Harmonic Equiangular Tight Frames Comprised of Regular Simplices

Air Force Institute of Technology AFIT Scholar

Faculty Publications

2-2020

Matthew C. Fickus Air Force Institute of Technology

Courtney A. Schmitt

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Recommended Citation Fickus, M. C., & Schmitt, C. A. (2020). Harmonic equiangular tight frames comprised of regular simplices. Linear Algebra and Its Applications, 586, 130–169. https://doi.org/10.1016/j.laa.2019.10.019

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To elaborate, letting “ETF(D,N)” denote an N-vector ETF for a D-dimensional space H, ETF(D, D) and ETF(D, D +1) correspond to orthonormal bases and regular simplices for H, respectively, and so exist for every D. Apart from these trivial examples, every other known infinite family of ETFs arises from some type of combinatorial design. Real ETFs in particular are equivalent to a subclass of strongly regular graphs [35, 39, 29, 45], and such graphs are well studied [10, 11, 13]. This equivalence has been partially generalized to the complex setting in various ways, including approaches that exploit properties of roots of unity [8, 6], abelian distance-regular covers of complete graphs [14, 22], and association schemes [30]. Infinite families N of ETFs whose redundancy D is either nearly or exactly two arise from the related concepts of conference matrices, Hadamard matrices, Paley tournaments and Gauss sums [42, 29, 37, 41]. Other constructions are more flexible, allowing one to prescribe the order of magnitude of D and N D almost independently, including harmonic ETFs and Steiner ETFs. As detailed in the next section, harmonic ETFs are equivalent to difference sets in finite abelian groups [44, 42, 47, 17]. Meanwhile, Steiner ETFs arise from balanced incomplete block designs [26, 25]. This construction has recently been generalized to yield new infinite families of ETFs arising from projective planes that contain hyperovals, Steiner triple systems, and group divisible designs [24, 21, 19]. By construction, a Steiner ETF is comprised of regular simplices in the sense that its vectors can be partitioned into subsequences, each of which is a regular simplex for its span. Every harmonic ETF arising from a McFarland difference set is known to be unitarily equivalent to a Steiner ETF, and so also has this structure [31]. In a recent paper [20], it was shown that other harmonic ETFs, including those arising from the complements of certain Singer and twin prime power difference sets, are comprised of regular simplices despite not being unitarily equivalent to any Steiner ETF. There, it was further shown that when an ETF is comprised of regular simplices, the subspaces spanned by these simplices form a particular type of optimal packing in Grassmannian space known as an equi-chordal tight fusion frame (ECTFF), achieving the simplex bound of [12]. Here, we continue this investigation into harmonic ETFs that are comprised of regular simplices. In the next section, we establish notation and review known concepts that we shall use later on. In Section 3, we better characterize the properties of difference sets that lead to ETFs comprised of regular simplices; see Theorem 3.3. In Theorem 3.5, we then characterize when the subspaces spanned by these simplices form a special type of ECTFF known as an equi-isoclinic tight fusion frame (EITFF). This occurs for some, but not all, of the ETFs considered in [20]. We further show that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix C, namely an (S + 1) (S + 1) circulant matrix whose diagonal entries are zero, × whose off-diagonal entries are unimodular and for which C∗C = SI. In Section 4, we refine this analysis further, showing in Theorem 4.2 that a special class of these ETFs arise from difference sets that are a Gordon-Mills-Welch sum of a relative difference set and a smaller difference set. We further show in Theorem 4.4 that these resulting relative difference sets yield collections of regular simplices that are mutually unbiased in the quantum-information-theoretic sense, as well as complex circulant conference matrices in a way that is related to, but distinct from, the method of Section 3. We then show that two known families of difference sets yield ETFs with these extraordinary properties, namely the complements of certain Singer difference sets (Theorem 4.5), and the complements of certain twin prime power difference sets (Theorem 4.6). Overall, these two methods yield (S + 1) (S + 1) circulant conference matrices when either S = Q +1 where Q × is a prime power or S = Q + 2 where Q and Q + 2 are twin prime powers with Q 3 mod 4. ≡

2 2. Background Let z be the complex conjugate of z C. More generally, let A denote the adjoint of ∗ ∈ ∗ an operator A between two complex Hilbert spaces. For any N-element indexing set , let N y , y := [y (n)] y (n) be the standard inner product on C := y : C . For any h 1 2i n 1 ∗ 2 N { N → } M-element indexing∈N set , we can regard a linear operator from C to C as a matrix whose P M N M entries are indexed by , namely as a member of C := A : C , a space we M×N M×N { M×N → } equip with the Frobenius (Hilbert-Schmidt) inner product, A1, A2 Fro := Tr(A1∗A2). h i H C H The synthesis operator of a sequence of vectors ϕn n in Hilbert space is Φ : N , { } ∈N H C → Φy := n y(n)ϕn. Its adjoint is the analysis operator Φ∗ : N , (Φ∗x)(n) = ϕn, x . In ∈N → h i the special case where H = C , Φ is the matrix whose nth column is ϕ , and Φ∗ is its P M M×N n conjugate-transpose. Composing these operators yields the frame operator ΦΦ∗ : H H, N × M C C → ΦΦ∗x = n ϕn, x ϕn and the Gram matrix Φ∗Φ : N N whose (n,n′)th entry ∈N h i N ×N → is (Φ∗Φ)(n,n ) = ϕ , ϕ ′ . We sometimes also regard each vector ϕ as a degenerate synthesis P ′ h n n i n operator ϕ : C H, ϕ (y)= yϕ , an operator whose adjoint is the linear functional ϕ : H C, n → n n n∗ → ϕn∗ x = ϕn, x . Under this notation, the frame operator of ϕn n is ΦΦ∗ = n ϕnϕn∗ . h i H { } ∈N ∈N We say that ϕn n is a (C-)tight frame for when ΦΦ∗ = CI for some C > 0. In this { } ∈N P K case, when the vectors ϕn n are regarded as members of some (larger) Hilbert space which H { } ∈N contains = span ϕn n as a (proper) subspace, we say that ϕn n is a tight frame for its { } ∈N { } ∈N span; elsewhere in the literature, such sequences are sometimes called “tight frame sequences.” Here H C K C the analysis operator Φ∗ : N extends to an operator Φ∗ : N and ϕn n is a tight → H → { } ∈N frame for its span precisely when ΦΦ∗x = Cx for all x = span( ϕn n )=C(Φ). As shown ∈ 2 { } ∈N 2 in [24], this is equivalent to having either ΦΦ∗Φ = CΦ, (ΦΦ∗) = CΦΦ∗ or (Φ∗Φ) = CΦ∗Φ. In particular, ϕn n is a C-tight frame for some D-dimensional space if and only if its Gram { } ∈N matrix Φ∗Φ has eigenvalues C and 0 with multiplicity D and N D, respectively. − H A Naimark complement of an N-vector C-tight frame ϕn n for a D-dimensional space K { } ∈N is any sequence ψn n of vectors in some space such that Φ∗Φ + Ψ∗Ψ = CI. Since Ψ∗Ψ has { } ∈N eigenvalues C and 0 with multiplicity N D and D, respectively, ψn n is a C-tight frame for − { } ∈N its (N D)-dimensional span. Being deﬁned in terms of Gram matrices, Naimark complements − are unique up to unitary transformations. They exist whenever N > D: one way to construct one H CD is to regard as , and take ψn n to be the columns of the (N D) matrix Ψ whose { } ∈N − ×N rows, when taken together with the rows of Φ, form an equal-norm orthogonal basis for CN .

2.1. Equi-chordal and equi-isoclinic tight fusion frames

When ϕn n is a sequence of unit norm vectors, its frame operator ΦΦ∗ = n ϕnϕn∗ is { } ∈N the sum of the orthogonal projection operators onto their 1-dimensional spans. More generally,∈N if P n n is any sequence of M-dimensional subspaces of H, its fusion frame operator is the sum {U } ∈N of the corresponding orthogonal projection operators Pn n . In particular, n n is a tight H { } ∈N {U } ∈N fusion frame (TFF) for if there exists C > 0 such that CI = n Pn. Here, the tight fusion MN ∈N frame constant is necessarily C = D since CD = Tr(CI)= n Tr(Pn)= MN. As such, any P∈N sequence n n of M-dimensional subspaces of H satisfies {U } ∈N P 2 2 2 MN(MN D) MN ′ M N ′ 0 Pn D I = Pn, Pn Fro D = Tr(PnPn ) D − , ≤ − Fro ′ h i − ′ − n n n n n =n X∈N X∈N X∈N X∈N X6 and achieves equality in this bound if and only if n n is a TFF for H. At the same time, any ∈N {U′ } ′ ′ such n n also satisfies n n′=n Tr(PnPn ) N(N 1) maxn=n Tr(PnPn ), and achieves {U } ∈N ∈N 6 ≤ − 6 P P 3 equality in this bound if and only if it is equi-chordal, namely when the (squared) chordal distance 2 dist ( , ′ ) := 1 P P ′ 2 = M Tr(P P ′ ) between any pair of subspaces is the same. c Un Un 2 k n − n k − n n Combining these two inequalities gives that any such n n satisfies {U } ∈N M(MN D) ′ 2 ′ − max Tr(PnPn )= M min distc( n, n ), (2) D(N 1) ≤ n=n′ − n=n′ U U − 6 6 and achieves equality in this bound if and only if it is a TFF for H that is also equi-chordal, M(D M)N H ′ 2 ′ namely an ECTFF for . When rewritten as minn=n distc( n, n ) D(N− 1) , namely as the 6 U U ≤ − simplex bound of [12], we see that an ECTFF n n has the property that the minimum chordal {U } ∈N distance between any pair of these subspaces is as large as possible. In particular, with respect to the chordal distance, an ECTFF is an optimal packing in the Grassmannian space that consists of all M-dimensional subspaces of H. Continuing, for any given sequence n n of M-dimensional subspaces of H, we for each {U } ∈N n let En be the synthesis operator of an orthonormal basis en,m m for n, and so En∗ En = I ∈N { } ∈M U and Pn = EnEn∗ . Here, since n Pn = n EnEn∗ = n m en,men,m∗ , we have that ∈N ∈N ∈N ∈M n n is a TFF for H if and only if the concatenation (union) en,m n ,m of these bases is a {U } ∈N { } ∈N ∈M H P ′ P P P 2 M 2 tight frame for . Moreover, Tr(PnPn ) = Tr(En∗ EnEn∗ ′ En′ )= En∗ En′ Fro = m=1 σn,n′,m where M k k σ ′ are the singular values of the cross-Gram matrix E E ′ , arranged without { n,n ,m}m=1 M × M n∗ nP loss of generality in decreasing order. Here, since the induced 2-norm of such a cross-Gram matrix ′ ′ ′ M satisfies σn,n ,1 = En∗ En′ 2 En 2 En 2 = 1, there exists an increasing sequence θn,n ,m m=1 π k ′ k ≤ k k′ k k { } in [0, 2 ] such that σn,n ,m = cos(θn,n ,m). These principal angles determine the chordal distances be- 2 M M tween subspaces: dist ( , ′ )= M Tr(P P ′ )= M cos2(θ ′ )= sin2(θ ′ ). c Un Un − n n − m=1 n,n ,m m=1 n,n ,m An EITFF is a special type of ECTFF whose principal angles are constant. To elaborate, ′ M 2 ′ 2 ′ P 2 P Tr(PnPn )= m=1 cos (θn,n ,m) M cos (θn,n ,1)= M En∗ En′ 2 for any n = n′. Moreover, n ′ ≤ k k 6 ′ M U and n achieve equality here if and only if they are isoclinic in the sense that θn,n ,m m=1 is U P { } 2 constant over m. This happens precisely when, for some σ ′ 0, we have E E ′ E ′ E = σ ′ I, n,n ≥ n∗ n n∗ n n,n ′ 2 or equivalently that PnPn Pn = σn,n′ Pn. When combined with (2), these facts imply

MN D 1 ′ 2 2 ′ − max Tr(PnPn ) max En∗ En′ 2 = 1 min dists ( n, n ), (3) D(N 1) ≤ M n=n′ ≤ n=n′ k k − n=n′ U U − 6 6 6 2 2 2 where dist ( , ′ ) := 1 E E ′ = sin (θ ′ ) is the (squared) spectral distance between s Un Un − k n∗ n k2 n,n ,1 Un and n′ [16]. Moreover, n n achieves equality throughout (3) if and only if it is an ECTFF for U {U } ∈N H where each pair of subspaces is isoclinic, or equivalently, a TFF for H that is equi-isoclinic [34] in the sense that θ ′ is constant over all n = n and m, namely when there is some σ 0 n,n ,m 6 ′ ≥ such that P P ′ P = σ2P for all n = n . In particular, every EITFF is an optimal packing in n n n n 6 ′ Grassmannian space with respect to both the chordal distance and the spectral distance. In the special case where M = 1, we can let E = ϕ be any unit vector in the line , both (2) n n Un and (3) reduce to the Welch bound (1), and ϕn n achieves equality in this bound if and only { } ∈N if it is a tight frame for H that is also equiangular, namely an ETF for H. An ETF(D, D) equates to D orthonormal vectors. Meanwhile, when D < N, any ETF(D,N) ϕn n has a Naimark { } ∈N complement ψn n which itself is tight and satisﬁes Ψ∗Ψ = CI Φ∗Φ, and so normalizing { } ∈N − these vectors yields an ETF(N D,N). In particular, an ETF(S,S + 1) exists for any positive − integer S, being equivalent to a Naimark complement of an ETF(1,S + 1), namely to a sequence of S + 1 unimodular scalars. We refer to any ETF(S,S + 1) as a regular S-simplex. In light of the 1 Welch bound (1), any S + 1 linearly dependent unit vectors with coherence S necessarily form a regular simplex for their span. 4 ECTFFs and EITFFs that consist of subspaces of dimension M 2 have received some atten- ≥ tion in the literature [34, 28, 12, 48, 33, 9, 32, 18, 5] but not nearly as much as ETFs. EITFFs seem particularly tricky to construct. One approach is to start with a given EITFF and take a Naimark complement of the tight frame formed by concatenating any orthonormal bases for the subspaces that comprise it. Doing so converts an EITFF for a D-dimensional space that consists of N subspaces of dimension M into an EITFF for an (MN D)-dimensional space that consists − of N subspaces of dimension M. Another known method for constructing EITFFs harkens back to [34]: if δm m is any orthonormal basis for an M-dimensional space H, and for each m , (m) { } ∈M ∈ M ϕn n is any ETF(D,N) for K, then the subspaces { } ∈N (m) n n , n := span δm ϕn m (4) {U } ∈N U { ⊗ } ∈M form an EITFF for the MD-dimensional space K H that consists of N subspaces of dimension (m) (m′) (m) (m′) ′ ′ ⊗ M. Indeed, δm ϕn , δm ϕn′ = δm, δm ϕn , ϕn′ for all n,n′ and m,m′ , h ⊗ ⊗ i h(m) ih i ∈ N ∈ M implying that for each n , δm ϕn m is an orthonormal basis for n, and moreover ∈ N { ⊗ } ∈M U that for any n = n′, the corresponding cross-Gram matrix En∗ En′ is diagonal with diagonal entries N6 D 1 of modulus [ − ] 2 . As such, n n achieves equality in (3) where “D” is MD. D(N 1) {U } ∈N Yet another method− for constructing EITFFs is to replace each entry of the synthesis operator Φ of a complex ETF(D,N) with its 2 2 representation as an operator from R2 R2, namely to × x y → apply the one-to-one ring homomorphism z = x + iy [ − ] to every entry of Φ; the resulting 7→ y x pairs of columns form orthonormal bases for N subspaces of R2D, each of dimension M = 2, that form an EITFF for R2D [28]. Recently, it was shown that applying the related yet distinct mapping x y z = x + iy [ y x ] to the entries of a symmetric complex conference matrix of size N yields a 7→ − matrix which, when suitably scaled and added to an identity matrix, yields the Gram matrix of an EITFF for RN that consists of N subspaces of dimension M = 2 [18]. Such a conference matrix can be obtained from a real symmetric conference matrix of size N + 1 [5], which itself equates to a 1 real ETF( 2 (N + 1),N + 1). For example, the well-known real ETF(3, 6) yields 5 planes that form an EITFF for R5. Notably, to date, [18, 5] give the only known method for constructing EITFFs in which the dimension of the subspaces does not divide the dimension of the space they span, and so are veriﬁably not of type (4).

2.2. Harmonic equiangular tight frames and difference sets A character on a finite abelian group is a homomorphism γ : T = z C : z = 1 . G G → { ∈ | | } The (Pontryagin) dual of is the set ˆ of all characters of , and is itself a group under pointwise G G G multiplication. In the general setting, we shall denote the group operations on and ˆ as addition G G and multiplication, respectively. It is well known that since is finite, ˆ is isomorphic to , and G G G moreover that γ : γ ˆ is an equal-norm orthogonal basis for CG, meaning its synthesis operator ˆ { ∈ G} F : C C is invertible with F 1 = 1 F where G is the order of . This operator is usually G → G − G ∗ G regarded as the ( ˆ)-indexed character table of whose (g, γ)th entry is F(g, γ) = γ(g). Its G× G G adjoint is the discrete Fourier transform (DFT) on , (F x)(γ)= γ, x . G ∗ h i Since FF∗ = GI, the rows of F are equal-norm orthogonal. Of course, any subset of these rows also has this property: if is any nonempty D-element subset of , then letting Φ be the ˆ Φ D 1 ΦΦ G I G ( )-index defined by (d, γ)= √ γ(d), we have ∗ = D . Regarding the γth column of D× G 1 D C Φ as the unit norm vector ϕγ = √ γ D, we equivalently have that ϕγ γ ˆ is a tight frame D ∈ { } ∈G for CD. Such tight frames are dubbed harmonic frames, and have a circulant Gram matrix with ′ 1 1 1 1 ϕγ, ϕγ = D d (γ− γ′)(d)= D (F∗χ )(γ(γ′)− ) where χ is the characteristic function of . h i ∈D D D D P 5 The convolution of x , x C is x x C , (x x )(g) := ′ x (g )x (g g ), and 1 2 ∈ G 1 ∗ 2 ∈ G 1 ∗ 2 g 1 ′ 2 − ′ satisfies [F (x x )](γ) = (F x )(γ)(F x )(γ) for all γ ˆ. Meanwhile, the∈G Fourier transform of ∗ 1 ∗ 2 ∗ 1 ∗ 2 ∈ G P the involution x˜(g) := [x( g)]∗ of x CG is the pointwise complex conjugate of F∗x˜. In particular, − 2 ∈ [F∗(χ χ˜ )](γ)= (F∗χ )(γ) for all γ ˆ where, for any g , D ∗ D | D | ∈ G ∈ G

(χ χ˜ )(g)= χ (g′)χ (g′ g)=#[D (g + D)]=# (d, d′) : g = d d′ D ∗ D D D − ∩ { ∈D×D − } g′ is the number of timesX∈Dg can be written as a difference of members of . D Now let be any subgroup of of order H. The Poisson summation formula states that H G F∗χ = Hχ ⊥ where ⊥ := γ ˆ : γ(h) = 1, h is the annihilator of . It is well known H H H { ∈ G ∀ ∈ H} H that is a subgroup of ˆ, and that is isomorphic to the dual of / , via the identification H⊥ G H⊥ G H of γ with the mapping g γ(g); here and throughout, we denote the cosets g + and γ ∈H⊥ 7→ H H⊥ as simply g and γ, respectively. In particular, has order G . A subset of is an -relative H⊥ H D G H difference set (RDS) of if every g / can be written as a difference of members of the same G ∈H D number of times, while no nonzero member of the “forbidden subgroup” can be written in this H way, namely if and only if there exists a scalar Λ such that χ χ˜ = Dδ0 + Λ(1 χ ), where D ∗ D − H 1 = χ is the all-ones vector. Taking Fourier transforms, this equates to having G 2 F∗χ = F∗[Dδ0 + Λ(1 χ )] = D1 + Λ(Gδ1 Hχ ⊥ ). | D| − H − H 2 2 D(D 1) In particular, we necessarily have D = (F∗χ )(1) = D + Λ(G H), that is, Λ = G −H ; | D | − alternatively, this follows from the fact that each of the G H members of c appears the− same − H number of times in the difference table of , namely the ( )-indexed matrix whose (d, d )th D D × D ′ entry is d d . As such, under this assumption, we have that is an -RDS if and only if − ′ D H D(D 1) 2 D ΛH, γ ⊥, γ = 1, Λ= G −H , (F∗χ )(γ) = − ∈H 6 − (5) | D | D, γ/ ⊥, ∈H namely if and only if the corresponding harmonic tight frame ϕγ γ ˆ satisfies { } ∈G D(D 1) 1 √D ΛH, γ = γ′, γ = γ′, Λ= − , ϕ , ϕ ′ = − 6 G H (6) |h γ γ i| D √D, γ = γ . − 6 ′ Following the literature [36], we denote such a set as an RDS( G ,H,D, Λ). D H In the particular case where = 0 , a relative difference set is simply called a difference set. H { } D(D 1) Here, every g = 0 can be written as a difference of members of in exactly Λ = − ways. 6 D G 1 Moreover, since 0 = ˆ and − { }⊥ G 2 1 D(D 1) D(G D) D D(G 1) 2 D ΛH = D G −1 = G −1 = S2 , S := [ G −D ] , − − − − − we see that (5) and (6) reduce to having

D ′ 1 (F∗χ )(γ) = S , γ = 1, i.e., ϕγ, ϕγ = S , γ = γ′. (7) | D | ∀ 6 |h i| ∀ 6 1 Since S also happens to be the Welch bound for G vectors in a D-dimensional space, we obtain C that is a difference set for if and only if ϕγ γ ˆ is an ETF for D. When is a difference D D2G { } ∈G c D set, the quantity D Λ= 2 is known as the order of . The complement of any difference − S D D set for is another difference set for of the same order, and the resulting harmonic ETFs D G G 6 are Naimark complementary. It is also straightforward to verify that shifting or applying a group automorphism of to a difference set of yields another difference set for . G D G G When is an -RDS for , then for any subgroup of of order K, applying the quotient D H G K H map g g + to produces an ( / )-RDS for / , transforming an RDS( G ,H,D, Λ) into an 7→ K D H K G K H RDS( G , H ,D,KΛ). Indeed, (d + ) (d + ) = g + if and only if d d = g + k for some H K K − ′ K K − ′ k . For g , this occurs exactly D times, namely when k = g, d is arbitrary and ∈ K ∈ K − ∈ D d = d. Meanwhile, for g / , this occurs exactly KΛ times, namely Λ times for each k . ′ − ∈ H ∈ K Finally, for g , no such (d, d ) exist, regardless of the value of k. In particular, quotienting ∈ H\K ′ an -RDS by produces a D-element difference set = g : g for / , transforming an H G D H G D { ∈ D} G H RDS( H ,H,D, Λ) into an RDS( H , 1,D,HΛ).

3. Equi-isoclinic subspaces arising from harmonic ETFs

3.1. Fine diﬀerence sets It was recently shown that certain harmonic ETFs are comprised of regular simplices [20]. To see this from basic principles, let be a D-element diﬀerence set in an abelian group of order G, 1 D G D(G 1) 1 let ϕ , ϕ (d) := γ(d) be the corresponding harmonic ETF for C , and let S = [ − ] 2 { γ }γ ˆ γ √D D G D be the reciprocal∈G of the corresponding Welch bound. If S is an integer, then any subsequence−

ϕγ γ of ϕγ γ ˆ that consists of S + 1 linearly dependent vectors is a regular simplex for its { } ∈S { } ∈G C span: being linearly dependent, ϕγ γ is contained in some S-dimensional subspace of D; at { } ∈S 1 U the same time, the coherence of ϕγ γ is , meaning it achieves the Welch bound for any S + 1 { } ∈S S vectors in , and thus is an ETF(S,S + 1) for . Moreover, if is disjoint from a subgroup of U U D H , then the vectors indexed by any coset of its annihilator are trivially dependent, since they G H⊥ sum to zero: for any γ ˆ and d , the Poisson summation formula gives ∈ G ∈ D

′ 1 1 ⊥ 1 G ϕγ (d)= (γγ′)(d)= γ(d)(Fχ )(d)= γ(d) H χ (d) = 0. (8) √D √D H √D H γ′ γ ⊥ γ′ ⊥ ∈XH X∈H

′ ′ ⊥ Altogether, we see that every subsequence ϕγ γ γ of the ETF ϕγ γ ˆ is a regular simplex { } ∈ H { } provided the underlying diﬀerence set is disjoint from a subgroup of order∈G H = G , that is, D H S+1 whose annihilator has order S + 1. In [20], it was further shown that several known families H⊥ of diﬀerence sets have this property. To make these concepts easier to discuss moving forward, we now give this property a name:

Definition 3.1. A D-element difference set for an abelian group of order G is fine if there D G G D(G 1) 1 exists a subgroup of that is disjoint from and is of order H = where S = [ − ] 2 . H G D S+1 G D When this occurs with a specific , we say is -fine. − H D H Below we show that such difference sets are “fine” in the sense that they can “pass through D a fine sieve,” that is, are disjoint from a subgroup of that is as large as any such subgroup can H G be. We further show that when a difference set is fine, every nonidentity coset of the corresponding subgroup intersects in the same number of points. Here, it helps to introduce the following H D notation: if is any subgroup of a finite abelian group , and is any subset of , let H G D G := ( g), g . (9) Dg H∩ D− ∀ ∈ G

7 Example 3.2. As a simple example of a complement of a Singer difference set, let be the cyclic G group Z and let be 6, 11, 7, 12, 13, 3, 9, 14 ; the rationale behind this unusual ordering of the 15 D { } elements of will eventually become apparent. Computing the difference table of , we see that D D each of the 14 nonzero elements of can be written as a difference of members of in exactly D(D 1) 8(7) G D Λ= G −1 = 14 = 4 ways: − 6 11 7 12 13 3 9 14 − 6 0 10 14 9 8 3 12 7 11 5 0 4 14 13 8 2 12 7 1 11 0 10 9 4 13 8 12 6 1 5 0149 313 . 13 7 2 6 1 010414 3 127116 5 0 9 4 9 3 13 2 12 11 6 0 10 14 8 3 7 2 1115 0 D(G 1) 1 8(14) 1 Thus, is a difference set for . Here, S = [ − ] 2 = [ ] 2 = 4 is an integer. Moreover, S+1 = D G G D 7 5 divides G = 15, and so the cyclic group contains− a unique subgroup of order H = G = 3, G S+1 namely = 0, 5, 10 . Being disjoint from , we see that is fine in the sense of Definition 3.1. H { } H D Moreover, since the cosets of partition , we can partition into its intersections with these H G D cosets, namely according to congruency modulo 5:

= 6, 11, 7, 12, 13, 3, 9, 14 = 6, 11 7, 12 13, 3 9, 14 . D { } ∅ ⊔ { } ⊔ { } ⊔ { } ⊔ { } D 8 Here, we note that every nontrivial member of this partition has cardinality S = 4 = 2. Below, we show this is not a coincidence, showing that this property is equivalent to being fine, in general. D For reasons that will eventually become apparent, we elect to express this partition in terms of subsets of itself, as opposed to subsets of cosets of . In particular, in the notation of (9), H H we have = = = , = = = = 5, 10 , = = = = 0, 5 , and D0 D5 D15 ∅ D1 D2 D8 D4 { } D6 D7 D13 D9 { } = = = = 0, 10 . Under this notation, the portion of that lies in the gth coset D11 D12 D3 D14 { } D of is ( + g)= g + [ ( g)] = g + . H D∩ H H∩ D− Dg As evidenced by this example, the set g+ only depends on the coset of to which g belongs: Dg H when g = g + and g = g + are equal, we have g + = (g + )= (g + )= g + ′ . H ′ ′ H Dg D∩ H D∩ ′ H ′ Dg On the other hand, itself depends on one’s choice of coset representative: when g = g , we have Dg ′ = (g g)+ ′ , which is not equal to ′ in general. Dg ′ − Dg Dg Theorem 3.3. If is a difference set for an abelian group of order G and is any subgroup of D G G D(G 1) 1 H of order H that is disjoint from , then H where S = [ − ] 2 . Moreover, the following G D ≤ S+1 G D are equivalent: − G (i) H = S+1 , that is, is -fine; D D H (ii) (F∗χ )(γ)= for all γ ⊥, γ = 1; D − S ∈H 6 D D (iii) #( g)= S for all g , where g := ( g), i.e., χ χ = S χ c . D 6∈ H D H∩ D− D ∗ H H As a consequence, if is fine then S is necessarily an integer that divides D, implying the order D2 D D Λ= 2 of is necessarily a perfect square. − S D

8 Proof. Recall that since is a diﬀerence set for , the DFT of its characteristic function satisﬁes D D G (F∗χ )(1) = D and (F∗χ )(γ) = S for all γ = 1. The fact that is disjoint from along with D | D | 6 H D the Poisson summation formula implies

1 H ⊥ H 0= χ ,χ = G F∗χ , F∗χ = G χ , F∗χ = G D + (F∗χ )(γ) , h H Di h H Di h H Di D γ ⊥ 1 ∈HX\{ } where is the G -element annihilator of . Multiplying by GS and rearranging then gives H⊥ H H HD S S = [ (F∗χ )(γ)], (10) − D D γ ⊥ 1 ∈HX\{ } G namely an expression for S as a sum of H 1 unimodular numbers. In particular, applying the − G G triangle inequality to (10) immediately gives S H 1, namely the claim that H S+1 . G ≤ − ≤ (i ii) If H = S+1 , (10) expresses S as a sum of S unimodular numbers, implying each of ⇔ S these numbers is 1, that is, D (F∗χ )(γ) = 1 for all γ ⊥, γ = 1, namely (ii). Conversely, if S − D ∈ H 6 G (ii) holds, then (F∗χ )(γ) = 1 for all γ ⊥, γ = 1 and so (10) gives S = 1, namely (i). − D D ∈H 6 H − (ii iii) Since (F∗χ )(1) = D, (ii) implies that the pointwise product of F∗χ and χ ⊥ is ⇒ D(S+1) D D H ⊥ D ⊥ (F∗χ )χ = S δ1 S χ . Since (ii) implies (i), we can further simplify this product as D H − H ⊥ DG D ⊥ D G ⊥ (F∗χ )χ = SH δ1 S χ = S ( H δ1 χ ). The Poisson summation formula then gives D H − H − H

⊥ D ⊥ D D F∗(χ χ )= H(F∗χ )χ = S (Gδ1 Hχ )= S F∗(1 χ )= S F∗χ c , D ∗ H D H − H − H H D namely that χ χ = S χ c . Since D ∗ H H

(χ χ )(g)= χ (g′)χ (g g′)=#[ (g + )] = #[ ( g)] = #( g) D ∗ H D H − D∩ H H∩ D− D g′ X∈G for any g , this equates to having #( )= D for all g / . ∈ G Dg S ∈H (iii i) The cosets of partition , and so can be partitioned as = g / (g + g). ⇒ H G D D ⊔ ∈G H D Since is disjoint from , = for the unique coset representative g that lies in . Combining D H Dg ∅ H these facts with (iii) gives D = #( )= #( )=0+( G 1) D , namely (i). D g / Dg H − S For the final conclusions, we now assume∈G His fine, meaning there exists some subgroup of P D G H G that is disjoint from and satisfies (i)–(iii). Here, (i) implies S = H 1 is necessarily an integer, D D2 − and (iii) then gives that S divides D. Thus, the order D Λ= 2 of is a perfect square. − S D For the interested reader, some alternative proofs of these facts are given in Appendix A.

3.2. A new representation of harmonic ETFs arising from fine difference sets As summarized in Theorem 7.5 of [20], several types of difference sets are known to be fine, including McFarland difference sets, the complements of twin prime power difference sets, and the appropriately shifted complements of “half” of all Singer difference sets. Moreover, by Theorem 6.2 of [20], every ETF that is comprised of regular simplices—including every harmonic ETF arising from a fine difference set—gives rise to an ECTFF. Below, we show that some of these ECTFFs are EITFFs while others are not. To do this, it helps to introduce some more notation.

As before, letting ϕγ γ ˆ be the harmonic ETF arising from a diﬀerence set that is ﬁne { } ∈G ˆ D with respect to , we have that for any γ , the subsequence ϕγ′ γ′ γ ⊥ indexed by the γth H ∈ G { } ∈ H 9 coset of is a regular simplex for its span. Here, to facilitate our work below, we elect to instead H⊥ index the vectors in every such regular simplex by a common set, namely ⊥. To be precise, for ˆ H any γ , let Φγ be the synthesis operator of ϕγγ′ γ′ ⊥ , that is, ∈ G { } ∈H ⊥ 1 Φ CD×H , Φ (d, γ′) := ϕ ′ (d)= γ(d)γ′(d). (11) γ ∈ γ γγ √D

This benefit comes at a small price: though both ϕγ′ γ′ γ ⊥ and ϕγγ′ γ′ ⊥ depend on γ, { } ∈ H { } ∈H the former only depends on the coset γ = γ ⊥ to which γ belongs, whereas Φγ is representative H ′ −1 dependent. That said, when γ = γ , we have Φ = Φ ′ Tγ(γ ) where, for any γ , Tγ is the ′ γ γ ∈H⊥ “translation by γ operator over ⊥”, namely the ( ⊥ ⊥)-indexed permutation matrix defined γ H 1 H ×H by T (γ1, γ2) = 1 if and only if γ1γ2− = γ. As such, the column space of Φγ is independent of coset representative. In particular, the following notation for these subspaces is well defined:

′ ′ ⊥ γ γ ˆ/ ⊥ , γ := span ϕγγ γ = C(Φγ). (12) {U } ∈G H U { } ∈H As mentioned above, the results of [20] imply that if is -ﬁne then (12) is an ECTFF for C . D H D Below, we show that this ECTFF is an EITFF for C if and only if = ( g) is a diﬀerence D Dg H∩ D− set for for every g . This is nontrivial since the techniques of [20] do not easily generalize to H ∈ G this harder problem. There, the key idea is that since ϕγγ′ γ′ ⊥ is an ETF(S,S +1) for γ where D(G 1) 1 { } ∈H SU S = [ − ] 2 , the orthogonal projection operator onto can be written as P = Φ Φ∗ . G D Uγ γ S+1 γ γ Here, since− ϕ is an ETF for C , we also have ϕ , ϕ = 1 for any γ , γ { γ}γ ˆ D |h γγ1 γγ2 i| S 1 2 ∈ H⊥ provided γ = γ . Together,∈G these facts imply that for any γ = γ , 6 ′ 6 ′ S2 2 S2 2 P P ′ Φ Φ ′ Tr( γ γ )= (S+1)2 γ∗ γ Fro = (S+1)2 ϕγγ1 , ϕγγ2 = 1. (13) k k ⊥ ⊥ |h i| γ1 γ2 X∈H X∈H

This means these N = H subspaces of CD of dimension M = S achieve equality in (2) and so form C 2 G(D 1) an ECTFF for D: solving for G in S = G −D gives − 2 (S 1)D G (S 1)D S(SH D) G = S2− D , i.e., H = S+1 = S−2 D , i.e., D(H −1) = 1. (14) − − −

One could conceivably continue this approach to characterize when (12) is an EITFF for CD: having S 2 ′ 2 Pγ = S+1 ΦγΦγ∗ , the goal is to characterize when there exists σ such that Pγ Pγ Pγ = σ Pγ for all γ = γ . We did not pursue this approach, and it seems more complicated than our alternative. 6 ′ We instead construct orthonormal bases for (12) that permit the singular values of their cross- Gram matrices to be computed explicitly. To be precise, for any γ ˆ, we obtain an isometry ∈ G Eγ so that Φγ = EγΨ where Ψ is the synthesis operator of a harmonic regular S-simplex that naturally arises in this context:

( / ) 0 ⊥ 1 Ψ C G H \{ }×H , Ψ(g, γ)= γ(g). (15) ∈ √S This matrix Ψ is well-deﬁned: if g = g then g g implying γ(g)= γ(g )γ(g g )= γ(g ) for ′ − ′ ∈H ′ − ′ ′ all γ ⊥. Moreover, the columns ψγ γ ⊥ of Ψ form a regular S-simplex: if γ = γ′, ∈H { } ∈H 6 1 1 1 1 1 1 ψ , ψ ′ = (γ− γ′)(g)= + (γ− γ′)(g)= . h γ γ i S − S S − S g / 0 g / ∈GXH\{ } ∈GXH

10 Here, we have used the fact that g / γ(g) = 0 for all γ ⊥, γ = 1, something which follows ∈G H ∈H 16 from multiplying this equation by γ(g0) 1 = 0. Thus, Ψ∗Ψ = S [(S + 1)I J]. This in turn S+1 P − 6 − implies ΨΨ∗ = S I, a fact which is also straightforward to prove directly. It is not surprising that Φγ = EγΨ for some isometry Eγ: though we do not rely on this fact in our proof below, the interested reader can use (8) to verify that Φγ and Ψ have the same Gram matrix, which in turn implies that such an isometry necessarily exists. What is remarkable however is that this Eγ is necessarily simple and sparse: if Φγ = EγΨ then multiplying this equation by S Ψ∗ gives E = Φ Ψ∗ and so for any d , g / 0 , γ S+1 γ ∈ D ∈ G H\{ } S √S E (d, g)= Φ (d, γ′)[Ψ(g, γ′)]∗ = γ(d) γ′(d g), γ S+1 γ (S+1)√D − γ′ ⊥ γ′ ⊥ X∈H X∈H at which point the Poisson summation formula and the fineness of implies D √ √ √ γ(d), d = g, S ⊥ S H S Eγ(d, g)= γ(d)(Fχ )(d g)= γ(d) G χ (d g)= (S+1)√D H − (S+1)√D H − √D 0, d = g. 6 Moreover, if Eγ is an isometry, then its columns are unit norm, giving yet a third way of proving that (i iii) in Theorem 3.3. Below, for the sake of an elementary self-contained proof, we instead ⇒ take the above expression for Eγ as a given, and use Theorem 3.3 to show that it is an isometry that satisfies Φγ = EγΨ. Both here and later on, we shall also make use of the following fact: Lemma 3.4. Let be a difference set for that is -fine; see Definition 3.1. Then a D -element D G H S subset of is a difference set for if and only if B H H 2 2 D ⊥ ˆ (F∗χ )(γ) = S3 [1 + (S 1)χ ](γ), γ . | B | − H ∀ ∈ G Moreover, in this case, S3 necessarily divides D2.

(S 1)D S(D S) 1 D D D D2 Proof. By (14), H 1= S−2 D 1= S2 −D , which in turn implies H 1 S ( S 1) = S S3 . As − − − − − − −D D2 such, is a diﬀerence set for if and only if for every g ,# (b, b ) : g = b b = 3 ; B H ∈H { ′ ∈B − ′} S − S since Theorem 3.3 gives that S divides D, this is only an integer when S3 divides D2. Moreover, D C since is a S -element subset of , this equates to having χ G satisfy B H B ∈ D S , g = 0, χ χ D D2 ( ˜ )(g)=# (b, b′) : g = b b′ =  S 3 , g 0 , B ∗ B { ∈B − } − S ∈ H\{ }  0, g / , ∈H D 2  namely to having χ χ˜ = S3 [Dδ0+(S D)χ ]. Taking Fourier transforms, and again using (14) B ∗ B − H and the Poisson summation formula, this equates to our claim:

2 2 2 D ⊥ D ⊥ ˆ (F∗χ )(γ) = S3 [D1 + H(S D)χ ](γ)= S3 [1 + (S 1)χ ](γ), γ . | B | − H − H ∀ ∈ G Theorem 3.5. Let be a difference set for that is -fine; see Definition 3.1. Define , Φ D G H Dg γ and Ψ by (9), (11) and (15), respectively. For any γ ˆ, let ∈ G

( / ) 0 √S γ(d), d = g, E CD× G H \{ }, E (d, g)= (16) γ ∈ γ √D 0, d = g. 6 Then: 11 (a) For any γ, γ ˆ, E E ′ is a diagonal matrix whose diagonal entries are given by ′ ∈ G γ∗ γ S 1 (E∗ E ′ )(g, g)= (γ− γ′)(d), g / 0 . γ γ D ∀ ∈ G H\{ } d Xd∈D=g

Moreover, for any γ ˆ, E is an isometry, that is, E E = I, and satisfies Φ = E Ψ. ∈ G γ γ∗ γ γ γ (b) The sequence of subspaces ⊥ given in (12) is an EITFF for C if and only if every {Uγ }γ ˆ/ D is a difference set for . In∈ thisG H case, S3 necessarily divides D2. Dg H (c) If every is a difference set for , then for every γ / , the ( / )-circulant matrix Dg H ∈H⊥ G H 3 ( / ) ( / ) S 2 C C G H × G H , C (g, g′)= γ(d), γ ∈ γ D d d=X∈Dg g′ − 0, g = g , is a conference matrix, that is, satisfies C C = SI where C (g, g ) = ′ γ∗ γ | γ ′ | 1, g = g . 6 ′ Proof. We first prove (a). For any γ, γ ˆ and g, g / 0 , ′ ∈ G ′ ∈ G H\{ } S 1 ′ ′ (Eγ∗ Eγ )(g, g′)= [Eγ (d, g)]∗Eγ (d, g′)= D (γ− γ′)(d). d d X∈D g=X∈Dd=g′

When g = g , the above sum is empty, implying (E E ′ )(g, g )=0. That is, E E ′ is diagonal. 6 ′ γ∗ γ ′ γ∗ γ Moreover, since d : d = g = d : d g + = (g + ) = g + , continuing the { ∈ D } { ∈ D ∈ H} D∩ H Dg above equation gives that the gth diagonal entry of this matrix is

S 1 S 1 S 1 1 E E ′ F χ ( γ∗ γ )(g, g)= D (γ− γ′)(d)= D (γ− γ′)(g + h)= D (γ− γ′)(g)( ∗ g )(γ(γ′)− ). D d h g Xd∈D=g X∈D

In the special case where γ = γ′, combining these facts with Theorem 3.3(iii) gives that Eγ∗ Eγ is a E E S F S diagonal matrix whose gth diagonal entry is ( γ∗ γ)(g, g) = D ( ∗χ g )(1) = D #( g) = 1. That D D is, for any γ ˆ we have E E = I. For the ﬁnal claim of (a), note that since is disjoint from ∈ G γ∗ γ D , any d lies in exactly one nonidentity coset of , implying that for any γ , H ∈ D H ′ ∈H⊥ √S 1 (E Ψ)(d, γ′)= E (d, g)Ψ(g, γ′)= γ(d)Ψ(d, γ′)= γ(d)γ′(d)= Φ (d, γ′). γ γ √D √D γ g / 0 ∈GXH\{ }

For (b), recall that γ is the column space Φγ. By (a), γ = C(Φγ) = C(Eγ Ψ) C(Eγ). U S U ⊆ Moreover, multiplying Φ = E Ψ by Ψ∗ gives E = Φ Ψ∗ and so C(E ) C(Φ ) = . γ γ γ S+1 γ γ ⊆ γ Uγ Thus, C(E )= . When combined with the fact that E E = I, this implies that the columns of γ Uγ γ∗ γ Eγ form an orthonormal basis for γ. As discussed in Section 2, we thus have that γ γ ˆ/ ⊥ , a U {U } ∈G H sequence of N = H subspaces of CD of dimension M = S, is an EITFF for CD if and only if for MN D 1 SH D 1 1 all γ = γ , every singular value of E E ′ is equal to [ − ] 2 = [ − ] 2 = , cf. (3) and (14). ′ γ∗ γ D(N 1) D(H 1) √S 6 − − Moreover, since Eγ∗ Eγ′ is diagonal, its singular values are the absolute values of its diagonal entries. C Altogether, we have that γ γ ˆ/ ⊥ is an EITFF for D if and only if {U } ∈G H S 1 1 E E ′ F χ ( γ∗ γ )(g, g) = D ( ∗ g )(γ(γ′)− ) = , γ = γ′, g / 0 . (17) | | | D | √S ∀ 6 ∈ G H\{ } 12 2 F χ 2 D More simply, this equates to having ( ∗ g )(γ) = S3 for all g / and γ / ⊥. Here, combining | D | ∈H ∈H Theorem 3.3(iii) with the fact that g is a subset of gives that for any g / and γ ⊥, D D H ∈ H ∈ H F χ ⊥ C ( ∗ g )(γ)=#( g)= S . As such, γ γ ˆ/ is an EITFF for D if and only if D D {U } ∈G H

2 D2 S, γ ⊥ D2 F χ 1 χ ⊥ ˆ ( ∗ g )(γ) = S3 ∈H = S3 [ + (S 1) ](γ), γ , g / . | D | 1, γ / ⊥ − H ∀ ∈ G ∈H ∈H By Lemma 3.4, this occurs if and only if for every g / we have that is a difference set for ∈ H Dg , and moreover, this requires S3 to divide D2. Also, since is disjoint from we have = H D H Dg ∅ for all g , which is trivially a difference set for . As such, ⊥ is an EITFF for C if ∈H H {Uγ }γ ˆ/ D and only if every is a difference set for . ∈G H Dg H For (c), we first make a broader observation: for any difference set and subgroup of a D H finite abelian group , and any γ / , the set of all translations of the vector G ∈H⊥ / S x CG H, x (g) := γ(d), γ ∈ γ D d Xd∈D=g is orthonormal, that is, x˜ x = δ . To see this, recall the identification of with the dual of γ ∗ γ 0 H⊥ / given in Section 2. In particular, for any γ ˆ, the DFT of x at γ is G H ∈ G γ ′ ∈H⊥ 1 S 1 (F∗ / xγ )(γ′)= (γ′)− (g)xγ (g)= D (γ′)− (g) γ(d). G H g / g / d ∈GXH ∈GXH Xd∈D=g

To continue, note that since γ , we have (γ ) 1(g) = (γ ) 1(d) whenever d = g and so ′ ∈H⊥ ′ − ′ − S 1 S 1 S 1 (F∗ / xγ)(γ′)= D ((γ′)− γ)(d)= D ((γ′)− γ)(d)= D (F∗χ )(γ− γ′). G H D g / d d ∈GXH Xd∈D=g X∈D

S 1 In particular, for any γ / ⊥ and γ′ ⊥, (7) gives (F∗ / xγ)(γ′) = D (F∗χ )(γ− γ′) = 1. ∈ H ∈ H | | | D | Thus, F (x˜ x )= F x 2 = 1 = F δ , and so x˜ G xH = δ as claimed. ∗ / γ ∗ γ | ∗ / γ| ∗ / 0 γ ∗ γ 0 As anG H aside, note thatG H when is fine,G H combining this fact with (a) gives an independent, D C alternative proof of the result from [20] that γ γ ˆ/ ⊥ is an ECTFF for D, cf. (13): the {U } ∈G H disjointness of and gives x −1 ′ (0) = 0, and so whenever γ = γ we have D H γ γ 6 ′ 2 2 2 2 E∗ E ′ = (E∗ E ′ )(g, g) = x −1 ′ (g) = x −1 ′ = 1; k γ γ kFro | γ γ | | γ γ | k γ γ k g / 0 g / ∈GXH\{ } ∈GXH in light of (14), these N = H subspaces of CD of dimension M = S thus achieve equality in (2). In particular, in the special case where is fine and every is a difference set for , for any D Dg H γ / we have x (0) = 0 while (17) gives x (g) = (E E )(g, g) = 1 for any g = 0. Since C ∈H⊥ γ | γ | | 1∗ γ | √S 6 γ is a ( / )-circulant matrix with C (g, g )= √Sx (g g ), this implies the diagonal entries of C G H γ ′ γ − ′ γ are zero while its off-diagonal entries are unimodular. Moreover, Cγ∗ Cγ = SI since

(C∗ C )(g, g′)= S [x (g′′ g)]∗x (g′′ g′)= S(x˜ x )(g g′)= Sδ (g g′). γ γ γ − γ − γ ∗ γ − 0 − g′′ / X∈G H 13 We give the diﬀerence sets that satisfy the condition of (b) a name:

Definition 3.6. We say a difference set for a finite abelian group is an amalgam if it is -fine D G H for some subgroup of —see Definition 3.1—and moreover for every g , := ( g) H G ∈ G Dg H∩ D− is a difference set for . H We emphasize that the above proofs of Theorems 3.3 and 3.5 are self-contained: though these results were strongly motivated by those of [20], our proofs here do not rely on facts from [20] in any formal way. In particular, though [20] implies that Φγ is the synthesis operator of a regular simplex for its span, we do not assume this fact in our proofs above. In fact, we instead provide an alternative proof of this fact, directly proving Φγ = EγΨ where Eγ is an isometry and Ψ is the synthesis operator of a regular simplex.

Example 3.7. As a continuation of Example 3.2, recall that = 6, 11, 7, 12, 13, 3, 9, 14 is a D { } difference set for = Z . To form the corresponding harmonic ETF, we extract the corresponding G 15 8 rows from the 15 15 character table of . Here, we regard ˆ as Z15, identifying n Z15 with the × 2πi G G ∈ character g ωng where ω = e 15 . (As such, throughout this example, we use additive notation ˆ 7→ on .) That is, the columns ϕn n Z15 of G { } ∈ ω0 ω6 ω12 ω3 ω9 ω0 ω6 ω12 ω3 ω9 ω0 ω6 ω12 ω3 ω9 ω0 ω11 ω7 ω3 ω14 ω10 ω6 ω2 ω13 ω9 ω5 ω1 ω12 ω8 ω4  ω0 ω7 ω14 ω6 ω13 ω5 ω12 ω4 ω11 ω3 ω10 ω2 ω9 ω1 ω8  1  ω0 ω12 ω9 ω6 ω3 ω0 ω12 ω9 ω6 ω3 ω0 ω12 ω9 ω6 ω3  Φ =   √  ω0 ω13 ω11 ω9 ω7 ω5 ω3 ω1 ω14 ω12 ω10 ω8 ω6 ω4 ω2  8    ω0 ω3 ω6 ω9 ω12 ω0 ω3 ω6 ω9 ω12 ω0 ω3 ω6 ω9 ω12     ω0 ω9 ω3 ω12 ω6 ω0 ω9 ω3 ω12 ω6 ω0 ω9 ω3 ω12 ω6     ω0 ω14 ω13 ω12 ω11 ω10 ω9 ω8 ω7 ω6 ω5 ω4 ω3 ω2 ω1      G D 1 1 form an ETF(8, 15), having ϕ , ϕ ′ = [ − ] 2 = for all n = n , and thus achieving equality |h n n i| D(G 1) 4 6 ′ in the Welch bound (1). Letting S = 4 be the− reciprocal of this bound, we further have that is D fine, being disjoint from the unique subgroup of of order H = G = 3, namely = 0, 5, 10 . G S+1 H { } As such, any 8 5 submatrix of Φ whose columns are indexed by a coset of have the property × H⊥ that these columns sum to zero àla (8) and moreover form a regular simplex for their 4-dimensional span. Here, under our identification of ˆ with Z , corresponds to those n Z that have the G 15 H⊥ ∈ 15 property that ωnh = 1 for all h = 0, 5, 10 , namely 0, 3, 6, 9, 12 . ∈H { } { } Here, for any n Z , the matrix Φ defined by (11) is the 8 5 submatrix of Φ whose columns ∈ 15 n × are indexed by the nth coset of , beginning with n, namely Φ = ϕ ϕ ϕ ϕ ϕ . H⊥ n n n+3 n+6 n+9 n+12 Each Φ is unique, but Φ and Φ ′ are related via a 5 5 permutation matrix whenever n n . n n n × − ′ ∈H Moreover, any choice of coset representatives of / yields a partition of the ETF’s vectors. G H For example, concatenating Φ0, Φ1 and Φ2 yields the matrix obtained by perfectly shuffling the

14 columns of Φ: ω0 ω3 ω6 ω9 ω12 ω6 ω9 ω12 ω0 ω3 ω12 ω0 ω3 ω6 ω9 ω0 ω3 ω6 ω9 ω12 ω11 ω14 ω2 ω5 ω8 ω7 ω10 ω13 ω1 ω4  ω0 ω6 ω12 ω3 ω9 ω7 ω13 ω4 ω10 ω1 ω14 ω5 ω11 ω2 ω8  1  ω0 ω6 ω12 ω3 ω9 ω12 ω3 ω9 ω0 ω6 ω9 ω0 ω6 ω12 ω3  Φ Φ Φ =   . (18) 0 1 2 √  ω0 ω9 ω3 ω12 ω6 ω13 ω7 ω1 ω10 ω4 ω11 ω5 ω14 ω8 ω2  8    ω0 ω9 ω3 ω12 ω6 ω3 ω12 ω6 ω0 ω9 ω6 ω0 ω9 ω3 ω12     ω0 ω12 ω9 ω6 ω3 ω9 ω6 ω3 ω0 ω12 ω3 ω0 ω12 ω9 ω6     ω0 ω12 ω9 ω6 ω3 ω14 ω11 ω8 ω5 ω2 ω13 ω10 ω7 ω4 ω1      Meanwhile, removing the ﬁrst row of the character table of Z = / and then normalizing 5 ∼ G H columns yields the synthesis operator (15) of a particularly nice regular 4-simplex:

ω0 ω3 ω6 ω9 ω12 1 ω0 ω6 ω12 ω3 ω9 Ψ =   . (19) √4 ω0 ω9 ω3 ω12 ω6  ω0 ω12 ω9 ω6 ω3      By Theorem 3.5(a), each Φ can be decomposed as Φ = E Ψ where E is the 8 4 isometry n n n n × deﬁned in (16). In particular, the matrices Φ0, Φ1, Φ2 in (18) factor as products of

1000 ω6 0 0 0 ω12 0 0 0 1000 ω11 0 0 0 ω7 0 0 0  0100   0 ω7 0 0   0 ω14 0 0  1  0100  1  0 ω12 0 0  1  0 ω9 0 0  E =   , E =   , E =   , (20) 0 √  0010  1 √  0 0 ω13 0  2 √  0 0 ω11 0  2   2   2    0010   0 0 ω3 0   0 0 ω6 0         0001   0 0 0 ω9   0 0 0 ω3         0001   0 0 0 ω14   0 0 0 ω13              with Ψ, respectively; here, the rows and columns of E are indexed by = 6, 11, 7, 12, 13, 3, 9, 14 n D { } and / 0 = 1, 2, 3, 4 , respectively. That is, each Φ is a regular simplex, being an isomet- G H\{ } { } n ric embedding of Ψ into a particular 4-dimensional subspace of C , namely into = C(E ). D Un n Continuing, Theorem 3.5(a) also implies every cross-Gram matrix En∗ En′ is diagonal. Here, since ω5 + ω10 = 1, − ω 0 0 0 ω2 0 0 0 1 0 ω2 0 0 1 0 ω4 0 0 E∗E = E∗E =   , E∗E =   . (21) 0 1 1 2 −2 0 0 ω8 0 0 2 −2 0 0 ω 0  0 0 0 ω4   0 0 0 ω8          This diagonality is crucial, since it means the singular values of En∗ En′ —the cosines of the principal angles between and ′ —are the absolute values of its diagonal entries. In particular, (21) Un Un implies that every principal angle θ between any pair of the subspaces , , satisfies cos(θ)= 1 , U0 U1 U2 2 meaning these three subspaces are equi-isoclinic. In fact, Theorem 3.5(b) gives that they form an EITFF for C since is an amalgam: from Example 3.2, recall that = when g while for D D Dg ∅ ∈H g / , is either 5, 10 , 0, 5 or 0, 10 , each of which is a difference set for = 0, 5, 10 . ∈H Dg { } { } { } H { } 15 Since is an amalgam, Theorem 3.5(c) also applies: for any n / , we construct the first D ∈ H⊥ column of the 5 5 circulant conference matrix C by reshaping the diagonal entries of E E into × n 0∗ n a 4 1 vector, padding it with a leading 0 entry, and scaling the result by √S = 2. For example, in × the n = 1 and n = 2 cases, the cross-Gram matrices in (21) yield the circulant conference matrices

0 ω4 ω8 ω2 ω 0 ω8 ω ω4 ω2 ω 0 ω4 ω8 ω2 ω2 0 ω8 ω ω4 C =  ω2 ω 0 ω4 ω8  , C =  ω4 ω2 0 ω8 ω  . 1 − 2 −  ω8 ω2 ω 0 ω4   ω ω4 ω2 0 ω8       ω4 ω8 ω2 ω 0   ω8 ω ω4 ω2 0          Circulant conference matrices are interesting objects, and there does not seem to be much literature regarding them. Perhaps this is due to the long-known fact [40, 15, 43] that the only real-valued instances of such a matrix are [ 0 1 ]. The complex circulant conference matrices we obtain here ± 1 0 are conceivably useful in certain applications where complex circulant Hadamard matrices are used, like waveform design for radar. While it is famously conjectured that real-valued N N × circulant Hadamard matrices only exist when N 1, 4 —the circulant Hadamard conjecture— ∈ { } infinite families of complex circulant Hadamard matrices are known; such objects are equivalent to the constant-amplitude, zero-autocorrelation (CAZAC) sequences of [3], and arise from quadratic chirps as well as Björck-Saffari sequences [4], for example. Later on, we show that the above example is but the first member of an infinite family of known fine difference sets that are amalgams and so generate EITFFs and circulant conference matrices via Theorem 3.5. But first, we consider a subclass of amalgams that can be factored in terms of a certain type of relative difference set (RDS). As we shall see, such RDSs lend themselves to a related and yet distinct method for constructing circulant conference matrices.

4. Composite diﬀerence sets and amalgams

4.1. Composite difference sets In the previous section, we showed that the ECTFF (12) arising from a fine difference set is D an EITFF for C if and only if is an amalgam, that is, if and only if for every g , the set D D ∈ G = ( g) is a difference set for . In this section, we show that there are an infinite number Dg H∩ D− H of fine difference sets that are amalgams, as well as an infinite number that are not. In fact, some but not all of these amalgams have an even stronger property, namely that any two nontrivial are Dg translations of each other. For instance, from Example 3.2, recall that = 6, 11, 7, 12, 13, 3, 9, 14 D { } is a difference set for Z that is -fine where = 0, 5, 10 and, for g / , is either 5, 10 , 15 H H { } ∈ H Dg { } 0, 5 or 0, 10 . In particular, there is a choice of representatives of the nonidentity cosets of / { } { } G H such that the corresponding are equal: = = = = 5, 10 . Applying this rationale Dg D1 D2 D8 D4 { } in general, we see that if all nontrivial are translates of some subset of , then there is a Dg B H set of representatives of the nonidentity cosets of / such that = for all a . This in A G H Da B ∈A turn implies that can be written as = + := a + b : a , b , and in fact can be D D A B { ∈A ∈ B} partitioned as = a (a + ). This means the characteristic function of factors as: D ⊔ ∈A B D

χ = χa+ = δa χ = δa χ = χ χ . D B ∗ B ∗ B A ∗ B a a a X∈A X∈A X∈A In the specific example above, = 1, 2, 8, 4 and = 5, 10 . We now give such sets a name: A { } B { } 16 Definition 4.1. We say a difference set for a finite abelian group is composite if it is -fine D G H for some subgroup of —see Definition 3.1—and moreover there exist an S-element subset of H G A and a difference set for such that χ = χ χ . G B H D A ∗ B Below, we show that the set here is necessarily an -relative difference set (RDS) for with A H G particularly simple parameters. Here, recall that applying the quotient map g g to any -RDS 7→ H produces a difference set for / . As we shall see, quotienting the RDS arising from a composite G H A difference set yields / 0 . For example, = 1, 2, 8, 4 is a 0, 5, 10 -RDS for Z since its G H\{ } A { } { } 15 difference table is 12 8 4 − 1 0 14 8 12 2 1 0 9 13 , 8 76 0 4 4 3 2 11 0 and quotienting it by yields the nonzero members of / = Z . We further show that the cross- H G H ∼ 5 Gram matrices of the isometries (16) arising from a composite difference set have the remarkable property that their triple products are scalar multiples of the identity. For example, for the cross- Gram matrices (21) arising from = 6, 11, 7, 12, 13, 3, 9, 14 , we have (E E )(E E )(E E ) is: D { } 0∗ 1 1∗ 2 2∗ 0 ω 0 0 0 ω 0 0 0 ω13 0 0 0 1000 1 0 ω2 0 0 0 ω2 0 0 0 ω11 0 0 1 0100       =   . −8 0 0 ω8 0 0 0 ω8 0 0 0 ω14 0 −8 0010  0 0 0 ω4   0 0 0 ω4   0 0 0 ω7   0001                  As we shall see, this implies that the subspaces (12) arising from a composite difference set are more than equi-isoclinic, having orthogonal projection operators Pn n for which Pn1 Pn2 Pn3 { } ∈N is always a scalar multiple of Pn1 Pn3 . Theorem 4.2. Assume is a composite difference set for , and take , and as in Defini- D G H A B tion 4.1. For any γ ˆ, define E as in (16) and let ζ C , ζ (b) := √S γ(b). Then: ∈ G γ γ ∈ B γ √D D S 1 (a) has cardinality , and is an RDS(S + 1,H,S, − ) that is disjoint from . B S A H H Moreover, G D divides D 1. − − (b) is an amalgam—see Definition 3.6—with = ( a)= for every a . D Da H∩ D− B ∈A (c) The orthogonal projection operators Pγ γ ˆ/ ⊥ onto the subspaces (12) satisfy { } ∈G H ζ , ζ P P P = ζ , ζ ζ , ζ P P , γ , γ , γ ˆ, (22) h γ1 γ3 i γ1 γ2 γ3 h γ1 γ2 ih γ2 γ3 i γ1 γ3 ∀ 1 2 3 ∈ G

1, γ = γ′, where ζγ, ζγ′ = 1 |h i| , γ = γ′. ( √S 6 Proof. For (a), recall from Section 2 that a D-element subset of is an -RDS for if and only D G H SG(S 1) S 1 if it satisﬁes (5); here, since has cardinality S and G = H(S + 1), Λ is necessarily − = − A G H H and is an -RDS for if and only if − A H G

2 1, γ ⊥, γ = 1, (F∗χ )(γ) = ∈H 6 (23) | A | S, γ/ ⊥. ∈H 17 To show this holds, note that since χ = χ χ we have (F∗χ )(γ) = (F∗χ )(γ)(F∗χ )(γ) D A ∗ B D A B for all γ ˆ. Letting γ = 1 gives D = S#( ) and so is a diﬀerence set for of cardinality D ∈ G 2 BD2 B H S . Lemma 3.4 then gives that (F∗χ )(γ) = S3 for any γ / ⊥. When combined with the fact D | B | ∈H from (7) that (F∗χ )(γ) = S for all γ = 1, this implies | D | 6 ∗ 2 2 3 2 (F χD)(γ) D S (F∗χ )(γ) = | ∗ |2 = 2 2 = S, γ / ⊥. (F χB)(γ) S D | A | | | ∀ ∈H D D Meanwhile, the fact that is a S -element subset of implies (F∗χ )(γ) = S for any γ ⊥. B H B ∈ H As such, for any γ , γ = 1, combining this with Theorem 3.3(ii) gives ∈H⊥ 6 ∗ (F χD)(γ) D S (F∗χ )(γ)= (F∗χ )(γ) = S D = 1. A B − − Thus, (23) indeed holds, meaning is an S-element -RDS for the group of order G = H(S +1). A S 1 H G In particular, is an RDS(S + 1,H,S, − ). Moreover, is disjoint from since A H A H

G 1 ⊥ H χH , χ = H F∗χ , F∗χ = χ , F∗χ = S + (F∗χ )(γ)= S + S( 1) = 0. h Ai h H Ai h H Ai A − γ ⊥ γX∈H=1 6 S 1 S2 1 D 1 Here, for any g / , the set (a, a ) : g = a a has cardinality − = − = − , and ∈H { ′ ∈A×A − ′} H G G D so G D divides D 1. − − − S 1 For (b), note that since is an -RDS(S + 1,H,S, − ) that is disjoint from , quotienting A H H H it by yields an S-element diﬀerence set = a : a for the group / of order S + 1 that H A { ∈ A} G H does not contain 0. Thus, = / 0 , and so is a set of representatives of the nonidentity A G H\{ } A cosets of . Moreover, since χ = χ χ = a δa χ = a χa+ , the set can be H D A ∗ B ∈A ∗ B ∈A B D partitioned as a (a + ) where is a subset of . This implies that for any g / , taking the ⊔ ∈A B B PH P ∈H unique a such that a = g, we have ∈A

= ( g)= (a′ + ) g = [(a′ g)+ ] = (a g)+ Dg H∩ D− H∩ B − H∩ − B − B a′ a′ G∈A G∈A is a difference set for , being a shift of the difference set . Since is empty whenever g , H B Dg ∈H we thus have that every is a difference set for , namely that is an amalgam. Moreover, in Dg H D the special case where g = a, the above equation becomes = . Da B For (c), note ζ consists of S + 1 copies of the harmonic ETF that arises from being a { γ}γ ˆ B ∈G ′ S 1 S 1 ˆ difference set for . Indeed, ζγ , ζγ = D b (γ− γ′)(b)= D (F∗χ )(γ(γ′)− ) for any γ, γ′ , H h i B ∈ G and so Lemma 3.4 gives ∈B P 1 2 2 1 2 1, γ(γ′)− ⊥, ′ S ζγ, ζγ = D2 (F∗χ )(γ(γ′)− ) = 1 1 ∈H |h i| | B | , γ(γ′)− / ⊥. S ∈H Moreover, for any a , we have d : d = a = a + = a + = a + b : b . Combining ∈A { ∈ D } Da B { ∈ B} these facts with Theorem 3.5(a), we find that for any γ, γ ˆ, E E ′ is a diagonal matrix where ′ ∈ G γ∗ γ S 1 S 1 1 1 ′ ′ (Eγ∗ Eγ )(a, a)= D (γ− γ′)(a + b)= D (γ− γ′)(a)(F∗χ )(γ(γ′)− ) = (γ− γ′)(a) ζγ, ζγ B h i b X∈B

18 for any a . As such, for any γ , γ , γ ˆ, E E E E and E E E E E E are ∈ A 1 2 3 ∈ G γ∗3 γ1 γ∗1 γ3 γ∗1 γ2 γ∗2 γ3 γ∗3 γ1 diagonal matrices whose ath diagonal entries are

1 1 (E∗ E E∗ E )(a, a) = (γ− γ )(a)(γ− γ )(a) ζ , ζ ζ , ζ , γ3 γ1 γ1 γ3 3 1 1 3 h γ3 γ1 ih γ1 γ3 i 1 1 1 (E∗ E E∗ E E∗ E )(a, a) = (γ− γ )(a)(γ− γ )(a)(γ− γ )(a) ζ , ζ ζ , ζ ζ , ζ , γ1 γ2 γ2 γ3 γ3 γ1 1 2 2 3 3 1 h γ1 γ2 ih γ2 γ3 ih γ3 γ1 i respectively. Here, the a-dependent terms perfectly cancel, yielding

2 E∗ E E∗ E = ζ , ζ I, E∗ E E∗ E E∗ E = ζ , ζ ζ , ζ ζ , ζ I. (24) γ3 γ1 γ1 γ3 |h γ1 γ3 i| γ1 γ2 γ2 γ3 γ3 γ1 h γ1 γ2 ih γ2 γ3 ih γ3 γ1 i 1 E E As such, right-multiplying the second equation by ζ ,ζ γ∗1 γ3 gives h γ3 γ1 i

ζ , ζ E∗ E E∗ E = ζ , ζ ζ , ζ E∗ E , h γ1 γ3 i γ1 γ2 γ2 γ3 h γ1 γ2 ih γ2 γ3 i γ1 γ3 as claimed. (The interested reader can verify that this single property actually implies both state- ments of (24) as special cases.) Moreover, since every Eγ satisﬁes Eγ∗ Eγ = I, this equates to having ζ , ζ E E E E E E = ζ , ζ ζ , ζ E E E E , namely to having (22). In the h γ1 γ3 i γ1 γ∗1 γ2 γ∗2 γ3 γ∗3 h γ1 γ2 ih γ2 γ3 i γ1 γ∗1 γ3 γ∗3 special case where γ = γ = γ , this reduces to the fact that (12) is an EITFF for C , as previously 1 3 6 2 D observed in Theorem 3.5(b).

There is a precedent for orthogonal projection operators that satisfy (22). To elaborate, from (m) Section 2, recall that if δm m is an orthonormal basis for H and for each m , ϕn n { } ∈M (m) ∈ M { } ∈N is any ETF(D,N) for K, then n n , n := span δm ϕn m is an EITFF for K H. In {U } ∈N U { ⊗ } ∈M ⊗ the special case where these M ETFs are identical, we have that for any n , the orthogonal ∈ N projection operator onto is Un

P = (δ ϕ )(δ ϕ )∗ = δ δ∗ ϕ ϕ∗ = I ϕ ϕ∗ . n m ⊗ n m ⊗ n m m ⊗ n n ⊗ n n m m X∈M X∈M In particular, for any n ,n ,n , 1 2 3 ∈N

P P = I (ϕ ϕ∗ ϕ ϕ∗ )= ϕ , ϕ (I ϕ ϕ∗ ), n1 n3 ⊗ n1 n1 n3 n3 h n1 n3 i ⊗ n1 n3 P P P = I (ϕ ϕ∗ ϕ ϕ∗ ϕ ϕ∗ )= ϕ , ϕ ϕ , ϕ (I ϕ ϕ∗ ), n1 n2 n3 ⊗ n1 n1 n2 n2 n3 n3 h n1 n2 ih n2 n3 i ⊗ n1 n3 and so ϕ , ϕ P P P = ϕ , ϕ ϕ , ϕ P P . The similarity between this and (22) h n1 n3 i n1 n2 n3 h n1 n2 ih n2 n3 i n1 n3 is not a coincidence. To explain, note that for any fine difference set, concatenating the matrices E ⊥ from Theorem 3.5 over any choice of representatives of the cosets of and then { γ}γ ˆ/ H⊥ perfectly∈G H shuffling columns yields a D HS block-diagonal matrix, specifically an S S array of × × blocks of size D H. For example, applying this process to (20) yields an 8 12 block diagonal S × × matrix, namely a 4 4 array whose four 2 3 diagonal blocks are × × 1 ω6 ω12 1 ω7 ω14 1 ω13 ω11 1 ω9 ω3 , , , . 1 ω11 ω7 1 ω12 ω9 1 ω3 ω6 1 ω14 ω13 Disregarding scalar multiples of columns, this is simply four copies of the harmonic ETF that arises from the difference set = 5, 10 for = 0, 5, 10 . To see the degree to which this behavior B { } H { } holds in general, note that for any choice of representatives of the nonidentity cosets of / , G H 19 dividing every column of Eγ by the corresponding value of γ(g) yields the matrix whose (d, g)th entry is

1 1 √S γ(d), d = g, √S γ(d g), d g g, γ− (g)Eγ (d, g)= γ− (g) √ = √ − − ∈ D D 0, d = g. D 0, d g / g. 6 − ∈ D Moreover, when d g , the fact that is a subset of implies that the value of γ(d g) − ∈ Dg Dg H − only depends on the coset of to which γ belongs. At the same time, ˆ/ is isomorphic to H⊥ G H⊥ the dual of via the mapping that identiﬁes γ with the restriction of γ to . And, under this H (g) H identiﬁcation, the synthesis operator of the harmonic tight frame ϕ ⊥ that arises from { γ }γ ˆ/ regarding as a subset of is ∈G H Dg H ⊥ (g) g ˆ/ (g) √S Φ CD ×G H , Φ (h, γ) := γ(h). ∈ √D Comparing the previous two equations gives

(g) 1 Φ (d g, γ), d g g, γ− (g)Eγ (d, g)= − − ∈ D 0, d g / . − ∈ Dg

As such, the union of the columns of Eγ over any choice of representatives of the cosets of ˆ/ ⊥ (g) G H is equivalent—via permutations and unimodular scalings—to a union of δg ϕγ γ ˆ/ ⊥ over { ⊗ } ∈G H any choice of representatives of the nonidentity cosets of / ; here, δg is the gth standard basis / 0 G H in the S-dimensional space CG H\{ }. Theorem 3.5 gives that (12) is an EITFF for CD if and only (g) g if each is a difference set for , namely if and only if every ϕ ⊥ is an ETF for C . Dg H { γ }γ ˆ/ D Altogether, we see that every harmonic EITFF produced by Theorem 3.5∈G isH but a disguised version of the classical EITFF construction given in (4). In the special case that is composite, then taking our set of representatives of / 0 to D (a) G H\{ } be , the fact that = for all a implies all ϕ ⊥ are equal, being the harmonic A Da B ∈ A { γ }γ ˆ/ ETF arising from the difference set of . In this case, the∈ unionG H of the columns of E over any B H γ choice of representatives of the cosets of ˆ/ is thus equivalent to a union of all tensor products G H⊥ of the standard basis of CA with this harmonic ETF for CB. From this perspective, it is thus not surprising that an EITFF that arises from a composite difference set obeys (22). For example, since = 6, 11, 7, 12, 13, 3, 9, 14 is composite with = 1, 2, 8, 4 , dividing the D { } A { } four columns of each E in (20) by ωn,ω2n,ω8n,ω4n , respectively, and then concatenating and n { } shuffling the resulting matrices yields a direct sum of four copies of the 2 3 synthesis operator of × the harmonic ETF that arises from = 5, 10 being a difference set for = 0, 5, 10 , namely B { } H { } 1 1 ω5 ω10 Φ(1) = Φ(2) = Φ(8) = Φ(3) = . √ 1 ω10 ω5 2 Here, it is simply a coincidence that the harmonic ETF arising from happens to itself be a regular B simplex. Later on, we provide an explicit construction of a composite difference set where turns B out to be the complement of an arbitrary Singer difference set.

4.2. Simplicial relative difference sets From Theorem 4.2, recall that any composite difference set for yields a relative difference S 1 D G set , specifically an RDS(S +1,H,S, − ) that is disjoint from . As we now explain, any relative A H H 20 difference set of this type has some remarkable properties, regardless of whether it arises from a composite difference set in this way. Here, quotienting such an RDS by produces the difference H set that consists of all S nonidentity members of the group / of order S + 1. That is, any such G H is a set of representatives of the nonidentity cosets of . We give such sets a name: A H Definition 4.3. Let be a subgroup of a finite abelian group . A subset of is a simplicial H G A G -RDS if it is an -RDS for that is also a set of representatives of the nonidentity cosets of . H H G S 1 H Equivalently, is an -RDS(S + 1,H,S, − ) that is disjoint from . A H H H We remark that if is a simplicial RDS for , then if is any subgroup of the corresponding A G K S 1 subgroup of , then quotienting by transforms the simplicial RDS(S + 1,H,S, − ) into H G K H a simplicial RDS(S + 1, H , S, K (S 1)). Below we show that the harmonic tight frame arising K H − from a simplicial RDS is comprised of regular simplices, and moreover, that these simplices are mutually unbiased in the quantum-information-theoretic sense. For example, extracting the rows of the character table of = Z indexed by members of the simplicial RDS = 1, 2, 8, 4 , and G 15 A { } grouping the resulting normalized columns according to cosets of gives H⊥ ω0 ω3 ω6 ω9 ω12 ω1 ω4 ω7 ω10 ω13 ω2 ω5 ω8 ω11 ω14 1 ω0 ω6 ω12 ω3 ω9 ω2 ω8 ω14 ω5 ω11 ω4 ω10 ω1 ω7 ω13 Ξ0 Ξ1 Ξ2 =   , √4 ω0 ω9 ω3 ω12 ω6 ω8 ω2 ω11 ω5 ω14 ω1 ω10 ω4 ω13 ω7  ω0 ω12 ω9 ω6 ω3 ω4 ω1 ω13 ω10 ω7 ω8 ω5 ω2 ω14 ω11      namely three modulated versions of the regular 4-simplex whose synthesis operator Ψ is given 1 in (19). Here, any columns from distinct simplices have an inner product of modulus 2 . Below, we further prove that every simplicial RDS for yields a complex / -circulant con- G G H ference matrix. In the special case where is a simplicial RDS arising from a composite difference A set, this construction reduces to a unimodular scalar multiple of the construction given in Theo- rem 3.5(c). However, as later examples will demonstrate, both this construction here and that of Theorem 3.5 are nontrivial generalizations of this common case, and each is capable of producing instances of circulant conference matrices that the other is not. Here, beginning with any simplicial RDS, such as = 1, 2, 8, 4 for = Z15 for example, we modulate its characteristic function by a A { } G 2πi character of that does not lie in , e.g. (0,ω,ω2, 0,ω4, 0, 0, 0,ω8, 0, 0, 0, 0, 0, 0) where ω = e 15 . G H⊥ We then periodize this vector into one that is indexed by / , e.g. (0,ω,ω2,ω8,ω). As we prove G H below, this new vector is orthogonal to each of its translates. As such, the ( / )-circulant matrix G H with this vector as its first column, e.g.,

0 ω4 ω8 ω2 ω ω 0 ω4 ω8 ω2   ω2 ω 0 ω4 ω8 ,  ω8 ω2 ω 0 ω4     ω4 ω8 ω2 ω 0      is a conference matrix. Since this construction is valid for any character that does not lie in , H⊥ we may, for example, raise each entry of the above matrix to any power not divisible by 3 to obtain another such matrix.

Theorem 4.4. Let be a subgroup of a ﬁnite abelian group . Let be a subset of with H G A G #( )= S = G 1. Let ξ C , ξ (a) := 1 γ(a). Then, the following are equivalent: H γ γ ˆ A γ √S A − { } ∈G ⊆ 21 (i) is a simplicial -RDS for ; see Deﬁnition 4.3. A H G (ii) (F∗χ )(γ)= 1 for all γ ⊥, γ = 1 while (F∗χ )(γ) = √S for all γ / ⊥. A −ˆ ∈H 6 | A | ∈H (iii) For each γ , ξγ′ γ′ γ ⊥ is a regular S-simplex whose vectors sum to zero, and moreover ∈ G { } ∈ H these simplices are mutually unbiased in the sense that ξ , ξ ′ is constant over all γ = γ . |h γ γ i| 6 ′ 1 Moreover, in this case, ξ , ξ ′ = whenever γ = γ , and for any γ / , |h γ γ i| √S 6 ′ ∈H⊥

/ / Cˆ CG H×G H, Cˆ (g, g′) := γ(a), γ ∈ γ a a=X∈Ag g′ − is a circulant conference matrix. In the special case where arises from a composite diﬀerence set A D via Theorem 4.2, Cˆ is a unimodular scalar multiple of the matrix C constructed in Theorem 3.5(c).

G S(S 1) S 1 Proof. Since S = H 1, we have G = H(S +1) and moreover G −H = H− . Thus, is an -RDS − S 1 − A H for if and only if it is an -RDS(S + 1,H,S, − ). Moreover, in this case, (5) and (6) thus give G H H that is an -RDS if and only if A H 1 , γ = γ , γ = γ , 2 1, γ ⊥, γ = 1, S ′ ′ (F∗χ )(γ) = ∈H 6 i.e., ξγ, ξγ′ = 1 6 (25) | A | S, γ/ ⊥, |h i| , γ = γ . ∈H ( √S 6 ′ Moreover, is disjoint from if and only if A H

G 1 ⊥ 0= H χH , χ = H F∗χ , F∗χ = χ , F∗χ = S + (F∗χ )(γ). (26) h Ai h H Ai h H Ai A γ ⊥ Xγ∈H=1 6 S 1 In particular, is a simplicial -RDS if and only if it is an -RDS(S +1,H,S, − ) that is disjoint A H H H from , namely if and only if it satisfies both (25) and (26). H (i ii) In light of the above facts, it suffices to prove that (ii) holds if and only if satisfies ⇔ A both (25) and (26). Here, (ii) immediately implies both (25) and (26). Conversely, if (25) and (26) hold, then (F∗χ )(γ) = √S for all γ / ⊥ and moreover (F∗χ )(γ) γ ⊥,γ=1 is a sequence of | A | ∈H { A } ∈H 6 S unimodular numbers that sum to S, implying (F∗χ )(γ)= 1 for all γ ⊥, γ = 1. − A − ∈H 6 (i iii) Again, it suffices to prove that (iii) holds if and only if satisfies both (25) and (26). ⇔ A ′ ′ ⊥ ˆ ′ 1 Here, ξγ γ γ is a regular S-simplex for each γ if and only if ξγ , ξγ = S for all γ = γ′ { } ∈ H ∈ G |h i| 6 such that γ = γ′. Moreover, if a collection of regular S-simplices are mutually unbiased, that is, if ′ 1 ξγ , ξγ is constant over all γ, γ′ γ = γ′, then this constant is necessarily √ : since ξγ γ ˆ, is a |h i| C 6 G S { } ∈G unit norm tight frame for A, its synthesis operator Ξ satisfies ΞΞ∗ = S I and so

G2 G2 2 2 2 2 1 2 2 = Tr[( I) ] = Tr[(ΞΞ∗) ]= Ξ∗Ξ = G[(1 )+ S( ) ]+ ξ , ξ ′ , S S k kFro S |h γ γ i| γ ˆ γ′ ˆ X X′ ∈G γ =∈Gγ 6 2 1 implying the average value of ξ , ξ ′ over all G(G S 1) choices of γ = γ , γ = γ is : |h γ γ i| − − 6 ′ 6 ′ S 2 G(S+1) 1 ′ 2 1 G 1 G(G S 1) ξγ, ξγ = G(G S 1) [ S S ]= S . − − ′ |h i| − − − γ ˆ γ =γ X∈G X6

22 In particular, (25) holds if and only if ξγ′ γ′ γ ⊥ γ / ⊥ is a collection of H mutually unbiased {{ } ∈ H } ∈G H ˆ regular S-simplices. Meanwhile (26) equates to each ξγ′ γ′ γ ⊥ summing to zero: for any γ , { } ∈ H ∈ G

′ 1 1 ⊥ 1 G ξγ (a)= √ γ(a)γ′(a)= √ γ(a)(Fχ )(a)= √ γ(a) H χ (a) S S H S H γ′ γ ⊥ γ′ ⊥ ∈XH X∈H is zero for all a if and only if is disjoint from , namely if and only if (26) holds. ∈A A H For the final conclusions, now assume is a simplicial -RDS, take any γ / , and define A H ∈ H⊥ Cˆ as in the statement of the result. When g = g , the fact that is disjoint from means γ ′ A H a : a = g g = a : a is empty, and so every diagonal entry of Cˆ is 0. Mean- { ∈A − ′} { ∈A ∈ H} γ while, if g = g , the fact that is a set of representatives of the nonidentity cosets of implies ′ A H that a : a = g g is a singleton set, meaning Cˆ (g, g ) is unimodular, being a “sum” of { ∈A − ′} γ ′ a single unimodular number. As such, all that remains to be seen is that Cˆ ∗Cˆ = SI. Here, (Cˆ Cˆ )(g, g ) = (y˜ y )(g g ), where y C / is the first column of Cˆ , defined by y (g) being γ∗ γ ′ γ ∗ γ − ′ γ ∈ G H γ 0 when g = 0, and as being γ(a) whenever g = 0, where a is the unique member of such that 6 A a = g. It thus suffices to show that y˜ y = Sδ . Taking Fourier transforms over the group / , γ ∗ γ 0 G H this further equates to having (F y )(γ ) 2 = S for all γ in the dual of / which, as noted in | ∗ / γ ′ | ′ G H Section 2, is naturally identified withG H . This is indeed the case: for any γ , H⊥ ′ ∈H⊥ 1 1 1 (F∗ / yγ )(γ′)= (γ′)− (g)yγ (g)= (γ′)− (a)γ(a) = (F∗χ )(γ− γ′), G H A g / a ∈GXH X∈A and combining this with (25) and the fact that γ / gives (F y )(γ ) 2 = S. ∈H⊥ | ∗ / γ ′ | In the special case where is an RDS that arises from a compositeG H difference set in the manner A of Theorem 4.2(a), then for any g / , taking the unique a such that a = g we have ∈ H ∈ A a : d = g = a while Theorem 4.2(b) gives d : d = g = a + = a + . As such, for { ′ ∈A } { } { ∈ D } Da B any g = g , the construction of the circulant conference matrix C of Theorem 3.5(c) reduces to 6 ′ γ 3 3 3 S 2 S 2 S 2 1 ˆ Cγ(g, g′)= D γ(d)= D γ(a + b)= D F∗(χ )(γ− ) γ(a)= zCγ (g, g′) B d b d=X∈Dg g′ X∈B − 3 S 2 1 where z = D F∗(χ )(γ− ) is constant over all g = g′. Since every diagonal entry of both Cγ and B 6 Cˆ is zero, this implies C = zCˆ where z = 1 by Lemma 3.4. γ γ γ | | 4.3. Constructions of fine difference sets, composite difference sets and amalgams We now discuss how the known fine difference sets from [20] fit into the framework discussed here. We shall see that some fine difference sets are amalgams while others are not, and that some amalgams are composite difference sets while others are not.

4.3.1. Singer difference sets F F For any prime power Q, let Q denote the finite field of order Q and let Q× denote its multi- F F plicative group, which is well known to be cyclic. For any integer J 2, let trQJ /Q : QJ Q, J 1 Qj ≥ → trQJ /Q(x) := j=0− x be the field trace, which is a well-known, nontrivial linear functional of F J , regarded as a J-dimensional vector space over the field F . In this setting, the affine hyper- Q P Q plane = x F× : tr J (x) = 1 is a well-known RDS for the cyclic group = F× of order E { ∈ QJ Q /Q } G QJ 23 J Q 1 [36]. Here, the sets x : x F× are the distinct affine hyperplanes of F J that do not − { E ∈ QJ } Q contain 0. The affine hyperplanes and x are equal if and only if x = 1, and are parallel if E E and only if x F×, x = 1. In any other case, these two affine hyperplanes intersect in an affine ∈ Q 6 subspace of codimension 2. Thus, for any x F× , ∈ QJ J 1 Q − , x = 1, #[ (x )] = 0, x F×, x = 1, E ∩ E  ∈ Q 6 J 2  Q , x / F×. − ∈ Q F QJ1 J 1 J 2 As such, letting = Q×, is a -RDS( Q−1 ,Q 1,Q − ,Q − ) for . Quotienting by thus J 1 K E K − − G K produces a Q -element difference set = x F× /F× : tr J (x) = 0 for / = F× /F×; the − E { ∈ QJ Q Q /Q 6 } G K QJ Q complement of is the classical Singer difference set. When J 4 is even, a shift of this difference E ≥ set is known to be fine [20]. Below, we show that this fine difference set is in fact composite, and the RDS from which it arises is part of a larger family of RDSs to which Theorem 4.4 applies. To elaborate, in the special case where J = 2, the aforementioned construction reduces to

Q = x F×2 : tr 2 (x)= x + x = 1 (27) E { ∈ Q Q /Q } being a -RDS(Q + 1,Q 1, Q, 1) for = F×2 where = F×. Since quotienting by produces K − G Q K Q E K a Q-element difference set D for the group / of order Q + 1, we can always shift if necessary G K E so that its quotient avoids 0 , that is, so that is disjoint from . In fact, when Q is even, { } Q E K Q no shift is necessary: every x F satisfies x = x and so tr 2 (x) = x + x = x + x = 0, ∈ Q Q /Q implying is disjoint from . Moreover, when Q is odd, such a shift can be computed explic- K E F (Q+1)/2 Q 1 (Q2 1)/2 itly: letting α be a generator of Q×2 , β = α− satisfies β − = α− − = 1; as such, Q Q − tr 2 (βx) = (βx) + (βx) = β(x x ) for all x F×2 implying Q /Q − ∈ Q (Q+1)/2 1 Q 1 α = β− = x F×2 : tr 2 (βx) = 1 = x F× : x x = β− E E { ∈ Q Q /Q } { ∈ Q2 − } Q is a -RDS for that is disjoint from = F× = x F×2 : x x = 0 . K G K Q { ∈ Q − } Regardless, for any prime power Q, we see that there exists a -RDS(Q + 1,Q 1, Q, 1) K − that is disjoint from , namely an RDS that is simplicial in the sense of Definition 4.3. By K Theorem 4.4(iii), the corresponding (Q2 1)-vector harmonic tight frame is a union of Q + 1 − mutually unbiased regular Q-simplices. Theorem 4.4 also yields a circulant conference matrix of size Q + 1. As we next explain, some quotients of some RDSs of this type naturally arise from composite difference sets in the manner of Theorem 4.2. Here, for any prime power Q and J 2, we consider the Singer-complement difference set that ≥ arises from an affine hyperplane in a 2J-dimensional extension of FQ, namely

= x F×2 /F× : tr 2J (x) = 0 (28) D { ∈ Q J Q Q /Q 6 } 2J 1 Qj where trQ2J /Q(x) = j=0− x . Such difference sets constitute “half” of all Singer-complement difference sets. As noted in [20], the remaining “half” of these difference sets—those that arise in P odd-dimensional extensions of FQ—do not seem to be fine in general, and in fact cannot be fine when Q is an odd power of a prime since in such cases S is not an integer. From above, we know 2J 1 F F that is a difference set of cardinality D = Q − for the cyclic group = Q×2J / Q× of order QD2J 1 G G = Q −1 . It follows that − 2J−1 2J−1 1 J Q 1 Q 1 D(G 1) 2 J G Q 1 G 1= Q( Q 1− ), G D = Q 1− , S := [ G −D ] = Q ,H := S+1 = Q −1 . − − − − − − 24 For to be fine, it must be disjoint from a subgroup of of order H. (To be precise, we shall see D H G that there is always a shift of (28) for which this occurs.) Since is cyclic and S+1 divides G, there F F G is exactly one such subgroup, namely = Q×J / Q×. If is composite, Theorem 4.2(a) implies that H D S 1 J QJ 1 J it factors in terms of an -RDS with parameters (S + 1,H,S, − ) = (Q + 1, − ,Q ,Q 1). H H Q 1 − There is a natural candidate for such an RDS: taking “Q” in (27) to be QJ gives that−

x F×2 : tr 2J J (x) = 1 (29) { ∈ Q J Q /Q } J J J is a -RDS(Q + 1,Q 1,Q , 1) where = F× ; quotienting this by F× produces an -RDS K − K QJ Q H with the desired parameters, namely

= x F×2 /F× : y xF× s.t. tr 2J J (y) = 1 = x F×2 /F× : tr 2J J (x) F× . A { ∈ Q J Q ∃ ∈ Q Q /Q } { ∈ Q J Q Q /Q ∈ Q} Further recall from Theorem 4.2 that when is composite, we expect = (a 1 ) = for D Da H∩ − D B all a . Here, for any a , writing a = y where y F×2 satisfies tr 2J J (y) F×, we have ∈A ∈A ∈ Q J Q /Q ∈ Q 1 = (F× /F×) y− x F×2 /F× : tr 2J (x) = 0 Da QJ Q ∩ { ∈ Q J Q Q /Q 6 } 1 = y x F× /F× : tr 2J (x) = 0 { − ∈ QJ Q Q /Q 6 } = z F× /F× : tr 2J (yz) = 0 . (30) { ∈ QJ Q Q /Q 6 } To continue simplifying this, we recall from finite field theory that finite field traces factor over 2J intermediate fields: the freshman’s dream implies that for any x FQ ∈ J 1 J 1 QJ − QJ Qj − QJ+j trQJ /Q(trQ2J /QJ (x)) = trQJ /Q(1 + x )= (1 + x ) = (1 + x )=trQ2J /QJ (x). Xj=0 Xj=0 J When combined with the fact that tr 2J J (y) F×, and the fact that tr 2J J is linear in Q Q /Q ∈ Q Q /Q while tr J is linear in F , this implies that for any representative z F× of a coset z F× /F×, Q /Q Q ∈ QJ ∈ QJ Q

trQ2J /Q(yz)=trQJ /Q[trQ2J /QJ (yz)] = trQJ /Q[z trQ2J /QJ (y)] = [trQ2J /QJ (y)][trQJ /Q(z)].

When taken together with the fact that tr 2J J (y) = 0, this simpliﬁes (30) as Q /Q 6

= z F× /F× : [tr 2J J (y)][tr J (z)] = 0 = z F× /F× : tr J (z) = 0 . Da { ∈ QJ Q Q /Q Q /Q 6 } { ∈ QJ Q Q /Q 6 } That is, for every a , we have = where = z F× /F× : tr J (z) = 0 is, by definition, ∈A Da B B { ∈ QJ Q Q /Q 6 } the complement of the canonical Singer difference set in the group = F× /F×. Here, the fact H QJ Q that is an -RDS for while is a subset of implies that the sets a a are disjoint. As A H G B H { B} ∈A such, for each a we have a = a a = (a ) . Thus, contains the disjoint union ∈ A B D D∩ H ⊆ D DJ J 1 2J 1 a a . Moreover, as the cardinality of a a is #( )#( )= Q (Q − )= Q − = #( ), ⊔ ∈A B ⊔ ∈A B A B D D is this disjoint union, meaning χ = χ χ where has cardinality S = QJ . D A ∗ B A Comparing this against Definition 4.1, all that remains to be shown is that is fine, namely D that it is disjoint from = F× /F×. Though this is not always the case, it is always possible to H QJ Q shift so as to gain this property without losing the others. In fact, since χ = χ χ where D F D A ∗ B B is a subset of and where is obtained by quotienting (29) by Q×, it suffices to shift (29) so that H F A it becomes disjoint from Q×J . We have already seen how to do this: no shift is necessary when Q (QJ +1)/2 F is even, and when Q is odd, we can multiply (29) by α where α is a generator of Q×2J . We summarize these facts as follows: 25 Theorem 4.5. For any prime power Q, the classical RDS (27) can be shifted to produce a simplicial RDS(Q+1,Q 1, Q, 1). Moreover, for any J 2, the complement of the Singer difference set in the 2 − Q J 1 ≥ 2J 1 cyclic group of order Q −1 can be shifted so as to produce a Q − -element composite difference G − J QJ 1 J set for , and the resulting factors and are a simplicial RDS(Q + 1, Q −1 ,Q ,Q 1) and D G A B QJ 1 − − the complement of the Singer difference set for the Q −1 -element subgroup of , respectively. − H G For any prime power Q and J 2, applying Theorem 4.2 to these composite difference sets ≥ yields an EITFF whose orthogonal projection operators satisfy (22), and also recovers the underly- J QJ 1 J QJ 1 ing RDS(Q + 1, − ,Q ,Q 1). Applying Theorem 4.4 to these RDSs produces − mutually Q 1 − Q 1 unbiased regular QJ−-simplices as well as circulant conference matrices of size QJ +1. In− the special case where Q = 2, J = 2, this construction yields the composite difference set for F 4 Z given in 2× ∼= 15 Example 3.2, and the resulting conference matrices are given in Example 3.7. Meanwhile, applying Theorem 4.4 directly to the (suitably shifted) version of the RDS(Q + 1,Q 1, Q, 1) given in (27) − gives Q 1 mutually unbiased regular Q-simplices as well as circulant conference matrices of size − Q+1. This latter construction is more general, as it is the only one that yields circulant conference matrices of size P +1 where P is prime. We note that a harmonic tight frame appearing from this RDS(Q + 1,Q 1, Q, 1) has recently appeared elsewhere in the frame literature: concatenating − it with the standard basis yields tight frames that achieve the orthoplex bound, and also provide an alternative solution to a reconstruction problem in quantum information theory that is usually solved with mutually unbiased bases [7]. The fact that complements of Singer difference sets factor in this way is not new. Indeed, it is the fundamental idea behind the now-classical method of Gordon, Mills and Welch for producing many nonequivalent difference sets with Singer-complement parameters [27, 36]. In fact, every known example of an RDS either quotients to the entire group or has parameters that match those G G QJ 1 J 1 of the complement of a Singer difference set [36], namely ( H ,H,D, Λ) where H = Q −1 , D = Q − and HΛ = QJ 2(Q 1). In light of Theorem 4.2(a), it is therefore not too surprising− that these − − are the only composite difference sets we have discovered so far.

4.3.2. Twin prime power diﬀerence sets 2 For any odd prime power Q, let := x : x F× and let := F× be the nonzero SQ { ∈ Q} NQ Q\SQ squares and nonsquares in F , with both sets having cardinality 1 (Q 1). When Q and Q + 2 are Q 2 − both powers of odd primes, the set

= ( 0 F× ) ( ) ( ) (31) D { }× Q+2 ⊔ SQ ×NQ+2 ⊔ NQ × SQ+2 is a difference set for = F F of cardinality D = Q +1+2 1 (Q 1)(Q +1) = 1 (Q + 1)2, G Q × Q+2 4 − 2 being the complement of (F 0 ) ( ) ( ), which is the well-known twin Q × { } ⊔ SQ × SQ+2 ⊔ NQ ×NQ+2 prime power difference set for . Here G = Q(Q + 2) and so G 1 2 1 2 D(G 1) 2 G G 1= Q + 2Q 1, G D = 2 (Q + 2Q 1), S := [ G −D ] = Q + 1,H := S+1 = Q, − − − − − and so is fine, being disjoint from the subgroup = F 0 of order H [20]. By Theorem 3.3, D H Q × { } every nonidentity coset of thus intersects in exactly D = 1 (Q + 1) points. This fine difference H D S 2 set can only be an amalgam if the necessary condition of Theorem 3.5(b) is met, namely only if 3 3 2 1 4 S = (Q + 1) divides D = 4 (Q + 1) , that is, only if Q 3 mod 4. Here, it is notable that D 1 ≡ G −D = 1 regardless of whether Q is congruent to 1 or 3 modulo 4. That is, even though every − 26 composite difference set is an amalgam, there are at least some cases in which the necessary condition on composite difference sets given in Theorem 4.2(a) does not imply the necessary condition on amalgams given in Theorem 3.5(b). Put another way, in order for a composite difference set to D2 D 1 exist, both S3 and G −D are necessarily integers. When Q 3 mod− 4, we claim that is an amalgam. Since is disjoint from , it suffices to ≡ D D H show that for all g / , = ( g) is a difference set for . Here, any g / = F 0 ∈H Dg H∩ D− H ∈H Q × { } is of the form g = (x,y) where y =0 and so 6 g = x (F× y) ( x) ( y) ( x) ( y) . D− {− }× Q+2 − ⊔ SQ − × NQ+2 − ⊔ NQ − × SQ+2 − When y , the intersection of g with = F 0 is thus ∈ SQ+2 D− H Q × { } = ( x 0 ) ( x) 0 = (F x) 0 , Dg {− } × { } ⊔ NQ − × { } Q\SQ − × { } which is a difference set for , since F x is a difference set for F , being a shift of the H Q\SQ − Q complement of the difference set . Similarly, when y , SQ ∈NQ+2 = ( x 0 ) ( x) 0 = (F x) 0 , Dg {− } × { } ⊔ SQ − × { } Q\NQ − × { } which is also a difference set for . Overall, we see that is an amalgam when Q 3 mod 4. H D ≡ However, as we now explain, is not composite in general since F and F are only D Q\SQ Q\NQ shifts of each other when Q = 3. Indeed, when Q = 3, F = 0, 2 and F = 0, 1 are 3\S3 { } 3\N3 { } shifts of each other, and so in this case, (31) becomes the composite difference set

(0, 1), (0, 2), (0, 3), (0, 4) (1, 2), (1, 3) (2, 1), (2, 4) { } ⊔ { } ⊔ { } for Z Z . Inverting the isomorphism 1 (1, 1) gives an alternative construction of the composite 3 × 5 7→ difference set 6, 12, 3, 9, 7, 13, 11, 14 for Z given in Example 3.2. { } 15 Meanwhile, for any Q = 3, and are not shifts of each other meaning (31) is an amalgam 6 SQ NQ that is not a composite difference set. For an elementary proof of this fact, take any prime power Q and suppose that there exists x FQ such that Q = Q + x. Here, Q and Q F 1 ∈ N S S N partition Q× into two sets of cardinality 2 (Q 1). Specifically, Q and Q are the even and odd F − S N powers, respectively, of any generator α of Q×. As such, we cannot have x = 0, and moreover if x then dividing = + x by x gives = + 1; since 1 = 12 , this implies ∈ NQ NQ SQ SQ NQ ∈ SQ 0 1 = , a contradiction. Thus, x , and dividing = + x by x gives ∈ SQ − NQ ∈ SQ NQ SQ F F ′ Q = Q +1. This fact is thus also true for any divisor Q′ of Q: when Q×′ Q× we have Q Q N S ′ ′ ′ ′ ⊆ 1 S ⊆ S and so Q + 1 Q +1= Q Q ; since both Q and Q have cardinality 2 (Q′ 1), this S ⊆ S N ⊆ N S N −2 implies Q′ = Q′ + 1. This in turn implies that Q is prime: if not, letting Q′ = P where N S 2 2 Q = P J for some J 2, we have P 2 1 mod 4, and so 1 = β(P 1)/2 = (β(P 1)/4)2 for any ≥ ≡ − − − generator β of F×2 ; thus 0 2 +1= 2 , a contradiction. Moreover, when Q = P is prime, P ∈ SP NP we necessarily have that P 3 mod 4 or else 1 , implying 0 , a contradiction. To ≡ − ∈ SP ∈ NP summarize our progress so far, if and are shifts of each other, then Q is necessarily some SQ NQ prime P 3 mod 4, and = + 1. In particular, this implies that every square modulo P ≡ NP SP is followed by a nonsquare modulo P . Since and partition F× = 1, 2,...,P 1 , this is SP NP P { − } only possible if = 1, 3,...,P 2 and = 2, 4,...,P 1 are the odd and even numbers SP { − } NP { − } modulo P , respectively. As we have already seen, this all indeed happens in the special case where P = 3. However, it fails for all greater primes since 4 = 22 is a square. We summarize these facts as follows: 27 Theorem 4.6. Let Q and Q + 2 be odd twin prime powers, and let be the 1 (Q + 1)2 element D 2 subset (31) of = F F , namely the complement of the classical twin prime power difference G Q × Q+2 set. Then is a fine difference set, and is an amalgam if and only if Q 3 mod 4. Moreover, D ≡ D is a composite difference set only when Q = 3.

Applying Theorem 3.5(c) to these amalgams produces circulant conference matrices of size S +1 = Q + 2. These conference matrices are distinct from those that arise from any known simplicial RDS via Theorem 4.4 since, as mentioned above, all of those are of size Q+1 where Q is a prime power. For example, when Q = 11, applying Theorem 3.5(c) to (31) gives, to our knowledge, the only known construction of a 13 13 circulant conference matrix; since 12 is not a prime power, × the requisite RDS needed to apply Theorem 4.4 to produce such a matrix is not known to exist [36]. These diﬀerence sets also provide our only known construction of amalgams that are not cyclic: when Q = 27, for example, (31) is an amalgam for the group F F = Z Z Z . We further 27 × 29 ∼ 3 × 3 × 87 note that in the case where Q = 11, the fact that is not composite means we should not expect D the orthogonal projection operators onto the subspaces that constitute the corresponding EITFF to satisfy the property given in Theorem 4.2(b); an explicit computation in this case reveals that they indeed do not have this property.

4.3.3. McFarland difference sets FJ For any prime power Q and integer J 2, regard Q as a J-dimensional vector space over the F ≥ QJ 1 finite field Q of order Q. Let be any abelian group of order Q −1 + 1, and let k k ,k=0 be K FJ − {V } ∈K 6 any enumeration of the distinct hyperplanes of Q according to the nonzero members of . The H FJ set = k ,k=0 (k, v) : v k is then a McFarland difference set for the group = Q. D ⊔ ∈K 6 { ∈V } G K× Here, a direct calculation reveals

J J 1 J J 1 Q 1 J Q 1 D(G 1) 2 Q 1 G J D = Q − ( Q −1 ), G = Q ( Q −1 + 1), S := [ G −D ] = Q −1 ,H := S+1 = Q . − − − − As noted in [20], every such is fine, being disjoint from a subgroup of order H = QJ , namely D = 0 FJ . By Theorem 3.3, every coset of intersects in D = QJ 1 points; by inspection, H { }× Q H D S − these intersections are of the form k for some k , k = 0. However, { }×Vk ∈ K 6 2 −2 −2 D2 Q J (Q 1) J 2 QJ (Q 1) S3 = QJ 1− = Q − (Q 1) + QJ 1− − − − is never an integer since 0 < QJ 2(Q 1) < QJ 1. As such, any such difference sets are never − − − amalgams, and so are never composite. That said, even in this case, much of the machinery developed to prove Theorem 3.5 still provides insights into these fine difference sets that go beyond those provided by the techniques of [20]. For instance, when Q = 2 and J = 2, = 1000, 1001, 0100, 0110, 1100, 1111 is a 6- D { } element McFarland difference set for the group = Z4 of order 16. Here, D = 6, G = 16, G 2 S = 3, and is disjoint from the subgroup = 0000, 0010, 0001, 0011 of order H = G = 4. D H { } S+1 We identify ˆ with Z4, regarding n n n n as the character g g g g ( 1)g1n1+g2n2+g3n3+g4n4 . G 2 1 2 3 4 1 2 3 4 7→ − In particular, is identified with those n n n n such that ( 1)n3 = ( 1)n4 = 1, namely H⊥ 1 2 3 4 − − 0000, 1000, 0100, 1100 . Here, to form the corresponding 16 16 character table, we elect to order { } × these characters lexicographically, that is, as 0000, 1000, 0100, 1100, 0010,..., 1111 . Under these { } arbitrarily chosen orderings of and ˆ, the synthesis operator of the corresponding harmonic D G ETF(6, 16), formed by extracting the 6 rows of the character table that correspond to and then D 28 normalizing columns, is:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 − − − − − − − − 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 − − − − − − − − 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  Φ = − − − − − − − − . √6  1 1 1 1 1 11111 1 1 1 1 1 1   − − − − − − − −   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   − − − − − − − −   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1     − − − − − − − −  Here, for any n n n n in ˆ, the corresponding 6 4 submatrix (11) is 1 2 3 4 G ×

Φn1n2n3n4 = ϕn1n2n3n4 ϕ(n1+1)n2n3n4 ϕn1(n2+1)n3n4 ϕ(n1+1)(n2+1)n3n4 . As such, the above matrix Φ can be regarded as the concatenation of Φ0000, Φ0010, Φ0001 and Φ0011. Meanwhile, the matrix Ψ in (15) is obtained by removing the ﬁrst row of the character table of / = Z Z , giving the synthesis operator of a particularly nice tetrahedron: G H ∼ 2 × 2 1 1 1 1 1 − − Ψ = 1 1 1 1 . √  − −  3 1 1 1 1 − −   Theorem 3.5(a) then gives Φn1n2n3n4 = En1n2n3n4 Ψ where

1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 − −  0 1 0   0 1 0   0 1 0   0 1 0  E = , E = , E = , E = . 0000  0 1 0  0010  0 1 0  0001  0 1 0  0011  0 1 0     −     −   0 0 1   0 0 1   0 0 1   0 0 1           0 0 1   0 0 1   0 0 1   0 0 1             −   −    As such, our ETF(6, 16) is comprised of four tetrahedra, each embedded in a 3-dimensional subspace = = C(E ) of C . Moreover, Theorem 3.5(a) implies every corresponding Un3n4 Un1n2n3n4 n1n2n3n4 D cross-Gram matrix is diagonal. In fact, by inspection, the cross-Gram matrix of any pair of the above four isometries is one of the following three matrices:

1 0 0 0 0 0 0 0 0 0 0 0 , 0 1 0 , 0 0 0 .       0 0 0 0 0 0 0 0 1       From [20], we already knew that subspaces , , , are equi-chordal, implying all of {U00 U10 U01 U11} these cross-Gram matrices have the same Frobenius norm. Here, we can say more: taking the absolute values of these diagonal entries of these cross-Gram matrices reveals that the principal angles between any pair of these subspaces are 0, π , π , meaning in particular that any pair of { 2 2 } these subspaces intersect in a line. This is a hallmark feature of the subspaces spanned by the regular simplices that comprise a Steiner ETF. In fact, it is known that harmonic ETFs arising from McFarland diﬀerence sets are unitarily equivalent to certain Steiner ETFs arising from aﬃne geometries [31]. What is remarkable is that, even if this fact was not already known, the machinery of Theorem 3.5 would have naturally led one to this realization. 29 5. Conclusions

We now have three, increasingly nice types of difference sets, namely fine difference sets, amalgams and composite difference sets, as given in Definitions 3.1, 3.6 and 4.1, respectively. Every composite difference set is an amalgam, and every amalgam is fine. In terms of the sets defined Dg in (9), being fine equates to being empty when g , and having equal cardinality otherwise. Dg ∈H When is an amalgam, we further have that each with g / is a difference set for . When D Dg ∈H H is a composite, we even further have that any two such are translates. D Dg Properly shifted, the complements of “half” of all Singer difference sets are fine, and all of these are composite. Meanwhile, the complement of every twin prime power difference set is fine, and these are only amalgams when Q 3 mod 4, and only composite when Q = 3. No McFarland ≡ difference set is an amalgam. Overall, we see that there are an infinite number of composite difference sets, as well as an infinite number of fine difference sets that are not amalgams. Moreover, there is a family of amalgams that are not composite, but whether or not this family is infinite depends on a form of the twin prime conjecture. Every fine difference set yields an ECTFF in a natural way, and this ECTFF is moreover D an EITFF if and only if is an amalgam. When is moreover composite, the corresponding D D orthogonal projection operators behave even more nicely than usual, satisfying (22). Every composite difference set yields a simplicial RDS. Moreover, every amalgam and every simplicial RDS yields a circulant conference matrix. These constructions are two distinct generalizations of a common construction that applies to composite difference sets, with each generalization producing examples that the other does not: for any prime power Q, a simplicial RDS yields a circulant conference matrix of size Q+1; when Q and Q+2 are twin prime powers with Q 3 mod 4, ≡ the corresponding amalgam yields a circulant conference matrix of size Q + 2.

Acknowledgments

We thank the three anonymous reviewers as well as Profs. Dustin Mixon and John Jasper for their many helpful suggestions. The views expressed in this article are those of the authors and do not reﬂect the oﬃcial policy or position of the United States Air Force, Department of Defense, or the U.S. Government.

Appendix A. Alternate proofs of some parts of Theorem 3.3

For a combinatorial proof of some parts of Theorem 3.3, recall that when is a difference set D(D 1) D2 D for , every nonzero member of appears exactly Λ = − = D 2 times in its difference G G G 1 − S table. As such, exactly (H 1)Λ of these differences are nontrivial− members of . Moreover, any − H given g, g have the property that g g if and only if g, g lie in a common coset of . In ′ ∈ G − ′ ∈H ′ H particular, partitioning as = g / (g + g) leads to a corresponding partition of the nonzero D D ⊔ ∈G H D -valued entries of the difference table of : H D

(d, d′) : 0 = d d′ = (d, d′) (g + ) (g + ) : d = d′ . { ∈D×D 6 − ∈ H} { ∈ Dg × Dg 6 } g / ∈GGH 2 Counting these sets gives (H 1)Λ = g / Dg(Dg 1) = D + g / Dg where Dg := #( g). − ∈G H − − ⊔ ∈G H D Moreover, since is disjoint from we have that Dg = 0 for the unique coset representative g D HP G that lies in , that is, such that g = 0. Overall, we have H 1 nonnegative integers Dg g / ,g=0 H − { } ∈G H 6 30 such that D = D and D2 = D + (H 1)Λ = (D Λ) + HΛ. Applying the Cauchy- g=0 g g=0 g − − Schwarz inequality6 to this sequence6 thus gives P P D2 ( G 1)[(D Λ) + HΛ], (A.1) ≤ H − − D2 where equality holds if and only if Dg is constant over all g / . Now recall that Λ = D S2 2 D(G 1) GS2Λ S2 2 ∈ H GS2 − where S = (G −D) and so D2 = G( D 1) = S 1. Multiplying (A.1) by HD2 thus gives − − − GS2 GS2 G D2 G G GS2Λ G G 2 2 ( 1)( 2 + HΛ) = ( 1)( + 2 ) = ( 1)( + S 1), H ≤ HD H − S H − H D H − H − that is, S2 ( G 1)2, namely the claim in Theorem 3.3 that H G . Moreover, when con- ≤ H − ≤ S+1 dition (i) of Theorem 3.3 holds, reversing the above argument gives equality in (A.1), meaning D consists of S equal numbers that sum to D, implying condition (iii). { g}g / ,g=0 For∈G yetH another6 alternative proof of one conclusion of Theorem 3.3, note that if is disjoint D G ⊥ from a subgroup of of order H > S+1 , then (8) implies that ϕγ γ is a linearly dependent H G { } ∈H subsequence of the harmonic ETF ϕγ γ ˆ that consists of fewer than S +1 vectors; since ϕγ γ ˆ 1 { } ∈G { } ∈G has coherence S , this violates a fact from compressed sensing known as the spark bound [20].

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