Packings of partial difference sets

A packing of partial difference sets is a collection of disjoint partial difference sets in a finite group $G$. This configuration has received considerable attention in design theory, finite geometry, coding theory, and graph theory over many years, although often only implicitly. We consider packings of certain Latin square type partial difference sets in abelian groups having identical parameters, the size of the collection being either the maximum possible or one smaller. We unify and extend numerous previous results in a common framework, recognizing that a particular subgroup reveals important structural information about the packing. Identifying this subgroup allows us to formulate a recursive lifting construction of packings in abelian groups of increasing exponent, as well as a product construction yielding packings in the direct product of the starting groups. We also study packings of certain negative Latin square type partial difference sets of maximum possible size in abelian groups, all but one of which have identical parameters, and show how to produce such collections using packings of Latin square type partial difference sets.


Introduction
A recurring theme across many diverse areas of mathematics is the natural occurrence of structures for which a key aspect can take one of only two values. We mention several examples of this two-valued phenomenon. In coding theory, a projective two-weight code is a linear code whose codeword weights take one of exactly two distinct values [13]. In finite geometry, a projective two-intersection set is a point set in projective space whose intersection with each hyperplane has one of exactly two distinct sizes [13]; and an m-ovoid or a tight set in a polar space is a point set whose intersection with every tangent hyperplane of the polar space has one of exactly two distinct sizes [10,Chapter 2], [66,Section 4.5]. In graph theory, a strongly regular graph has exactly two distinct eigenvalues [10,Chapter 1]. In group theory, a rank 3 permutation group has exactly two nontrivial orbitals [13,Section 10]; and a symmetric Schur ring of dimension 3 over a group G is a partition of the nonidentity group elements into exactly two nontrivial subsets which, together with the identity element, form a subalgebra of C[G] [90]. In number theory, uniform cyclotomy is a partition of the nonzero elements of a finite field into cyclotomy classes such that exactly two distinct cyclotomic numbers occur [6].
This paper is concerned with the construction of two of the richest classes of partial difference sets, whose parameters are determined by only two integers. A partial difference set whose parameters (v, k, λ, µ) take the form n 2 , r(n − 1), n + r(r − 3), r(r − 1) has (n, r) Latin square type, and one whose parameters take the form n 2 , r(n + 1), −n + r(r + 3), r(r + 1) has (n, r) negative Latin square type. A partial difference set D in G is regular if 1 G / ∈ D and D = {d −1 | d ∈ D}. A great deal of research has been conducted into the existence of a collection of t > 1 disjoint regular (tc, c) Latin square type partial difference sets in an abelian group G of order t 2 c 2 . The number of elements of G avoided by the partial difference sets of such a collection is tc, so the collection has maximum possible size when c > 1. The principal motivation for this paper was the discovery of a widespread structural property that does not seem to have been previously recognized: in all previous constructions, possibly after some modification, the tc avoided elements of G form a subgroup U. We therefore refer to such a collection of partial difference sets as a (c, t) LP-packing in G relative to U. The presence of the associated subgroup U is key to the recursive lifting construction of LP-packings of increasing exponent that we present in Section 5. In general, lifting constructions (that increase the group exponent) have historically been more challenging to find than product constructions (that combine objects into the direct product of the starting groups). Table 1.1 illustrates how widely the concept of LPpackings has been previously studied, often implicitly, and in each case identifies the subgroup U explicitly. In some cases, identifying the crucial subgroup U from the original reference requires considerable effort.
We shall unify and extend many results for LP-packings, whose original derivation relied on a variety of sometimes delicate approaches, by means of a common framework that depends only on elementary methods. We shall show that we have considerable control over the choice of the associated subgroup U. In addition to the recursive construction of Section 5, we shall give a product construction for LP-packings in Section 4. We present the following result (to be proved as Corollary 5.4) as a showcase for our constructions. Theorem 1.1. Let p be prime, let s 1 , . . . , s v be nonnegative integers (not all zero), and let m = min{s i | s i > 0}. For each i = 1, . . . , v, let G i = Z 2s i p i and let U i be a subgroup of G i of order p is i . Let U i , and let n = |G|. Then for each j = 0, . . . , m − 1, there exists an n p m−j , p m−j LP-packing in G relative to U: a collection of p m−j disjoint regular n, n p m−j Latin square type partial difference sets in G avoiding U.
We shall show in Section 3 that a collection of t disjoint regular (tc, c) Latin square type partial difference sets with c = 1 does not have maximum possible size, because there exists a size t + 1 collection known equivalently as a (t, t + 1) partial congruence partition. However, we regard the (1, t) LP-packing as a more natural object than the larger partial congruence partition, because it forms the base case of the recursive construction of Section 5 and so allows a powerful generalization to nonelementary abelian groups.
We similarly refer to a collection of t − 1 > 0 disjoint regular (tc, c) negative Latin square type partial difference sets in an abelian group G of order t 2 c 2 , for which the (c − 1)(tc + 1) nonidentity avoided elements of G form a regular (tc, c − 1) negative Latin square type partial difference set in G, as a (c, t − 1) NLP-packing; such a collection has maximum possible size for all c 1. The NLP-packing structure was already known, and previous constructions are summarized in Table 1.2. We shall use a product construction in Section 6 to combine small examples of NLP-packings with the families of LP-packings constructed recursively here, and so extend previous constructions of families of NLP-packings. We mention that both LP-packings and NLP-packings give rise to group-based amorphic association schemes [87].
Our principal objective is to determine for which abelian groups G and subgroups U there exists a (c, t) LP-packing in G relative to U, and for which abelian groups G there exists a (c, t − 1) NLP-packing in G. We propose that the natural framework for studying regular (tc, c) Latin square type and negative Latin square type partial difference sets is as these respective collections, rather than as single examples.
The rest of the paper is organized as follows. Section 2 gives a historical overview of packings of partial difference sets of Latin square type and negative Latin square type in abelian groups. Section 3 introduces LP-packings, describes their relationship to partial congruence partitions, establishes constraints on their structure, and presents a product construction. Section 4 introduces an LP-partition as an auxiliary configuration in the construction of an LP-packing in a larger group from an LP-packing in a smaller group. Section 5 recursively constructs infinite families of LP-partitions and LP-packings of increasing exponent, inspired by a recursive lifting construction of difference sets using relative difference sets [33]. Section 6 introduces NLPpackings, and uses a product construction to combine an NLP-packing with an LP-packing to give an NLP-packing in the direct product of the starting groups. Section 7 examines how LPpackings and NLP-packings are related to hyperbolic and elliptic strongly regular bent functions, and Section 8 examines how these packings are related to reversible Hadamard difference sets. Section 9 proposes some open problems for future research.

Historical Overview
In this section, we give an overview of previous constructions of regular Latin square type and negative Latin square type partial difference sets in abelian groups. In many cases, these results produce only a single partial difference set, whereas our objective in later sections will be to construct disjoint collections. Let G be an abelian group. We firstly present some basic results involving the group ring Z[G] and character theory. For A ∈ Z[G] and g ∈ G, denote the coefficient of g in A by For a subset A of G, by a standard abuse of notation we also denote the group ring element g∈A g by A. A character χ of G is a group homomorphism from G to the multiplicative group of the complex field C. each nonidentity element of D exactly λ times and each nonidentity element of G − D exactly µ times. The partial difference set is regular if 1 G / ∈ D and D = D (−1) .
The condition D = D (−1) in Definition 2.1 is guaranteed to hold except in the special case λ = µ [60, Prop. 1.2], in which case the partial difference set is a (v, k, λ)-difference set in G (see Section 8). Provided that D = D (−1) , the condition 1 G / ∈ D is not restrictive [60, p. 222]; we shall be concerned only with partial difference sets that are regular. Let D be a k-subset of a group G of order v for which 1 G / ∈ D. Then, in group ring notation, D is a (v, k, λ, µ) PDS in G if and only if The following result is a consequence of the orthogonality properties of characters.

Proposition 2.2 (Fourier inversion formula). Let G be an abelian group and let
Then In particular, elements For a regular PDS D, we can use the relation D = D (−1) to rewrite (2.1) as . Applying a nonprincipal character χ to both sides then shows that the character sum χ(D) satisfies the quadratic equation Using Proposition 2.2, we therefore obtain the following characterization of a regular partial difference set in terms of its character sums.
We are interested in the following two types of parameters for partial difference sets.
The connection between Latin square type partial difference sets and Latin squares is "rather indirect" [52, p. 2]. We allow the case r = 0 in Definition 2.4, which corresponds to the empty set (see Remark 6.2 (iii)). We next give a character-theoretic description of the two parameter sets of interest.
Lemma 2.5. Let n 1 and r 0 be integers, and let ǫ ∈ {1, −1}. Let G be an abelian group of order n 2 , and let D be an r(n − ǫ)-subset of G. Then D is a regular (n, r) Latin square type PDS in G (when ǫ = 1), and is a regular (n, r) negative Latin square type PDS in G (when ǫ = −1), if and only if If D is a regular (n, r) Latin square type or negative Latin square type PDS in G, then the character set D = {χ ∈ G | χ(D) = ǫ(n − r)} is also a regular (n, r) Latin square type or negative Latin square type PDS in G, respectively.
Proof. The first statement follows directly from Lemma 2.3. The second statement is obtained from [60,Thm. 3.4]: let D + be the dual of D as defined in [60], and note that D = D + in the case ǫ = 1 and that D = G − 1 G − D + in the case ǫ = −1.
We now briefly survey the numerous previous constructions of Latin square type and negative Latin square type partial difference sets, which we classify into three classes that are restricted to elementary abelian groups and five that are not. The three construction classes that are restricted to elementary abelian groups arise from the following sources.
A fundamental construction employs nondegenerate quadratic forms over finite fields [60,Thm. 2.6]. This construction has been greatly extended to allow the quadratic form to be replaced by other functions, including bent and vectorial bent functions [14,15,17,20,44,72,86].
(3) Cyclotomic classes of a finite field.
We now describe the five construction classes that are not restricted to elementary abelian groups. We shall later use LP-packings and NLP-packings to streamline many of these constructions, using only elementary methods. In some cases, we shall give a unified treatment for previously known parameter sets; in other cases, we shall produce examples with entirely new parameter sets.
(1) Direct constructions via partial congruence partitions Definition 2.6. Let G be a group of order n 2 > 1. An (n, r) partial congruence partition of degree r in G is a collection {U 1 , . . . , U r } of order n subgroups of G such that If U 1 , . . . , U r is an (n, r) partial congruence partition in G, then r i=1 (U i − 1 G ) is a regular (n, r) Latin square type partial difference set in G [3, Sect. 4.1]. A central objective in the study of (n, r) partial congruence partitions in groups of order n 2 is to determine the largest possible degree r [3,55,56]. The following result determines this value for abelian p-groups of the form H × H. Then the largest integer r for which there exists an (n, r) partial congruence partition in G is p m + 1. Therefore there exists a regular (n, r) Latin square type PDS in G for each positive integer r p m + 1.

3) Lifting constructions via finite local rings
Finite local rings also provide constructions for Latin square type partial difference sets that lift examples from a group G × G to a larger group G ′ × G ′ [48,50,52]. We shall present in Theorem 4.4 a ring-free lifting construction that applies to a collection of Latin square type partial difference sets, rather than just a single one. This construction leads to the result stated in Theorem 1.1, which recovers the parameters of partial difference sets constructed via lifting in each of [48], [50,Sects. 4,5], and [52].

(4) Product constructions
Various delicate product constructions for Latin square type and negative Latin square type partial difference sets have been found [62,74,75,78,79,82]. The following examples occur in p-groups having arbitrarily large exponent.
p 2i and let n = |G|. Then there is a partition of G − 1 G into p regular partial difference sets in G, of which p − 1 are of (n, n p ) Latin square type and one is of (n, n p + 1) Latin square type.
We shall strengthen Result 2.9 using Corollary 5.4 (see Remark 5.5 (v)). (i) Let u, w and s 4 , s 6 , . . . , s 2v be nonnegative integers satisfying u + w Then there is a partition of G − 1 G into 4 regular partial difference sets in G, of which three are of (n, n 4 ) negative Latin square type and one is of (n, n 4 − 1) negative Latin square type. (ii) Let u and s 2 , s 4 , . . . , s 2v be nonnegative integers.
3 2i and let n = |G|. Then there is a partition of G − 1 G into 3 regular partial difference sets in G, of which two are of (n, n 3 ) negative Latin square type and one is of (n, n 3 − 1) negative Latin square type.

(5) Constructions in Galois domains
A final construction class uses cyclotomy [46] and character sums [64,70] over Galois domains (direct products of finite fields) to produce partial difference sets in the direct product of elementary abelian groups.

LP-Packings
In this section, we introduce a (c, t) LP-packing as a collection of t disjoint regular (tc, c) Latin square type PDSs, in an abelian group of order t 2 c 2 , whose union satisfies an additional property. We provide examples of LP-packings, describe their relationship to partial congruence partitions, establish constraints on their structure, and present a product construction.
A principal motivation for this paper was the realization that in many previous constructions of such collections, possibly after some modification as summarized in Table 1.1, the tc-set G − t i=1 P i of elements avoided by the partial difference sets forms a subgroup U of G. Our definition of a (c, t) LP-packing is based on this observation.
We next give a characterization of a (c, t) LP-packing involving character sums.
holds for all nonprincipal characters χ of G.
Proof. Let χ be a nonprincipal character of G.
Since χ(P i ) ∈ {−c, (t − 1)c} for each i by Lemma 2.5, this implies that as required. Now suppose that (3.1) holds. Then each P i is a c(tc − 1)-subset of G for which χ(P i ) ∈ {−c, (t − 1)c}, and so by Lemma 2.5 is a regular (tc, c) Latin square type PDS in G. It remains to show that t i=1 P i = G − U. We have from |P i | = c(tc − 1) and (3.1) that, for all χ ∈ G, By the Fourier inversion formula (Proposition 2.2), this implies that t i=1 P i = G − U, as required.
A (t, t + 1) partial congruence partition (see Definition 2.6) lies at the heart of many configurations [40], [41,Sect. 4]. We next show that this structure can equivalently be expressed as a (1, t) LP-packing.

Proposition 3.4. Let G be an abelian group of order
This follows directly from Definition 2.1.
The following result greatly constrains the degree of a partial congruence partition. By Result 3.5 (i), a (t, t + 1) partial congruence partition in an abelian group G can exist only when t = p s for some prime p and positive integer s, and G ∼ = Z 2s p . By Proposition 3.4, such a structure exists if and only if there is a (1, p s ) LP-packing in G relative to an order p s subgroup. We now show, in an example we shall refer to frequently in the rest of the paper, that this structure exists for all primes p and positive integers s by reference to a spread of a vector space over a finite field. Definition 3.6. Let p be prime and s a positive integer, and let V be a 2s-dimensional vector space over the field F p with identity p if and only if H i is an s-dimensional vector subspace of V . Then by Proposition 3.4 with t = p s , we see that p relative to H 0 ∼ = Z s p , and by Lemma 3.3 each nonprincipal character of V is principal on exactly one of H 0 , . . . , H p s . Remark 3.8. According to Definition 3.1, a (c, t) LP-packing involves a collection of t disjoint regular (tc, c) Latin square type PDSs in an abelian group G of order t 2 c 2 . Since this collection covers all but tc elements of G, and each PDS contains c(tc − 1) elements, the collection has maximum possible size when c > 1. The collection is not necessarily of maximum possible size when c = 1: for p prime and t = p s , the (t, t+1) partial congruence partition of Example 3.7 is a collection of t + 1 disjoint regular (t, 1) Latin square type PDSs in the elementary abelian group of order t 2 . However, as Result 3.5 shows, if we instead attempt to create a partial congruence partition in a non-elementary abelian group of order t 2 then the degree must drop from t + 1 to at most √ t + 1, covering a proportion of only about 1/ √ t of the elements of G. In contrast, in Section 5 we shall provide constructions of (c, t) LP-packings with c > 1 in various nonelementary abelian groups. For this reason, we regard a (c, t) LP-packing with c > 1 as a natural generalization of a (t, t + 1) partial congruence partition.
The characterization in Lemma 3.3, together with Lemma 2.5, shows that we can combine subsets from an LP-packing to obtain Latin square type PDSs with various parameters. Lemma 3.9. Suppose that {P 1 , . . . , P t } is a (c, t) LP-packing in an abelian group G of order t 2 c 2 relative to a subgroup U of order tc. Let I be a b-subset of {1, . . . , t}. Then Remark 3.10. Suppose that {P 1 , . . . , P t } is a (c, t) LP-packing in an abelian group G of order t 2 c 2 relative to a subgroup U of order tc.
(i) By Definition 3.1 and Lemma 3.9, we see that Latin square type and the last is of (tc, c + 1) Latin square type.
(ii) For each i, let P i = {χ ∈ G | χ(P i ) = (t − 1)c}. By Lemma 2.5, each P i is a regular (tc, c) Latin square type PDS in G, and by Lemma 3.3 each χ ∈ G \ U ⊥ belongs to exactly one P i and each χ ∈ U ⊥ belongs to no P i . By Definition 3.1, the collection { P 1 , . . . , P t } is therefore a (c, t) LP-packing in G relative to U ⊥ .
Combining the subsets of an LP-packing into equally-sized collections gives an LP-packing with fewer subsets. Lemma 3.11. Suppose there exists a (c, t) LP-packing in an abelian group G of order t 2 c 2 relative to a subgroup U of order tc, and suppose s divides t. Then there exists an (sc, t s ) LPpacking in G relative to U.
Proof. Let {P 1 , . . . , P t } be a (c, t) LP-packing in G relative to U, and let Then The following result, which is inspired by [36,Lemma 3.2], allows us to establish some constraints on LP-packings in Proposition 3.13. These constraints assist in finding examples computationally, as demonstrated in Example 3.14.
Lemma 3.12. Let G be a group of order n 2 and let U be a subgroup of order n. Let {g 0 , g 1 , . . . , For all x, y ∈ P , we have xy −1 ∈ U if and only if x, y belong to the same right coset of U. Therefore Now P is an (n 2 , r(n − 1), n + r 2 − 3r, r 2 − r) PDS in G, and so by (2.1)
Then there are c-subsets H ij of U for i = 1, . . . , t and j = 1, . . . , tc − 1 satisfying Proof. By Definition 3.1, each P i is a regular (tc, c) Latin square type PDS in G, and G − U is the disjoint union of the P i . Write By Lemma 3.12, for each i = 1, . . . , t and each j = 1, . . . , tc − 1 we may write Substitute in (3.4) and use P i = P , and the constraints H Example 3.14. Using Proposition 3.13, we obtain the following exhaustive search results for LP-packings in Z 3 4 relative to two nonisomorphic order 8 subgroups.
(i) There are exactly 1536 distinct (4, 2) LP-packings The presented form of P 1 shows that both P 1 and P 2 intersect the seven nonidentity cosets of the subgroup x 2 , y 2 , z 2 in x, y, z in exactly 4 elements.
The presented form of R 1 shows that both R 1 and R 2 intersect the seven nonidentity cosets of the subgroup x, y 2 in x, y, z in exactly 4 elements.
We conclude this section with a product construction for LP-packings. Theorem 3.15. For j = 1, 2, suppose there exists a (c j , t) LP-packing in an abelian group G j of order t 2 c 2 j relative to a subgroup U j of order tc j . Then there exists a (tc 1 c 2 , t) LP-packing in Proof. For j = 1, 2, let {P j,0 , . . . , P j,t−1 } be a (c j , t) LP-packing in G j relative to U j and define where the subscript i+ℓ is reduced modulo t. We shall use Lemma 3.3 to show that {K 0 , K 1 , . . . , . Let χ be a nonprincipal character of G 1 × G 2 , and let χ j = χ| G j for j = 1, 2. Then By Lemma 3.3, it remains to prove that For j = 1, 2, by Lemma 3.3 we are given that Since χ is nonprincipal on G 1 × G 2 , we may assume by symmetry that χ 2 is nonprincipal on G 2 . Again by symmetry, we need to consider only the following three cases, which together establish (3.5).
Case 1: χ 1 is principal on U 1 and χ 2 is principal on U 2 . Then χ 1 (P 1,i ) is constant over i (regardless of whether χ 1 is principal or nonprincipal on Case 2: χ 1 is principal on U 1 and χ 2 is nonprincipal on U 2 . Then Case 3: χ 1 is nonprincipal on U 1 and χ 2 is nonprincipal on U 2 . Then . . , P t } is a (c, t) LP-packing in G relative to U (see also Remark 3.10 (i)). Our construction has the advantage that, by identifying the role of the avoided subgroup U in the definition of an LP-packing, we are able to control the avoided subgroup U 1 × U 2 in Theorem 3.15.
We extend the construction of Theorem 3.15 using Lemma 3.11. Corollary 3.17. For j = 1, 2, suppose there exists a (c j , t j ) LP-packing in an abelian group G j of order t 2 j c 2 j relative to a subgroup U j of order t j c j , and suppose that t 1 divides t 2 . Then there exists a (t 2 c 1 c 2 , t 1 ) LP-packing in G 1 × G 2 relative to U 1 × U 2 .
Proof. By Lemma 3.11 with s = t 2 t 1 , there exists a ( t 2 t 1 c 2 , t 1 ) LP-packing in G 2 relative to U 2 . Then by Theorem 3.15, there exists a (t 2 c 1 c 2 , t 1 ) LP-packing in G 1 × G 2 relative to U 1 × U 2 .

LP-partitions
In this section, we introduce a (c, t) LP-partition as an auxiliary configuration in the construction of an LP-packing in a larger group from an LP-packing in a smaller group. We then show how to construct an LP-partition from a collection of LP-partitions in various factor groups. We shall apply these two constructions recursively in Section 5 to produce infinite families of LPpackings.
Definition 4.1 (LP-partition). Let t > 1 and c > 0 be integers. Let G be an abelian group of order t 2 c 2 , let V be a subgroup of G of order tc 2 , and let H V . A (c, t) LP-partition in G − V relative to H is a collection {R 1 , . . . , R t } of t disjoint (t − 1)c 2 -subsets of G whose union is G − V and for which the multiset equality holds for all nonprincipal characters χ of G.
(i) In Definition 4.1, we can deduce that the R i are disjoint and their union is G − V from the other conditions, by applying the Fourier inversion formula to the equation   a (1, t) LP-packing in G relative to U is identical to a (1, t) LP-partition in G − U relative to U. Let t = p s for a prime p and positive integer s, and let V be a 2s-dimensional vector space over F p with identity 1 V . Let {H 0 , . . . , H p s } be a spread of V ∼ = Z 2s p . Following Example 3.7, We next combine the subsets of an LP-partition with the subsets obtained by lifting an LPpacking, in order to produce an LP-packing in a larger group. Proof. Let {R 1 , . . . , R t } be a (tc, t) LP-partition in G − V relative to H, and let {P 1 , . . . , P t } be a (c, t) LP-packing in V /H relative to U/H. For i = 1, . . . , t, let P ′ i = {g ∈ V | gH ∈ P i } be the pre-image of P i under the quotient mapping from V to V /H. We shall use Lemma 3.3 to Let χ be a nonprincipal character of G. If χ is principal on H then it induces a nonprincipal character ψ on G/H. Since P ′ i is a union of cosets of the order t subgroup H, we obtain Therefore the value of {χ(P ′ 1 ), . . . , χ(P ′ t )} is as specified in the second column of the following table: . . , χ(R t )} is as given in the third column of the table, and the fourth column contains the sum of the second and third columns. Lemma 3.3 then implies that {P ′ 1 + R 1 , . . . , P ′ t + R t } is a (tc, t) LP-packing in G relative to U, as required.
Remark 4.5. Lifting constructions similar to that of Theorem 4.4 were proposed in [48], [50,Sects. 4,5], [52], each producing a single Latin square type partial difference set. These previous constructions make use of the delicate structure of finite local rings. In contrast, the construction of Theorem 4.4 applies simultaneously to a collection of such partial difference sets, and does not require ring theory.
We now construct an LP-partition by lifting and combining the subsets of a collection of LP-partitions in various factor groups. The construction makes use of two spreads of a vector space over a finite field (as introduced in Definition 3.6).
By Definition 4.1 and Remark 4.2 (i), it is now sufficient to show that, for all nonprincipal characters χ of G, (4.1) Let χ be a nonprincipal character of G. Since the H i correspond to a spread of Q, if χ is nonprincipal on Q, then χ is principal on exactly one of the H i (see Example 3.7). If χ is nonprincipal on V i and principal on H i , then χ induces a nonprincipal character ψ i on V i /H i ; since each S ′ ij is a union of cosets of the order t subgroup H i , we have for each i, j satisfying 1 i, j t that If χ is principal on G ′ then it induces a nonprincipal character τ on G/G ′ ; since the V i /G ′ form a spread of G/G ′ , we then have that τ is principal on exactly one of the V i /G ′ and so χ is principal on exactly one of the V i . We shall use the subgroup inclusions (4. 2) The conclusions of the following five cases collectively establish (4.1).
Case 1: χ is principal on V 0 . Then by (4.2), χ is principal on G ′ and on each H i . For each i = 1, . . . , t, therefore χ is nonprincipal on V i and principal on G ′ /H i . For each j, we obtain Case 2: χ is principal on G ′ and nonprincipal on V 0 . Then by (4.2), χ is principal on each H i . Therefore χ is principal on each G ′ /H i and on exactly one of V 1 , . . . , V t , say V I . For each j, we obtain Case 3: χ is principal on Q and nonprincipal on G ′ . Then by (4.2), χ is principal on each H i and nonprincipal on each V i . Therefore χ is principal on each Q/H i and nonprincipal on each Case 4: χ is principal on H 0 and nonprincipal on Q.
Using the same (4, 4) LP-partition {R 1 , R 2 , R 3 , R 4 }, we can similarly use Theorem 4.4 to construct a (4, 4) LP-packing in G relative to the subgroup U ′ = x 1 , x 2 ∼ = Z 2 4 and to U ′′ = Following the above procedure, and using 4) LP-packing in G relative to U ′ , and that

Recursive Construction of LP-partitions and LP-packings
In this section, we recursively construct infinite families of LP-partitions and LP-packings, as shown schematically in Figure 5 We begin with a technical lemma.
Proof. The proof is by induction on a 1. The case a = 1 is given by Example 4.3. Suppose that all cases up to a − 1 1 are true. Let Q and G ′ be the unique subgroups of G for which Q ∼ = Z 2s p and G ′ ∼ = Z 2s p a−1 . Since G/G ′ ∼ = Z 2s p , by Example 3.7 there exists a spread in G/G ′ one of whose elements we may take to be V 0 /G ′ . Let the remaining elements of this spread be V 1 /G ′ , . . . , V p s /G ′ , where V 1 , . . . , V p s are subgroups of G containing G ′ . Let φ be a group isomorphism from G/G ′ to Q for which φ(V 0 /G ′ ) = H 0 , and let H i = φ(V i /G ′ ) for i = 1, . . . , p s . Since {V 0 /G ′ , . . . , V p s /G ′ } is a spread of G/G ′ , it follows that {H 0 , . . . , H p s } is a spread of Q. Furthermore, for each i we have H i Q G ′ V i and therefore H i is a subgroup of V i , and V i /H i ∼ = G ′ .
Let i ∈ {1, . . . , p s }. By construction, we have H i p a−2 . Therefore by the inductive hypothesis there exists a (p (a−2)s , p s ) LP-partition in V i /H i − G ′ /H i relative to Q/H i . Then by Theorem 4.6 there exists a (p s · p (a−2)s , p s ) LP-partition in G − V 0 relative to H 0 , so the case a is true.
We now apply Theorem 4.4 iteratively, making use of the LP-partitions constructed in Theorem 5.2, to produce an infinite family of LP-packings in groups of increasing exponent relative to an arbitrary subgroup of the appropriate order. The initial LP-packing in an elementary abelian group is obtained from a spread. Proof. The proof is by induction on a 1. The case a = 1 follows from the construction of a spread in Example 3.7, because by a suitable choice of generators of G = Z 2s p we may assume that the element H 0 of the spread is U.
Suppose that all cases up to a − 1 1 are true. Since U is a subgroup of G of order p as , it has at most s direct factors isomorphic to Z p a and has rank at least s. We may therefore choose V 0 ∼ = Z s p a ×Z s p a−1 for which U V 0 G, and also assume that the subgroup of V 0 isomorphic to Z s p a is contained in the first s direct factors of G and has intersection of rank s with U. Therefore U contains a subgroup H 0 ∼ = Z s p for which H 0 is contained in the first s direct factors of G. We then have V 0 /H 0 ∼ = Z 2s p a−1 and |U/H 0 | = p (a−1)s . Then by Theorem 5.2, there exists a (p (a−1)s , p s ) LP-partition in G − V 0 relative to H 0 . By the inductive hypothesis, there exists a (p (a−2)s , p s ) LP-packing in V 0 /H 0 relative to U/H 0 . Therefore by Theorem 4.4, there exists a (p (a−1)s , p s ) LP-packing in G relative to U, so the case a is true.
The construction process illustrated in Figure 5.1 shares many features with the recursive construction of difference sets from relative difference sets introduced in [33], which is illustrated in [34] using an analogous representation to Figure 5.1. In both settings, an ingredient on the left side is lifted from a factor group and combined with an ingredient on the right side to form a larger example on the left side; and multiple instances of an ingredient on the right side are lifted from factor groups and combined to form a larger example on the right side. The ingredients in [33] corresponding to an LP-packing and an LP-partition are, respectively, a covering extended building set for constructing a difference set, and a building set for constructing a relative difference set. However, in [33] the ingredients are placed into distinct cosets of a subgroup and so can be combined without regard to their intersections as subsets of the subgroup. In contrast, in the construction of Theorem 4.4, the lifted subset P ′ j of an LP-packing is combined with the subset R j of an LP-partition by taking their union. Likewise, in the construction of Theorem 4.6, the subsets of the i th LP-partition are lifted to S ′ i1 , . . . , S ′ it , and then subset R j of the new LP-partition is formed as the union of S ′ 1j , . . . , S ′ tj (with one lifted subset derived from each of the t LP-partitions). This leads to an additional constraint, that the subsets P ′ j and S ′ 1j , . . . , S ′ tj must be disjoint for each j. We now extend Theorem 5.3 using constructions from Section 3 in order to obtain the result stated in Theorem 1.1. p be prime, let s 1 , . . . , s v be nonnegative integers (not all zero), and let m = min{s i | s i > 0} . For each i = 1, . . . , v, let G  (ii) Result 2.8 (iii) constructs a regular (n, r) Latin square type PDS in G for each positive integer multiple r of n p gcd(s 1 ,...,sv ) satisfying r n, whereas the case j = 0 of Corollary 5.4 gives such a PDS for each positive integer multiple r of n p m satisfying r n and whose parameters therefore differ when m > gcd(s 1 , . . . , s v ).

Corollary 5.4. Let
(iii) The LP-packing of [82, Cor. 6.2] described in Table 1.1 is a disjoint collection of p regular (n, n p ) PDSs in certain groups G constrained to have at most one direct factor with an exponent that is an odd power of p, whereas the case j = m − 1 of Corollary 5.4 gives such disjoint collections of PDSs without this constraint.
(iv) In each group for which the lifting approach in [48], [50,Sects. 4,5], [52] constructs a single PDS, Corollary 5.4 provides a collection of disjoint PDSs having the same parameters and avoiding an arbitrary order n subgroup U of G.
(v) Result 2.9 constructs a partition of G − 1 G into p regular PDSs in G, of which p − 1 are of (n, n p ) Latin square type and one is of n, n p + 1 Latin square type, where G is constrained to have s 2i+1 = 0 for each i > 0. In contrast, by applying Remark 3.10 (i) to the case j = 0 of Corollary 5.4 we obtain a partition of G − 1 G into p m regular PDSs in G, of which p m − 1 are of n, n p m Latin square type and one is of n, n p m + 1 Latin square type, without constraining G.
The LP-packings in Corollary 5.4 are all contained in p-groups of the form H × H, having even rank. We now extend this result to p-groups of odd rank, starting from the following infinite family of odd rank examples that was obtained from an elegant construction using Galois rings.
(We could likewise make use of the further odd rank example of Example 3.14 (ii), relative to a nonelementary abelian group.) Combine with the LP-packing of Result 5.6 using Corollary 3.17 to give an n p m , p m LP-packing in G relative to U. Then for j ∈ {0, . . . , m − 1}, apply Lemma 3.11 with s = p j to give an n p m−j , p m−j LP-packing in G relative to U. Remark 5.8. By Definition 3.1, the constructed LP-packing in Corollary 5.7 is a collection of p m−j disjoint regular n, n p m−j Latin square type PDSs in G avoiding U. By Lemma 3.9, we also obtain a regular Latin square type PDS in G with parameters n, r) and with parameters n, r + 1), for each positive integer multiple r of n p m−j satisfying r n. We can show that this improves on previous constructions along similar lines to Remark 5.5. For example, Result 2.8 (ii) constructs a single regular (n, r) Latin square type PDS in G for v = 1 and i = 1 and s 1 a positive integer multiple of w, for each positive integer multiple r of n p σ(p,w) satisfying r n. In contrast, the case j = 0 of Corollary 5.7 gives a disjoint collection of p m such PDSs in much more general groups G (having arbitrary v and s i ).

NLP-packings
In this section, we introduce a (c, t − 1) NLP-packing as a collection of t − 1 disjoint regular (tc, c) negative Latin square type PDSs, in an abelian group of order t 2 c 2 , whose union satisfies an additional property. This structure forms a counterpart to a (c, t) LP-packing. We provide examples of NLP-packings, and show how to combine them with LP-packings via a product construction to form NLP-packings in the direct product of the starting groups. In conjunction with the LP-packings of Corollaries 5.4 and 5.7, this produces infinite families of NLP-packings. However, we do not have a lifting result for NLP-packings similar to Theorem 4.4. Definition 6.1 (NLP-packing). Let t > 1 and c > 0 be integers, and let G be an abelian group of order t 2 c 2 . A (c, t − 1) NLP-packing in G is a collection {P 1 , . . . , P t−1 } of t − 1 regular (tc, c) negative Latin square type PDSs in G for which G − 1 G − t−1 i=1 P i is a regular (tc, c − 1) negative Latin square type PDS in G.
(i) We can rephrase Definition 6.1 to say that a (c, t − 1) NLP-packing in G is a partition of G − 1 G into t regular partial difference sets, of which t − 1 are of (tc, c) negative Latin square type and one is of (tc, c − 1) negative Latin square type. Previous results on NLP-packings use this phrasing.
(ii) In Definition 6.1, each P i is a c(tc+1)-subset of G, and G−1 NLP-packings have been studied in a series of previous papers, as summarized in Table 1.2. We now characterize a (c, t − 1) NLP-packing using character sums. Proof. Each P i is a c(tc + 1)-subset of G, and the P i are disjoint, so for all nonprincipal characters χ of G. For each nonprincipal character χ of G, condition (6.1) is equivalent to By Lemma 2.5 and Definition 6.1, each P i is a regular (tc, c) negative Latin square type PDS in G, and P is a regular (tc, c − 1) negative Latin square type PDS in G. By Lemma 6.3, each nonprincipal character χ of G belongs to exactly one of P 1 , . . . , P t−1 , P . By Definition 6.1, the collection { P 1 , . . . , P t−1 } is therefore a (c, t − 1) NLP-packing in G.
We give some small examples of NLP-packings. Example 6.6.
We next show that combining the subsets of an NLP-packing into equally-sized collections gives an NLP-packing with fewer subsets. Lemma 6.7. Suppose there exists a (c, t − 1) NLP-packing in an abelian group G of order t 2 c 2 , and suppose s divides t. Then there exists an (sc, t s − 1) NLP-packing in G.
We now extend the construction of Theorem 6.8 using Lemmas 3.11 and 6.7. Corollary 6.10. Suppose there exists a (c 1 , t 1 − 1) NLP-packing in an abelian group G 1 of order t 2 1 c 2 1 , and there exists a (c 2 , t 2 ) LP-packing in an abelian group G 2 of order t 2 2 c 2 2 relative to a subgroup U 2 of order t 2 c 2 .
The product constructions of Theorems 3.15 and 6.8 both combine two packings to form a packing in the direct product of the starting groups. Theorem 3.15 combines two LP-packings to form an LP-packing, whereas Theorem 6.8 combines an NLP-packing and an LP-packing to form an NLP-packing. We note in passing that we can likewise combine two NLP-packings to form a collection of regular Latin square type PDSs that partitions the nonidentity elements of the product group. . . , K t−1 } of t−1 disjoint regular (t 2 c 1 c 2 , tc 1 c 2 ) Latin square type PDSs in G 1 ×G 2 for which G 1 ×G 2 −1 G 1 ×G 2 − t−1 ℓ=1 K ℓ is a regular (t 2 c 1 c 2 , tc 1 c 2 + 1) Latin square type PDS in G 1 × G 2 .
Proof (Outline). For j = 1, 2, let {P j,0 , . . . , P j,t−2 } be a (c j , t − 1) NLP-packing in G j and define P j,t−1 = G j − t−2 i=0 P j,i . Define where the subscript i+ℓ is reduced modulo t. Model the rest of the proof on that of Theorem 6.8, distinguishing the case that χ| G 1 is principal on G 1 from the case that χ| G 1 is nonprincipal on G 1 , and applying Lemma 2.5.
Remark 6.12. In the construction of Proposition 6.11, if we could identify an order t 2 c 1 c 2 subgroup U of G 1 × G 2 contained in G 1 × G 2 − t−1 ℓ=1 K ℓ for which G 1 × G 2 − U − t−1 ℓ=1 K ℓ is a regular (t 2 c 1 c 2 , tc 1 c 2 ) Latin square type PDS in G 1 × G 2 , then we could conclude that {K 1 , . . . , K t−1 , G 1 × G 2 − U − t−1 ℓ=1 K ℓ } is a (tc 1 c 2 , t) LP-packing in G 1 × G 2 relative to U. However, we do not know how to identify such a subgroup U (see also Remark 7.3).
Then by Definition 6.1, {P 1 , . . . , P t−1 } is a (c, t − 1) NLP-packing in G from G to H. Remark 7.3. Proposition 7.2 shows the equivalence of an NLP-packing and an elliptic strongly regular bent function, but not the equivalence of an LP-packing and a hyperbolic strongly regular bent function f . In order to establish the converse to Proposition 7.2 (i), we would need to identify an order |G| subgroup U of G contained in f −1 (1 H ) for which f −1 (1 H )−U is a regular |G|, √

|G| |H|
Latin square type PDS in G. The additional conditions on f required to guarantee the existence of such a subgroup U could be substantial, as we discuss in Example 7.4 (i) below.
The following example uses strongly regular bent functions to construct collections of PDSs, and provides an alternative derivation of a special case of Corollary 6.13 (ii). . Let p be a prime, and let ζ p be a primitive p th root of unity. Let g be a function from F p n to F p , and let Tr be the trace function from F p n to F p . The Walsh transform W g : F p n → C of g is given by The function g is bent if |W g (b)| = p n 2 for all b ∈ F p n . A bent function g is weakly regular if there exists a function g * : F p n → F p satisfying W g (b) = µp n 2 ζ g * (b) p for some µ ∈ C with |µ| = 1 and for all b ∈ F p n , and is regular (a much stronger condition) if there exists a function g * : F p n → F p such that W g (b) = p n 2 ζ g * (b) p for all b ∈ F p n . Suppose f : F 3 2s → F 3 is a weakly regular bent function satisfying f (−x) = f (x) and f (0) = 0. Then exactly one of the following holds: (i) |f −1 (0)| = 3 2s−1 + 2 · 3 s−1 , and f is a hyperbolic strongly regular bent function from F 3 2s to F 3 : each of f −1 (1) and f −1 (2) is a regular (3 s , 3 s−1 ) Latin square type PDS in F 3 2s , and f −1 (0) \ {0} is a regular (3 s , 3 s−1 + 1) Latin square type PDS in F 3 2s .

Lemma 8.3.
Let c be a positive integer, let G be an abelian group of order 4c 2 , and let U be a subgroup of order 2c. Then P 1 , P 2 are disjoint 4c 2 , c(2c − 1), c(c − 1) reversible Hadamard difference sets in G whose union is G − U if and only if {P 1 , P 2 } is a (c, 2) LP-packing in G relative to U.
We now use LP-packings in 2-groups to produce infinite families of reversible Hadamard difference sets.
(i) By Lemma 8.3, an equivalent statement is that for w = 1 there exists a (c, 2) LP-packing in G relative to U. To show this when w is even, take p = 2 and (s 1 , s 2 ) = (u, w 2 ) and j = m−1 0 in Corollary 5.4. To show this when w is odd, note that σ(2, w) = w−1 2 1 and take p = 2 and (s 1 , s 2 ) = (u, 0) and j = m − 1 0 in Corollary 5.7.
(ii) In view of part (i), we may assume w = 1. In the case that u and each of the s i is zero, there is a trivial reversible Hadamard difference set D in Z 4 comprising just the identity element. Otherwise, take p = 2 and (s 1 , s 2 ) = (u, 0) and j = m − 1 0 in Corollary 5.4 to produce a ( c 2 , 2) LP-packing in Z 2u 2 × v i=3 G i relative to some subgroup of order c. Use Theorem 6.8 to combine this with the (1, 1) NLP-packing {Z 4 − D} in Z 4 (as given by Lemma 8.2) to produce a (c, 1) NLP-packing in G, then apply Lemma 8.2.
(ii) All abelian 2-groups in which a reversible Hadamard difference set is known to exist are recovered by Corollary 8.4 (ii); several other families of abelian groups are also known to contain reversible Hadamard difference sets (see [7, Chap. VI, Sect. 14]), and all these groups are recovered by [20,Prop. 3.12]) using elliptic strongly regular bent functions.
(iii) The construction of linking systems of reversible Hadamard difference sets in [35,Thm. 5.3] depends on the existence of collections of disjoint partial difference sets in abelian 2groups satisfying mutual structural properties. After some modification, we can reinterpret this construction as combining a (2 (a−1)s , 2 s ) LP-packing in an abelian group of order 2 2as , relative to some subgroup of order 2 as , with the complement of a (2 s − 1, 2 s−1 − 1, 2 s−2 − 1)-difference set in Z 2 s −1 [7, Chap. VI, Thm. 1.10] for s > 1 and a > 0.