfies all conditions of Definition 2.3, except for the requirement that H is normal (this would be impossible since A 5 is a simple group).

DPDFs and EPDFs partitioning PDSs
In this section, we will explore the special properties of DPDFs and EPDFs which partition PDSs.
The following key result was proved in [14]: Theorem 2.6.Let G be a group of order v. Let A = {A 1 , . . ., A s } be a family of disjoint subsets of G * , each of size k.Then any two of the following conditions implies the third: Moreover, if the PDS in Theorem 2.6 is proper, then it is regular.As mentioned in the introduction, results have been obtained ( [19], [20]) which significantly restrict the possibilities for regular PDSs.For cyclic groups, the following holds: Theorem 2.7.Let Z v be the cyclic group of order v. Let S ′ be a (v, s, k, λ 1 , µ 1 )-DPDF and a (v, s, k, λ 2 , µ 2 )-EPDF in Z v which partitions a proper PDS.Then (i) if v is a prime and v ≡ 3 mod 4 then no such S ′ exists; (ii) if v is a prime and v ≡ 1 mod 4 then S ′ partitions the set of non-zero quadratic residues or the non-residues modulo v; (iii) if v is a composite number then S ′ partitions a proper non-trivial subgroup H of Z v or its complement Z v \ H.
Proof.It is known ( [19], [20]) that if G is a cyclic group of order v and D is a regular PDS in G then either v is an odd prime such that v ≡ 1 mod 4 and D is the set of quadratic residues (or non-residues) modulo v; Examples of DPDFs and EPDFs partitioning a PDS of each type are given below: Example 2.8.(i) Let G = Z 13 ; the sets {1, 3, 9}, {4, 10, 12} form a (13, 2, 3, 0, 2)-DPDF and a (13, 2, 3, 2, 1)-EPDF which partition the quadratic residues mod 13 (see [14]).The following basic PDS result is useful (see [20]): Lemma 2.9.Let G be a group of order mn with identity e and subgroup H of order n.
(i) The sets H, H \{e}, G\H and (G\H)∪{e} are partial difference sets, with H \{e} and G \ H being regular.
In the case when a regular PDS D is a subgroup with the identity removed, then we can characterize any DPDF or EPDF which partitions D, as follows.
Theorem 2.10.Let G be a group of order mn and H be a subgroup of G of order n.Let D = H \ {0}.
Proof.For (i), we establish the DDF case; the EDF case then follows by Theorem 2.6.Let S ′ be an (mn, s, k, λ, µ)-DPDF partitioning D. By definition, Int(S ′ ) must comprise every element of D (i.e.every non-identity element of H) λ times and every non-identity element of G \ D (i.e.G \ H) µ times.Since S = D ⊆ H, all elements of Int(S ′ ) lie in H * , and so µ = 0. Thus Int(S ′ ) comprises λ copies of the non-identity elements of H, and S ′ partitions H * , so S ′ is a near-complete DDF in H. Correspondingly S ′ is also a near-complete EDF in H. Part (ii) is clear, using the natural embedding of H into G.
The group Z 10 contains the subgroup Since EDFs have been well-studied elsewhere (see [23] and references therein), we therefore focus on the situation when the PDS D is the complement of a subgroup of G.
The next result guarantees that there exists a DPDF/EPDF of this type, in any group G containing a normal subgroup H. Theorem 2.12.Let G be a group of order mn and H a normal subgroup of G of order n.Then the set of cosets of H in G, excepting H itself, forms an (mn, m − 1, n, 0, mn − n)-DPDF and an (mn, m − 1, n, mn − 2n, 0)-EPDF.
Proof.By Lemma 2.9, G \ H is an (mn, mn − n, mn − 2n, mn − n)-PDS.The cosets of H in G, other than H itself, partition G \ H and for each, its internal difference multiset comprises n copies of H * and 0 copies of G \ H.So these cosets from an (mn, m − 1, n, 0, mn − n)-DPDF and consequently an (mn, m − 1, n, mn − 2n, 0)-EPDF by Theorem 2.6.
We end this section with a result about the possible parameters for DPDFs/EPDFs which partition the complement of a subgroup.We first need a technical lemma which in fact applies more widely to any DPDF or EPDF; note however it will never apply to any of the DPDFs or EPDFs from [14] which partition a cyclotomic class.
Constructions producing DPDFs/EPDFs of type(a) in Theorem 2.14(ii)/(iii) are presented in Section 4 using RDSs, while Section 3 includes constructions giving DPDFs/EPDFs of type (b) in cyclic groups.

Cyclic DPDFs/EPDFs
We present constructions for infinite families of DPDFs/EPDFs such that G is a cyclic group, H is a subgroup of G and the DPDF/EPDF partitions G \ H.
In this section, we will use the group ring notation whereby λS indicates the multiset comprising λ copies of a set S. (In general we shall avoid this notation in the rest of the paper, to avoid confusion with multiplicative translates of a set.) We first introduce a family of subsets S i of Z 2m (m > 3 odd) which have the useful property that ∆(S i ) and ∆(S i , S j ) consist entirely of unions of S k 's and copies of H \ {0}.Proposition 3.1.Let m > 3 be an odd integer and let G = Z 2m .Let H = {0, m} be the order-2 subgroup of G.
In particular S 1 , S 2 , . . ., S m−1 Proof.Part (i) is immediate from the definition of S i .For part (ii), the fact that m is odd guarantees that all 4 elements are distinct.It is clear that as i runs through 1, . . ., m−1 2 , the sets S i account for all non-identity elements of G other than m.For part (iii), write i=1 ∆(S i ).Again by Proposition 3.1, the multiset ∆(S i ) comprises 4 copies of {m} and 2 copies of S 2i .So In either case this union equals 2 , 4, 2m − 6, 0)-EPDF.For m = 3, Theorem 3.2 can still be used but, rather than constructing a family of sets, it yields just one set {1, 2, 4, 5} which is an RDS.Next, we present a construction in Z 2p where p is a prime congruent to 1 mod 4. It uses the fact that the non-zero squares in GF (p) form a PDS when p ≡ 1 mod 4 ( [20]).
Theorem 3.4.Let p be a prime congruent to 1 mod 4. Let G be the additive group Z 2p and let H = {0, p} ≤ G. Define subsets A 0 , A 1 of Z 2p as follows: (ii) } be the quadratic residues modulo p, viewed as elements of Z 2p ; similarly, let N 2p = {b 1 , . . ., bp−1 2 } be the quadratic non-residues modulo p, viewed as elements of Z 2p .For later convenience, we order the elements of Q 2p in increasing order when viewed as integers, i.e. 0 < a The multiset of internal differences of a set is unchanged by translation of the set, so 2 ){p} (in this multiset, unlike in ∆(Q 2p ), we have a contribution from terms with indices i = j).A similar argument shows that ∆ 2 ){p}.Hence, to determine ∆(A 0 ), it suffices to determine ∆(Q 2p ).
Denote by Q p the non-zero quadratic residues modulo p viewed as elements of Z p .It is well-known (see [20]) that, as a subset of Z p , Q p is a (p, p−1 2 , p−5 4 , p−1 4 )-PDS (where the element {0} also occurs as an internal difference with frequency p−1 2 ).Due to the order imposed on the elements of Q 2p , it is clear that the elements D 2p > = {a i − a j : a i > a j } of ∆(Q 2p ) will be precisely the same integers as the corresponding elements D p > = {a i − a j : 2 copies of {p}.Combining all multisets which make up ∆(A 0 ) yield the result.A precisely analogous argument holds for the quadratic non-residues N 2p to yield part (ii) (since N p , the set of quadratic non-residues modulo p viewed as elements of Z p , is also a (p, p−1 2 , p−5 4 , p−1 4 )-PDS in Z p ). Finally, combining (i) and (ii) shows that the internal differences of {A Observe that Example 3.5(i) is the same DPDF/EPDF obtained in Example 3.3.This is because, in GF (5), the set of non-zero squares is {1, −1}.
In [18], a characterization is given for nontrivial reversible DDSs in cyclic groups (reversible means that D = D −1 for the DDS D).The result shows that (up to complementation and equivalence) there are only two possibilities for such a DDS D and group G.The first possibility is that G = Z 2p where p is an odd prime with p ≡ 1 mod 4 and D is precisely A 0 ∪ {0} from Theorem 3.4; D is a (p, 2, p, p − 1, p−1 2 )-DDS in G relative to H = {0, p}.Our proof of Theorem 3.4 demonstrates directly how the partial difference set of quadratic residues mod p gives the required properties for this DDS (whereas the proof in [18] follows from a structural characterization using Sylow subgroups, combined with parameter restrictions from [20], so is not constructive).
Finally, we present an infinite family of DPDF/EPDFs constructed via coset partitioning.
Proof.It is clear that the sets of S ′ partition G \ H. Since ∆(K) = 2(K \ {0}) and the multiset of internal differences is unchanged by translation, for any g + K (g ∈ G), the multiset ∆(g + K) = 2(K \ {0}).So by partitioning each a i + H (1 ≤ i ≤ d) into a i + K and b i + K, then computing ∆(a i + K) and ∆(b i + K), we obtain a collection of multisets comprising 4d copies of {6d + 2} in total.For each set of the form B j or C j , we have ∆(B j ) = ∆(C j ) = {3d + 1, 9d + 3} = H \ K.By partitioning each a j + H (d + 1 ≤ j ≤ 3d) into B j and C j , and for each computing ∆(B j ) and ∆(C j ), we obtain 4d copies of H \ K.
Hence Int(S ′ ) comprises 4d copies of H \ {0} and zero copies of G \ H, so S ′ is a DPDF with the stated parameters.Since G is a (12d + 4, 12d, 12d − 4, 12d)-PDS, S ′ is also an EPDF with the stated parameters.

RDS-based constructions for DPDFs/EPDFs
In this section, we will show how relative difference sets can naturally be used to construct DPDFs.
Relative difference sets were first introduced by Bose in [1], though they were not named as such; he presented his result as the "affine analogue" of Singer's Theorem on difference sets.The name and concept of RDS were formally introduced by Elliott and Butson in [8].The original construction of Bose for an RDS has parameters (q + 1, q − 1, q, 1), and is couched in terms of finite geometry; a formulation in terms of finite fields is given in [10].A more general result with parameters ( q r −1 q−1 , q − 1, q r−1 , q r−2 ) has been proved in various other ways, including via linear recurring sequences [8] and (in a particularly clear exposition) via linear functionals [17].

Extending the Bose RDS construction to DPDFs/EPDFs
Our first result demonstrates how Bose's original construction elegantly extends to a construction of a DPDF.Each of the component sets is a Bose RDS.We present it first in finite field terminology then outline the finite geometry viewpoint.Theorem 4.1.Let q be a prime power and let α be a primitive element of GF (q 2 ) with primitive polynomial f over GF (q).For each α i ∈ GF (q 2 ) (0 ≤ i ≤ q 2 − 2), there exist a i , b i ∈ GF (q 2 ) such that α i = a i + b i α.
Proof.Let c ∈ GF (q) * .To construct the set S c , we first form the multiplicative cosets of GF (q) * in GF (q 2 ) * , and express their elements in the form a + bα (a, b ∈ GF (q)) via the primitive polynomial.There are q 2 −1 q−1 = q + 1 cosets, each of the form C i where C 0 = α q+1 ∼ = GF (q) * and C i = α i C 0 (0 ≤ i ≤ q) (each of size q − 1).Each coset has the form Observe that, in each C i other than C 0 , there is a unique element whose coefficient of α is c, namely cb −1 i (a i + b i α).Hence for each c ∈ GF (q) * , we have As c runs through GF (q) * , the q − 1 sets {S c } partition the elements of GF (q 2 ) * \ C 0 .Now, each S c is a (multiplicative) (q + 1, q − 1, q, 1)-RDS in GF (q 2 ) * with respect to the multiplicative subgroup α q+1 .In fact, any one of these is a Bose RDS.To see that no element of C 0 arises as a (multiplicative) difference, observe that each element of S c is in a distinct coset of C 0 .It can be shown by direct calculation in the finite field that every element of GF (q 2 ) \ C 0 arises precisely once as a difference, by considering the elements of ∆(S c ) directly (details are left to the reader).Hence the family {S c } c∈GF (q) * form a (multiplicative) (q 2 −1, q−1, q, q, 0)-DPDF and (q 2 −1, q−1, q, (q−1)(q−2), q 2 −q)-EPDF.
Finally, take the set of powers of α to convert each S c to a set S ′ c = {i : It is clear that this yields a collection {S ′ c } c∈GF (q) * of additive (q + 1, q − 1, q, 1)-RDSs which form an additive DPDF and EPDF in Z q 2 −1 , with the same parameters as in the multiplicative case.
This has a natural interpretation in finite geometry.Consider a + bα ∈ GF (q 2 ) as the point (a, b) in the affine plane AG(2, q).The construction above may be viewed as taking sets which are lines in a given parallel class (in this case, the class with y = c).
A parallel class has q lines, each with q points, and the lines in the class partition the points of the affine plane.From Bose's paper, [1], each line with c = 0 in the parallel class gives an RDS with the required parameters, and taking all q − 1 such lines, we obtain the DPDF described.To see this directly, replace each point in AG(2, q) by the corresponding power of α via the above identification.Since (0, 0) does not correspond to a power of α, the line with c = 0 is missing a point: the differences between all remaining points on this line are all multiples of q + 1.This corresponds to our omitted subgroup.For any other line, the differences will be precisely one occurrence of each of the q(q − 1) elements of Z q 2 −1 that are not multiples of q + 1, corresponding to our RDS.
We note that the generalized RDS construction with parameters ( q r −1 q−1 , q−1, q r−1 , q r−2 ) cannot be used in this way to create a DPDF.

A general RDS construction for DPDFs/EPDFs
In this section, we present an important general approach to constructing DPDFs and EPDFs using RDSs.When dealing with groups which are not necessarily abelian, we will use multiplicative notation.
Proposition 4.3.Let G be a group of order mn and let H be a (not necessarily normal) subgroup of G of order n.
One natural method of producing a collection of disjoint sets with similar properties is to take an original set then form a collection of its translates by suitable group elements.The next result, based on a lemma in [10] In order for a set of translates of D to partition G \ H, we require k|mn − n. n .We present an explicit example of a DPDF/EPDF satisfying Proposition 4.3, formed from an RDS relative to a subgroup H, translated by the elements of H.It takes as its main ingredient the non-abelian RDS from [5] mentioned in a previous section.This RDS is notable as being the first example of an RDS in a finite simple group with a non-trivial forbidden subgroup.The following DPDF/EPDF construction is noteworthy since it is the first known non-abelian example of a proper DPDF or EPDF.Since the group is non-abelian, the result is written in multiplicative notation.(ii) A = {R, R ′ } is a (60, 2, 29, 28, 0)-DPDF and (60, 2, 29, 28, 58)-EPDF which partitions G \ H.
In general, it is not feasible to perform explicit verification of the properties required for Proposition 4.3: we will therefore establish results which guarantee that large classes of structures satisfy the requirements of Proposition 4.3.Henceforth we will assume that H is a normal subgroup of G.In particular, for any g ∈ G, any translate gD is a (m, n, k, λ)-RDS relative to H.
(ii) D cannot contain more than one representative from any coset gH (g ∈ H).In particular, k ≤ m. Proof.
By definition, the multiset ∆(D) comprises λ copies of G \ H and 0 copies of H. Since H is a normal subgroup, for any g ∈ G we have gHg −1 = H, and since gGg −1 = G we also have that g(G \ H)g −1 = G \ H.So the multiset ∆(gD) also has λ copies of G \ H and 0 copies of H. (ii) Suppose D contains elements Remark 4.9.Observe that the construction in Theorem 4.1 extending the Bose approach uses a component RDS with parameters (q+1, q−1, q, 1), which satisfies the requirements of Theorem 4.7.This is not a coincidence; we can view the Bose approach as an instance of Theorem 4.7, in the following way.

Applications of the general RDS construction
An RDS with parameters (n + 1, n − 1, n, 1) (and H normal in G) can always be used in the construction of Theorem 4.7.An RDS with these parameters is said to be affine; more detail about affine RDSs is given in [10] and [24], including non-abelian examples.
It is conjectured that in the abelian case, n must be a prime power.
The following existence result for a non-abelian affine RDS is from [10].
Proposition 4.10.Let n = p r where p is prime and (G, +) is the cyclic group of integers modulo p 2r − 1.We define a new addition on the elements of G. Let q = p h and suppose For a subset D of G, we define the multiset ∆(D) = {x − y : x = y ∈ D} and for sets D 1 , D 2 ⊆ G, we define the multiset ∆(D 1 , D 2 ) = {x − y : x ∈ D 1 , y ∈ D 2 }.(In multiplicative notation these are ∆(D) = {xy −1 with the property that the multiset of differences ∆(D) comprises each non-identity element of D precisely λ times, and each non-identity element of G \ D µ times.If λ = µ then D is simply called a (v, k, λ)-difference set (DS); otherwise the PDS is said to be proper.(ii)A (v, s, k, λ)-disjoint difference family (DDF) is a collection of disjoint k-subsets S ′ = {A 1 , . . ., A s } of G with the property that Int(S ′ ) comprises each non-identity element of G precisely λ times.If the disjointness condition is relaxed we obtain a difference family (DF).

Lemma 4 . 6 .
Let G be a group, H a normal subgroup of G and let be D an (m, n, k, λ)-RDS relative to H.

Theorem 4 . 7 .
Let G be a group of order mn, let H be a normal subgroup of G of order n, and suppose there exists an (m, n, m − 1, m−2 n )-RDS R in G relative to H. Then there exists a (mn, n, m − 1, m − 2, 0)-DPDF and an(mn, n, m − 1, (m − 2)(n − 1), (m − 1)n)-EPDF which partitions G \ H. Proof.Suppose we have an (m, n, m − 1, m−2 n )-RDS R. Since |R| = k = m − 1 = [G : H]−1,and by Lemma 4.6 R cannot contain a representative of more than one coset of H, there is precisely one coset of H with no representative in R. Without loss of generality, we may replace R by a suitable translate D := gR (g ∈ G), so that the coset without a representative in D is H itself. (This can be the trivial translation by the identity if H ∩R = ∅.)By a previous result, any translate of R is also an (m, n, m−1, m−2 n )-RDS and has its elements in distinct cosets of H. Hence D is a (m, n, m − 1, m−2 n )-RDS comprising a representative of each coset of H except H itself.Let D H = {hD : h ∈ H}; we shall show this is the desired DPDF/EPDF.Since D contains no element of H, any translate hD with h ∈ H must also have empty intersection withH (if h 1 ∈ (hD) ∩ H then h 1 = hd for some d ∈ D, i.e. d = h −1 h 1 ∈ H, impossible).By Lemma 4.4, the sets in D H are pairwise disjoint, i.e. their union comprises kn distinct elements of G. Hence the sets of D H partition the mn − n = kn elements of G \ H.By Lemma 4.6, each set in D H is an (m, n, m − 1, m−2 n )-RDS relative to H. Finally, by Proposition 4.3 D H is an (mn, n, m − 1, m − 2, 0)-DPDF and an (mn, n, m − 1, (m − 2)(n − 1), (m − 1)n)-EPDF.Example 4.8.Let G = Z 8 and consider the (4, 2, 3, 1)-RDS D = {1, 6, 7} relative to the subgroup H = {0, 4}.Note that the coset of H not represented in D is H itself. Then D H = {{1, 6, 7}, {5, 2, 3}} forms an (8, 2, 3, 2, 0)-DPDF and an (8, 2, 3, 2, 6)-EPDF.
, indicates what "suitable" means in this context.Lemma 4.4.Let G be a group, H a (not necessarily normal) subgroup of G and let D be an (m, n, k, λ)-RDS relative to H.Let g 1 = g 2 ∈ G.The translates g 1 D and g 2 D of D are disjoint if and only if g −1 2 g 1 ∈ H. Proof.Suppose there is an element in |g 1 D ∩ g 2 D|, i.e. g 1 d 1 = g 2 d 2 for some d 1 , d 2 ∈ D. Then g −1 2 g 1 = d 2 d −1 1 .If g −1 2 g 1 ∈ H then,since there is no non-identity element of H of the form d 2 d −1 1 , we must have g −1 2 g 1 = 1 i.e. g 1 = g 2 , a contradiction.So in this case g 1 D and g 2 D are disjoint.Otherwise g −1 2 g 1 ∈ G \ H; this element has λ > 0 representations in the multiset ∆(D).So there is at least one pair (d 1 , d 2