Kirkman Equiangular Tight Frames and Codes
aa r X i v : . [ c s . I T ] J un Kirkman Equiangular Tight Frames and Codes
John Jasper, Dustin G. Mixon and Matthew Fickus
Member, IEEE
Abstract —An equiangular tight frame (ETF) is a set of unitvectors in a Euclidean space whose coherence is as small aspossible, equaling the Welch bound. Also known as Welch-bound-equality sequences, such frames arise in various applications,such as waveform design and compressed sensing. At the moment,there are only two known flexible methods for constructingETFs: harmonic ETFs are formed by carefully extracting rowsfrom a discrete Fourier transform; Steiner ETFs arise from atensor-like combination of a combinatorial design and a regularsimplex. These two classes seem very different: the vectorsin harmonic ETFs have constant amplitude, whereas SteinerETFs are extremely sparse. We show that they are actuallyintimately connected: a large class of Steiner ETFs can beunitarily transformed into constant-amplitude frames, dubbedKirkman ETFs. Moreover, we show that an important class ofharmonic ETFs is a subset of an important class of KirkmanETFs. This connection informs the discussion of both types offrames: some Steiner ETFs can be transformed into constant-amplitude waveforms making them more useful in waveformdesign; some harmonic ETFs have low spark, making them lessdesirable for compressed sensing. We conclude by showing thatreal-valued constant-amplitude ETFs are equivalent to binarycodes that achieve the Grey-Rankin bound, and then constructsuch codes using Kirkman ETFs.
Index Terms —equiangular tight frame, Welch bound equalitysequence, Welch bound, Grey-Rankin bound
I. I
NTRODUCTION An equiangular tight frame (ETF) is a maximal packing of N lines in an M -dimensional Euclidean space. To be precise,the coherence of N unit vectors { ϕ n } n ∈N in such a space isthe maximal modulus of their inner products: µ := max n = n ′ |h ϕ n , ϕ n ′ i| . (1)In applications such as waveform design [25] and compressedsensing [7], one often wants vectors with low coherence. Here,the best one can hope to achieve is the Welch bound [29]: µ ≥ (cid:0) N − MM ( N − (cid:1) . (2)As detailed later on, equality in (2) is only achieved when the ϕ n ’s are an ETF [25], also known as a Welch bound equality(WBE) sequence.ETFs are not easily found. Indeed, despite over a decadeof active research, only five general constructions of ETFs areknown, and the first two of these are trivial: in any spaceof dimension M , we can always take an orthonormal basis( N = M ) or a regular simplex ( N = M + 1 ). All other knownconstructions involve combinatorial design. For instance, onecan build ETFs with N = 2 M using conference matrices J. Jasper is with the Department of Mathematics, University of Missouri,Columbia, MO 65211, USA.D. G. Mixon and M. Fickus are with the Department of Mathematics andStatistics, Air Force Institute of Technology, Wright-Patterson Air Force Base,OH 45433, USA, e-mail: [email protected] provided either N = 2 j +1 or N = p j + 1 where j is a positiveinteger and p is an odd prime [25]. This class of ETFs islimited in the sense that their redundancy N/M is necessarilytwo. Moreover, they are closely related to other ETFs whichare constructed using the two remaining methods [23], [24].The fourth known construction method is much more ver-satile, relying on the well-studied topic of
Abelian differencesets , namely subsets D of a commutative group G with theproperty that the size of the set { ( d, d ′ ) ∈ D : g = d − d ′ } isindependent of g ∈ G . For decades, it has been known that asubset D of G is a difference set precisely when the modulusof the discrete Fourier transform (DFT) of its characteristicfunction is a perfect spike [27]. As shown in [25], [30], [8],this implies that restricting the characters of G to D yields anETF of N = |G| vectors for a space of dimension M = |D| .In particular, Singer and McFarland difference sets [16] yieldETFs of almost arbitrary redundancy and size.The fifth known construction method involves Steiner sys-tems, namely a set B of blocks (subsets) of a finite set V which has the properties that (i) every block has the samenumber of points, (ii) every point is contained in the samenumber of blocks and (iii) any two points determine a uniqueblock. For many years, it has been known that such systemscan be used to build strongly regular graphs [12]. Moreover,strongly regular graphs with certain parameters are known tobe equivalent to real ETFs [15], [28]. In [10], these ideas aredistilled into a direct method for constructing real or complexETFs via a tensor-like combination of the incidence matrix ofSteiner system with a unimodular regular simplex.Like harmonic ETFs, these Steiner ETFs are extremely flex-ible, providing ETFs whose size and redundancy are arbitrary,up to an order of magnitude. However, whereas harmonicETFs have constant amplitude —the entries of their framevectors have constant modulus—Steiner ETFs are extremelysparse. This sparsity can be a detriment in applications: inradio communication and radar, constant-amplitude waveformsallow more energy to be transmitted by power-limited hard-ware. Moreover, though Steiner ETFs have optimal coherence,are thus good for coherence-based compressed sensing, theyhave terrible spark : a small number of Steiner ETF elementscan be linearly dependent [10]. As such, Steiner ETFs donot satisfy compressed sensing’s Restricted Isometry Property(RIP) in a way that rivals that of random matrices.In this paper, we provide a new method for unitarily trans-forming certain Steiner ETFs into constant-amplitude ETFs.This method only works when the underlying Steiner systemis resolvable , meaning that its blocks B can partitioned intoseveral collections of blocks {B r } r ∈R , where for any r , theblocks in B r form a partition of V . Such systems were firstmade famous in 1850 by Kirkman’s schoolgirl problem , andas such, we dub these frames
Kirkman ETFs . In the next section, we provide the basic mathematicalbackground on Steiner ETFs. In Section 3, we provide theKirkman construction itself, and then use the existing literatureon resolvable Steiner systems to construct several new familiesof constant-amplitude ETFs. It turns out that one of these“new” families—those arising from finite affine geometries—corresponds to one of the most important classes of harmonicETFs, namely those constructed via McFarland difference sets;as discussed in the fourth section, this identification allows us,for the first time, to seriously investigate the RIP propertiesof these McFarland ETFs. In Section 5, we identify a real-valued constant-amplitude ETF with a self-complementarybinary code, and in this context show that the Welch boundis equivalent to the Grey-Rankin bound of coding theory. Wethen use the results from the previous sections to explicitlyconstruct some Grey-Rankin-bound-equality codes.II. P
RELIMINARIES
Throughout this paper, lowercase, uppercase and calligraphydenote an element of a finite set, the number of elements inthat set, and the set itself, respectively. For example, m ∈ M where M = |M| . Also, C M := { x : M → C } denotes the M -dimensional inner product space consisting of all complex-valued functions over M , and bold lowercase and uppercasedenote vectors and operators/matrices, respectively.The synthesis operator of any vectors { ϕ n } n ∈N in C M isthe linear operator Φ : C N → C M , Φ y := P n ∈N y ( n ) ϕ n .In the special case M = { , . . . , M } and N = { , . . . , N } ,the synthesis operator is the M × N matrix Φ = [ ϕ · · · ϕ N ] .The analysis operator Φ ∗ : C M → C N is the adjoint of thesynthesis operator, meaning ( Φ ∗ x )( n ) = ϕ ∗ n x = h x , ϕ n i .In finite frame theory , one seeks ϕ n ’s that meet variousapplication-motivated constraints and whose frame operator ΦΦ ∗ : C M → C M , ΦΦ ∗ x = P n ∈N h x , ϕ n i ϕ n is as well-conditioned as possible. In particular, { ϕ n } n ∈N is a unitnorm tight frame (UNTF) if k ϕ n k = 1 for all n and if ΦΦ ∗ = A I for some constant A . Here, since M A =Tr( A I ) = Tr( ΦΦ ∗ ) = Tr( Φ ∗ Φ ) = P n ∈N k ϕ n k = N ,this constant A is necessarily the frame’s redundancy N/M .This paper is about equiangular tight frames (ETFs), namelyUNTFs { ϕ n } n ∈N which have the additional property that themagnitude of h ϕ n , ϕ n ′ i is independent of n and n ′ . It turnsout that such frames have minimal coherence (1). In short,letting { ϕ n } n ∈N be any unit vectors in C M , we have ≤ Tr[( ΦΦ ∗ − NM I ) ]= Tr[( Φ ∗ Φ ) ] − N M = X n ∈N X n ′ ∈N |h ϕ n , ϕ n ′ i| − N M ≤ N + N ( N − µ − N M . (3)Solving for µ yields the Welch bound (2). Moreover, equalityin (2) forces equality throughout (3). In particular, any unitnorm vectors { ϕ n } n ∈N which achieve the Welch bound arenecessarily equiangular (since |h ϕ n , ϕ n ′ i| = µ for all n = n ′ ),and also a tight frame, since the Frobenius norm of ΦΦ ∗ − NM I is zero, forcing ΦΦ ∗ = NM I . Conversely, any ETF is both tight and equiangular, yielding equality in the first and lastinequalities in (3), respectively, and thus equality in (2).In the next section, we convert some of the Steiner ETFsof [10] into constant-amplitude ETFs. In general, Steiner ETFsare built from special types of balanced incomplete blockdesigns known as (2 , K, V ) -Steiner systems . Such a systemconsists of a set of V points V along with a set of B blocks(subsets) of V , denoted B , with the property that every blockcontains exactly K points, every point is contained in exactly R blocks, and every pair of points is contained in exactly oneblock. Ordering the points and blocks, we can form a real, { , } -valued B × V incidence matrix B that indicates whichpoints belong to which blocks; the rows of this matrix mustsum to K , its columns must sum to some constant number R ,and the dot product of any two distinct columns must be . Forexample, consider the (2 , , -Steiner system that consists ofall -element subsets of a set of elements: B = . (4)Here, B = 6 and R = 3 .As shown in [10], every (2 , K, V ) -Steiner system generatesan ETF of N = V ( R + 1) vectors in a space of dimension M = B . The main idea is to take a tensor-like combinationof B with a unimodular regular simplex of R + 1 vectorsin R -dimensional space. For example, for the (2 , , -Steinersystem (4) in which R = 3 , we can construct such a simplex byremoving a row from a × matrix with orthogonal columnsand unimodular entries, such as a DFT or Hadamard matrix: F = + − + − + + − − + − − + . (5)Here and throughout, “ + ” and “ − ” denote and − , respec-tively. To construct an ETF from B and F , we replace eachof the R nonzero entries in any given column of B with acorresponding row from F , and replace each zero entry of B with a ( R + 1) -long row of zeros. We then normalize theresulting columns. In particular, Figure 1 gives the × ETF Φ obtained by “tensoring” (4) with (5) in this fashion.Since any finite-dimensional space always contains a uni-modular regular simplex, the only restrictions on the existenceof such ETFs arise from restrictions on Steiner systems them-selves. For example, the B and R parameters of a (2 , K, V ) -Steiner system are uniquely determined by K and V accordingto the necessary relationships that BK = V R, R ( K −
1) = V − . (6)The first identity follows from counting the total number of ’s in the incidence matrix B both row-wise and column-wise;the second follows from counting the number of ’s in B thatlie to the right of a in the first column. In particular, for a (2 , K, V ) -Steiner system to exist, both R = ( V − / ( K − and B = V ( V − / [ K ( K − must be integers. Φ = 1 √ + − + − + − + − − + − + − + − + + − − − − − − − − + − − + 0 0 0 0 0 0 0 0 + − − +0 0 0 0 + − − + + − − + 0 0 0 0 . Fig. 1. A Steiner ETF of vectors in -dimensional space obtained by “tensoring” the incidence matrix (4) of a (2 , , -Steiner system with a regularsimplex of vectors in -dimensional space (5) according to [10]. Here, the Welch bound (2) is / , and any two distinct columns have a dot product of thismagnitude. Indeed, grouping the columns as V = 4 sets of R + 1 = 4 vectors (as pictured), any two distinct columns from the same set (like the first andsecond columns of Φ ) have a dot product of − / since it corresponds to a dot product of distinct columns in our regular simplex (5). Meanwhile, any twocolumns from distinct sets (like the first and fifth columns of Φ ) have only one point of common support, since any two distinct points in our Steiner systemdetermine a unique block; as such, the dot product of such columns has value ± / . Such ETFs are sparse—many of the entries of Φ are zero—and alsohave low spark: the first four of these vectors are linearly dependent. In this paper, we show how to unitarily transform this matrix into a constant-amplitudeETF, a trick which “corrects” its sparsity but not its spark. Our approach will only work due to the fact that our Steiner system here is resolvable: the firstand second blocks (rows) in (4) yield a partition of the points (columns), as do its third and fourth blocks, and its fifth and sixth blocks. Our parameters must also satisfy B ≥ V ; known as Fisher’sinequality , this follows from the fact that the V × V matrix B T B is necessarily of full rank, since its off-diagonal entriesare while its diagonal entries are R = ( V − / ( K − > .These facts are important here since the parameters K and V indicate the dimensions of the resulting Steiner ETF. Indeed,in light of (6), the redundancy of a Steiner ETF is NM = V ( R +1) B = K V ( R +1) BK = K V ( R +1) V R = K (1 + R ) . Since (6) and Fisher’s inequality give
K/R = V /B ≤ , theredundancy of a Steiner ETF is essentially K . Moreover, forany fixed K , both M and N grow quadratically with V : M = B = V ( V − K ( K − , N = V ( R + 1) = V ( V − K − + 1) . In particular, in order to build ETFs of various sizes andredundancies, we need explicit constructions of (2 , K, V ) -Steiner systems which permit flexible, independent controlof both K and V . There are three known families of suchsystems [5], all arising from finite geometry.To be precise, for any prime power q and j ≥ , thereexist affine geometry-based Steiner systems with K = q and V = q j +1 . For j ≥ , there also exist projective geometry-based systems with K = q + 1 and V = ( q j +1 − q − .In either case, varying q and j controls the redundancy andsize of the ETF, respectively. The third family is Dennistondesigns [5] in which K = 2 i , V = 2 i + j + 2 i − j forsome ≤ i < j which arise from maximal arcs in projectivespaces [6]. Importantly, both the affine and Denniston designsare resolvable [11]. As we now discuss, this means we cantransform them into constant-amplitude ETFs.III. K IRKMAN
ETF S In this section, we introduce a method for unitarily trans-forming certain Steiner ETFs, like the one depicted in Figure 1,into constant-amplitude ETFs. This method requires the under-lying Steiner system to be resolvable , meaning its blocks B can be partitioned into disjoint subcollections {B r } r ∈R so thatthe blocks in any given B r form a partition for V . For example,the (2 , , -Steiner system given in (4) is resolvable: its firstand second blocks (rows) form a partition for our underlyingset of V = 4 points, as do its third and fourth, and its fifthand sixth. The main idea of this new method is to multiply thesynthesis matrix of a resolvable Steiner ETF, like Figure 1, by a block-Hadamard/DFT matrix to obtain a constant-amplitudeETF; see Figure 2.Not every Steiner system is resolvable. Indeed, if any subsetof the blocks B forms a partition of V , then K must divide V : each block contains K points and there are V points total.This requirement alone prohibits the famous (2 , , -Steinersystem known as the Fano plane from being resolvable. Whencoupled with the previous restriction that K − divides V − ,this new condition subsumes the previous requirement that K ( K − divides V ( V − . Moreover, since we necessarilyhave V ≡ K − and V ≡ K where K isrelatively prime to K − , the Chinese Remainder Theoremgives that these two conditions are equivalent to having V ≡ K mod K ( K − . For resolvable designs, it also turnsout [11] that Fisher’s inequality can be strengthened to Bose’scondition that B ≥ V + R − .Nevertheless, many Steiner systems are resolvable, such asthose arising from affine geometries over finite fields andDenniston designs [11]. It seems to be an open questionwhether or not projective geometries with K = q + 1 and V = ( q j +1 − / ( q − are resolvable when j is odd. At leastis some cases, the answer is yes: when q = 2 and j = 2 , thisis Kirkman’s schoolgirl problem . Since this famous problem isso closely associated with resolvable Steiner systems, we referto the constant-amplitude ETFs that arise from such systemsas
Kirkman ETFs .We now formally verify that every resovable Steiner systemgenerates a (constant-amplitude) Kirkman ETF that is unitarilyequivalent to a (sparse) Steiner ETF. Here, as usual, thequickest way to verify that certain vectors form an ETFis to show they satisfy the Welch bound (2) with equality.In this Steiner-system-induced setting where M = B and N = V ( R + 1) , the lower bound itself is simply /R ; notedin [10], this can be most easily seen by making repeated use ofthe identities (6) to show M ( N − / ( N − M ) = R . This isa special case of a known necessary integrality condition [26];if all the entries in an ETF are suitably-normalized roots ofunity, then M ( N − / ( N − M ) is necessarily an integer.Before stating the result, it is helpful to introduce somenotation. Note that in any resolvable (2 , K, V ) -Steiner system,the number of blocks in any single partition is V /K . Sincethe total number of blocks is B = V R/K , the number ofdistinct partitions of V is R . As such, we enumerate these Ψ = 1 √ + + 0 0 0 0+ − − − × √ + − + − + − + − − + − + − + − + + − − − − − − − − + − − + 0 0 0 0 0 0 0 0 + − − +0 0 0 0 + − − + + − − + 0 0 0 0 = 1 √ + − + − + − + − + − + − + − + − + − + − + − + − − + − + − + − ++ + − − + + − − + + − − + + − − + + − − − − + + + + − − − − + ++ − − + + − − + + − − + + − − ++ − − + − + + − − + + − + − − + Fig. 2. Constructing a constant-amplitude ETF of 16 vectors in -dimensional space by multiplying the Kirkman (resolvable Steiner) ETF of Figure 1 by aunitary block-DFT/Hadamard matrix. Here, the horizontal lines in Φ indicate the way in which the B = 6 blocks (rows) of the underlying Steiner system (4)can be broken up into R = 3 distinct partitions of a set of V = 4 elements. We obtain a new ETF Ψ by multiplying Φ on the left by a unitary block-diagonalmatrix consisting of R = 3 DFT/Hadamard matrices. Multiplying these two matrices blockwise, we see that the resulting ETF has constant-amplitude: everyentry of Ψ is a product of exactly one nonzero entry of Φ with an entry of a DFT/Hadamard matrix. Having constant-amplitude, this ETF Ψ is better suitedthan Φ for certain radio waveform design problems like CDMA [14]. Nonetheless, the second ETF is obtained by applying a unitary operator to the first,meaning they share many of the same linear algebraic properties. In particular, since the first four columns of Φ are linearly dependent, the first four columnsof Ψ are as well. In Section IV, we show that an important subclass of harmonic ETFs, namely those that arise from McFarland difference sets, can be builtwith this approach, and thus are unitarily equivalent to very sparse Steiner ETFs. This allows us, for the first time, to observe that such ETFs do not satisfycompressed sensing’s RIP to any degree that rivals random constructions. partitions using some R -element indexing set R , and writeour blocks as the disjoint union B = ⊔ r ∈R B r . Here, for any r ∈ R , the S := V /K blocks that lie in B r form a partitionfor V , and we index them with some S -element indexing set S . To be precise, for any r ∈ R , let B r = { b r,s } s ∈S where V = ⊔ s ∈S b r,s . We now state and prove our first main result: Theorem 1:
Let ( V , B ) be a resolvable (2 , K, V ) -Steinersystem: let {B r } r ∈R be a partition of B where for any r , B r = { b r,s } s ∈S is a partition of V . Let { f u } Ru =0 be aunimodular regular simplex in C R and let { h s } s ∈S be aunimodular orthogonal basis for C S . Then letting M = R×S and N = V ×{ , . . . , R } , the V ( R +1) vectors { ϕ u,v } ( u,v ) ∈N form a Steiner ETF for the B -dimensional space C M : ϕ u,v ( r, s ) := R − (cid:26) f u ( r ) , v ∈ b r,s , , v / ∈ b r,s . (7)Moreover, applying a unitary operator to this Steiner ETFyields the Kirkman ETF { ψ u,v } ( u,v ) ∈N defined by ψ u,v ( r, s ) := B − f u ( r ) h s ( r,v ) ( s ) , (8)where for any r ∈ R and v ∈ V , s ( r, v ) denotes the unique s ∈ S such that v ∈ b r,s . Proof:
We first prove that { ϕ u,v } ( u,v ) ∈N is an ETF. Todo this, it suffices to show that each ϕ u,v is unit norm andthat the inner product of any distinct two of these vectors hasmodulus equal to the Welch bound /R . In general, we have h ϕ u,v , ϕ u ′ ,v ′ i = X r ∈R X s ∈S ϕ u,v ( r, s )[ ϕ u ′ ,v ′ ( r, s )] ∗ . (9)In the special case where v = v ′ , note that for any r ∈ R , thefact that { b r,s } s ∈S is a partition of V implies there is exactlyone s ∈ S such that v ∈ b r,s . In light of (7), this fact reduces(9) in this case to h ϕ u,v , ϕ u ′ ,v i = R X r ∈R f u ( r )[ f u ′ ( r )] ∗ = R h f u , f u ′ i . When coupled with the fact that { f u } Ru =0 is a unimodularregular simplex, this implies that the ϕ u,v ’s have unit norm andsatisfy |h ϕ u,v , ϕ u ′ ,v i| = 1 /R whenever u = u ′ , as needed.To show that we also have |h ϕ u,v , ϕ u ′ ,v ′ i| = 1 /R whenever v = v ′ , recall that since ( V , B ) is a (2 , K, V ) -Steiner system,there is exactly one block b = b r ,s that contains both v and v ′ . Again recalling (7), this means that there is only onenonzero summand of (9), yielding |h ϕ u,v , ϕ u ′ ,v ′ i| = R | f u ( r ) || f u ′ ( r ) | = R . Thus, { ϕ u,v } ( u,v ) ∈N is an ETF, as claimed. For the secondconclusion, note that by (8), h ψ u,v , ψ u ′ ,v ′ i is: B X r ∈R X s ∈S f u ( r ) h s ( r,v ) ( s )[ f u ′ ( r ) h s ( r,v ′ ) ( s )] ∗ = B X r ∈R f u ( r )[ f u ′ ( r )] ∗ h h s ( r,v ) , h s ( r,v ′ ) i . (10)Fixing r for the moment, note that since { h s } s ∈S is aunimodular orthogonal basis for the space C S of dimension S = V /K = B/R , we have that h h s ( r,v ) , h s ( r,v ′ ) i = B/R when s ( r, v ) = s ( r, v ′ ) and is otherwise zero. Since s ( r, v ) denotes the unique s ∈ S such that v ∈ b r,s , this implies h h s ( r,v ) , h s ( r,v ′ ) i = BR X s ∈S b r,s ( v ) b r,s ( v ′ ) , (11)where b r,s : V → { , } ⊆ C is the characteristic functionof the block b r,s . For every r ∈ R , substituting (11) into (10)and then recalling (7) gives that h ψ u,v , ψ u ′ ,v ′ i = R X r ∈R X s ∈S f u ( r )[ f u ′ ( r )] ∗ b r,s ( v ) b r,s ( v ′ )= X r ∈R X s ∈S ϕ u,v ( r, s )[ ϕ u ′ ,v ′ ( r, s )] ∗ = h ϕ u,v , ϕ u ′ ,v ′ i . (12) Since (12) holds for all ( u, v ) , ( u ′ , v ′ ) ∈ N , we know that { ψ u,v } ( u,v ) ∈N is also an ETF, and moreover, is obtained byapplying a unitary transformation to { ϕ u,v } ( u,v ) ∈N . In truth,this transformation is a block-unitary transform, but we do notneed this specificity for the work that follows.For the remainder of this section, we consider the ramifi-cations of Theorem 1 on the existence of constant-amplitudeETFs. In particular, we first describe Kirkman ETFs that arisefrom known flexible families of resolvable (2 , K, V ) -Steinersystems, meaning they permit independent control of K and V . We then describe some inflexible families, meaning that K uniquely determines V , or vice versa. Finally, we conclude thissection with a discussion of the known asymptotic existenceresults for resolvable Steiner systems.For each of these families, we state whether or not constant-amplitude ETFs with those parameters have been found before.To be clear, the existence of Steiner ETFs of these sizes wasalready noted in [10]. However, before Theorem 1, the onlyknown method for constructing constant-amplitude ETFs wasto use difference sets [30], [8]. We also do our best to answera deeper question: whether or not a given Kirkman ETF isactually a harmonic ETF in disguise. Note that this wouldnecessarily imply that there exist difference sets D of M = B elements in Abelian groups G of order N = V ( R + 1) . This,in turn, requires that Λ := M ( M − N − = V ( V − K ) K ( K − is an integer, since Λ is the number of times any nonzeroelement of G may be written as a difference of two elementsin D . However, every Kirkman ETF automatically satisfies thisintegrality condition: if a resolvable (2 , K, V ) -Steiner systemexists, then V ≡ K mod K ( K − ; writing V = W K ( K −
1) + K gives Λ = W [ W ( K −
1) + 1] . Moreover, this impliesthat the degree M − Λ of such a difference set is necessarilythe perfect square M − Λ = [ W ( K −
1) + 1] , meaning thatthe necessary conditions of the Bruck-Ryser-Chowla Theoremare automatically satisfied whenever N is even [11]. As such,trying to show that a given Kirkman ETF is not harmonic canquickly lead to hard, open problems concerning the existenceof difference sets. A. Flexible Kirkman ETFs1) Affine geometries over finite fields:
For any j ≥ andprime power q , there exists a resolvable (2 , K, V ) -Steinersystem with K = q and V = q j +1 . Here, the points V inthis design are the vectors in F j +1 q where F q is the finite fieldof order q . Meanwhile, the blocks B are affine lines in thisspace, namely sets of the form { au + v : a ∈ F q } for somedirection vector u ∈ F j +1 q \{ } and initial point v ∈ F j +1 q .These systems play an important role in the theory of the nextsection, and we describe them more fully there. For now, themost important things to note are that (i) these systems are easyto construct explicitly, meaning the construction of Theorem 1can be truly implemented; (ii) the resulting Kirkman ETFsconsist of N vectors in an M -dimensional space where M = q j (cid:18) q j +1 − q − (cid:19) , N = q j +1 (cid:18) q j +1 − q − (cid:19) ; (13) and that (iii) the redundancy and size of this ETF can becontrolled by manipulating q and j , respectively. The existenceof constant-amplitude ETFs with these parameters is not new:harmonic ETFs with dimensions (13) can be constructed withMcFarland difference sets [8]. In the next section, we show thisis not a coincidence: we prove that every McFarland harmonicETF is a Kirkman ETF, and as such, is unitarily equivalent toa low-spark, sparse Steiner ETF.
2) Denniston designs:
For any positive integers i and j , i ≤ j , there exists a resolvable (2 , K, V ) -Steiner system with K = 2 i and V = 2 i + j +2 i − j . The construction is nontrivial:one constructs a maximal arc in the projective plane of order j using an irreducible quadratic form [6], and then constructsa resolvable design in terms of this arc [11]. The resultingKirkman ETF has M = (2 j + 1)(2 j + 1 − j − i ) vectors in aspace of dimension N = 2 i (2 j + 2)(2 j + 1 − j − i ) .Note that when i = j , these designs have the sameparameters as an affine geometry where q = 2 i . Meanwhile,for i < j , the constant-amplitude ETFs generated by thesedesigns seem to be new. For example, when i = 2 and j = 3 ,we find that there exists a constant-amplitude ETF of N = 280 vectors in a space of dimension M = 63 ; such an ETF is notfound in the existing literature [30], [8]. Such ETFs might beharmonic: we did not find any examples of i and j for whichit is known that there cannot exist an M -element differenceset in an Abelian group of order N ; since N is even, theBruck-Ryser-Chowla Theorem is toothless. We leave a morethorough investigation of this problem for future research. B. Inflexible Kirkman ETFs1) Round-robin tournaments:
For any positive integer V ,consider the (2 , , V ) -Steiner system that consists of everytwo-element subset of V = { , . . . , V } , such as the (2 , , -Steiner system whose incidence matrix is given in (4). When V is even, this system is resolvable via the famous round-robin schedule, which is sometimes used in tournament com-petitions, as it ensures that each competitor faces all othersexactly once while letting the entire tournament be as quick aspossible. The resulting family of constant-amplitude KirkmanETFs is inflexible, since the redundancy N/M of any suchframe is essentially two: M = V ( V − / , N = V .Some of the constant-amplitude ETFs generated by thesedesigns via Theorem 1 are new. To be clear, when V = 2 j +1 ,ETFs with these parameters are well-known [8], arising fromMcFarland difference sets in Abelian groups isomorphic to Z j +22 . However, when V is even but not a power of , some ofthe resulting Kirkman ETFs do not arise from difference sets.In particular, there does not exist a difference set of M = 45 elements in an Abelian group of order N = 100 [16], andso the (2 , , -Round Robin Kirkman ETF is not harmonic.In Section V, we exploit these ideas to build new examplesof real-valued constant-amplitude ETFs provided there existsa Hadamard matrix of size V ; this leads to new examples of(nonlinear) binary codes that achieve the Grey-Rankin bound.
2) Kirkman’s Schoolgirl Problem:
For any positive integer V ≡ , there exists a resolvable (2 , , V ) -Steiner(triple) system [21]. The resulting Kirkman ETFs have an approximate redundancy of , consisting of V ( V + 1) / vectors in a space of dimension V ( V − / . At least some ofthese frames are new constant-amplitude ETFs: when V = 15 ,for example (Kirkman’s original problem), the resulting ETFconsists of vectors in a space of dimension , andthere does not exist an Abelian difference set with thoseparamters [16].
3) Three-dimensional projective geometries:
For any primepower q , a resolvable (2 , q + 1 , q + q + q + 1) -Steiner systemexists [17]. The resulting family of Kirkman ETFs is inflexible:though both M = ( q +1)( q + q +1) and N = ( q + q +2)( q + q + q + 1) can grow arbitrarily large, the single parameter q determines both the size and redundancy of such frames.Note that when q = 2 this design is a (2 , , -Steiner triplesystem which, as noted above, generates an ETF which is notharmonic. As such, at least some of these frames are newconstant-amplitude ETFs.To be clear, projective geometries generate a flexible familyof Steiner
ETFs: for any j ≥ the projective geometry withparameters K = q + 1 and V = ( q j +1 − / ( q − generatesa Steiner ETF [10] with dimensions M = ( q j − q j +1 − q + 1)( q − , N = q j +1 − q − (cid:18) q j − q − − (cid:19) , and varying q and j independently generates ETFs of varioussizes and redundancies. However, we could only find refer-ences to projective geometries being resolvable—and thus ableto generate Kirkman ETFs—in special cases, like here where j = 3 . Note that in order to be resolvable we need K to divide V which requires j to be odd; it seems to be an open questionwhether such systems are resolvable for odd j ≥ .
4) Unitals:
For any prime power q , there exists a resolvable (2 , q + 1 , q + 1) -Steiner system [3]. This family, like the last,is inflexible, since q determines both M = q ( q − q + 1) and N = ( q +1)( q +1) . At least some of these are new constant-amplitude ETFs: taking q = 3 yields a constant-amplitude ETFof vectors in a space of dimension which, as notedearlier, is not in the literature. We did not find any examplesof any such ETFs which are provably not harmonic; note thatwhenever q is odd, N is even and so the conditions of theBruck-Ryser-Chowla Theorem are automatically satisfied. C. Existence results for Kirkman ETFs
A remarkable fact about resolvable Steiner systems is thatthey are known to exist asymptotically. That is, for any K ≥ it is know that there exists a positive integer V ( K ) such thatfor any V ≥ V ( K ) which satisfies V ≡ K mod K ( K − ,there exists a resolvable (2 , K, V ) -Steiner system [22]. Thus,for any such V , we know there exists a constant-amplitudeETF of N = V ( V − K + 2) / ( K − vectors in a spaceof dimension M = V ( V − / [ K ( K − . In particular,for any K ≥ , there exist constant-amplitude ETFs whoseredundancy N/M is arbitrarily close to K . In contrast, allknown examples of harmonic ETFs have redundancies whichare essentially a power of a prime. Unfortunately, the methodsused to demonstrate the existence of these systems are not constructive in any practical sense. Nevertheless, these exis-tence results encourage the search for explicit constructions ofconstant-amplitude ETFs with arbitrary redundancies.IV. C ONNECTING K IRKMAN
ETF
S AND HARMONIC
ETF S In this section, we show that an important class of harmonicETFs, namely those generated by McFarland difference sets,can also be constructed as Kirkman ETFs via Theorem 1. Asa corollary, we find that harmonic ETFs from this particularfamily have Steiner representations, making them less desir-able for certain compressed-sensing-related applications. Thisresult also demonstrates how truly rare it is to discover a newflexible family of ETFs.To be precise, as noted in the introduction, there are onlythree known approaches for constructing nontrivial ETFs: viaconference matrices [25], difference sets [24], [30], [8] andSteiner systems [10]. In the previous section, we refined theSteiner-based approach so as to produce constant-amplitudeKirkman ETFs. Moreover, nearly all of the particular in-stances of these constructions are inflexible. Indeed, ETFsgenerated by conference matrices have redundancy NM = 2 ;surveying [30], [8] as well as a comprehensive list of Abeliandifference sets [16], we see that harmonic ETFs generated byPaley, Hadamard, twin prime power and Davis-Jedwab-Chendifference sets all have an approximate redundancy of , whilethose generated by other cyclotomic or Spence difference setshave approximate redundancies of either , or ; as notedearlier, families of Steiner systems with a fixed K yield ETFswhose approximate redundancy is K , and those produced fromunital designs are also inflexible.There are only five known flexible families of ETFs: har-monic ETFs arising from (i) Singer difference sets [30] and(ii) McFarland difference sets [8], and Steiner ETFs arisingfrom (iii) affine geometries, (iv) Denniston designs and (v)projective geometries [10]. Also, as shown in the previoussection, classes (iii) and (iv) arise from resolvable Steinersystems, meaning we can apply a unitary operator to themto produce constant-amplitude ETFs. In this section, we showthat modulo such unitaries, (ii) is a special case of (iii). Thatis, we show that in truth there are only four known flexiblefamilies of ETFs; three of these four, namely (i), (ii/iii) and(iv), have constant-amplitude representations; another threeof these four, namely (ii/iii), (iv) and (v), have very sparserepresentations.To show that every McFarland harmonic ETF is a specialexample of an affine Kirkman ETF, let j ≥ , let q be a primepower, and consider G × V where G is any Abelian group oforder ( q j +1 − / ( q −
1) + 1 and V is the additive group offinite field F q j +1 . We form a harmonic ETF over this groupvia the approach of [8] by letting D be a McFarland differenceset in G × V . McFarland’s approach [18] is clever, and iseerily similar to Goethals and Seidel’s method for constructingstrongly regular graphs from Steiner systems [12]; thoughmade independently, we show these two approach are related,since both can lead to the same ETFs.The first step in forming a McFarland difference set is toparametrize the distinct hyperplanes in F q j +1 , regarded as the ( j + 1) -dimensional vector space F j +1 q over the field F q . Notethat each hyperplane is the null space of some nontriviallinear functional over F j +1 q . Moreover, there are q j +1 − such functionals, since each can be uniquely represented by anonzero × ( j + 1) matrix. Also, any two such null spacesare equal precisely when their corresponding functionals arenonzero scalar multiples of each other. As such, there are R := ( q j +1 − / ( q − distinct hyperplanes overall.To find an explicit expression for these hyperplanes, regard F q j +1 as an extension of F q , and let tr q j /q : F q j +1 → F q be theassociated field trace , namely the sum of the automorphismsof F q j +1 that fix F q . Regarding F q j +1 as the vector space F j +1 q ,it is well known that this trace is a nontrivial linear functional,and so its null space S = { v ∈ F q j +1 : tr q j /q ( v ) = 0 } (14)is one example of a hyperplane in F j +1 q , and thus has cardi-nality S = q j . To find the remaining hyperplanes, let γ bea primitive element of F q j +1 , meaning γ generates its cyclicmultiplicative group. Since the mappings v tr q j /q ( γ − r v ) are distinct for every r = 0 , . . . , q j +1 − , every nontrivial lin-ear functional on F j +1 q can be represented this way. Moreover,since the nonzero elements of F q form a ( q − -element sub-group of this multiplicative group of F q j +1 , these functionalsare distinct, even modulo scalar multiplication, provided werestrict the exponent r of γ to the set { , . . . , R − } . Assuch, the R distinct hyperplanes of F q j +1 can be written asthe null spaces of the mappings v tr q j /q ( γ − r v ) , that is, as { γ r S} r ∈R where S is the canonical hyperplane (14).These hyperplanes in hand, we are ready to construct aMcFarland difference set D in G × V , where G is any Abeliangroup of order R + 1 and V is the additive group of F q j +1 .To be precise, letting { g r } Rr =0 be any enumeration of G andletting R := { , . . . , R − } , McFarland [18] showed that D = { ( g, v ) : ∃ r ∈ R such that g = g r , v ∈ γ r S} (15)is a difference set in G×V . Moreover, as discussed in [8], thesedifference sets, like all Abelian difference sets, yield harmonicETFs. Our goal is to show that these McFarland harmonicETFs can also be constructed by applying Theorem 1 in thespecial case where the underlying resolvable Steiner system isan affine geometry. To accomplish this, we next find explicitexpressions for the harmonic ETF that arises from (15).To be precise, [8] gives that the restrictions of the charactersof
G×V to D , suitably normalized, form an ETF for C D . Here,a character of a finite Abelian group is a homomorphism fromthat group into the unit circle in the complex plane. It is wellknown that the characters of any finite Abelian group form aunimodular orthogonal basis over that group, and moreover,that the characters of the direct product G × V are simplythe tensor products of the characters of G with those of theadditive group of F q j +1 .We denote the characters of ( R + 1) -element group G as { χ u } Ru =0 . To form the characters of V , usually called the addi-tive characters of F q j +1 , recall that F q j +1 is an extension of F q ,which in turn is an extension of its base field F p = h i ∼ = Z p ,where the prime p is the characteristic of F q . As such, wehave another trace function tr q j /p : F q j +1 → F p . Moreover, it is well known that the characters { e v } v ∈V of V can be for-mulated in terms of this trace: e v ( v ′ ) := exp( π i P tr q j /p ( vv ′ )) for all v ′ ∈ V . Overall, for any u = 0 , . . . , R + 1 and v ∈ V ,we see that the ( u, v ) th character of G × V is the function ( g, v ′ ) χ u ( g ) e v ( v ′ ) .To form an ETF with the approach of [8], we restrict thedomain of these characters to the difference set (15), andthen normalize the resulting functions. Note that since D isparametrized in terms of R and S with ( g, v ) = ( g r , γ r s ) , weregard these restricted characters as functions over R × S ; forany u = 0 , . . . , R + 1 and v ∈ V , consider ψ u,v ∈ C R×S , ψ u,v ( r, s ) := D − χ u ( g r ) exp( π i P tr q j /p ( vγ r s )) . (16)Note that these ETFs have the exact dimensions (13) ofKirkman ETFs generated via affine geometries: they consistof N = |G × V| = V ( R + 1) = q j +1 [( q j +1 − / ( q −
1) + 1] vectors in a space of dimension M = |R × S| = SR = q j ( q j +1 − / ( q − . Noting the similarity between the formulafor these restricted characters (16) and the formula (8) ofthe Kirkman ETFs from Theorem 1, it becomes even morereasonable that the two types of ETFs are, in fact, the same.To formally make this identification, we construct thesesame vectors { ψ u,v } via the approach of Theorem 1. Inparticular, we show these vectors arise from an affine geometryover the finite field F q which, as stated in the previous sectionhas K = q and V = q j +1 . Indeed, note that the vectors ofany Kirkman ETF generated via such a system lie in a spaceof dimension B = V R/K = q j ( q j +1 − / ( q −
1) = D , andso the normalization factors in both (16) and (8) are identical.Moreover, the fact that { f u } Ru =0 , f u ( r ) := χ u ( g r ) formsa unimodular simplex in C R follows from the fact that thecharacters { χ u } Ru =0 form a unimodular orthogonal basis in C G where G = { g r } Rr =0 = { g r } r ∈R ∪ { g R } . Indeed, for any u and r we have | f u ( r ) | = | χ u ( g r ) | = 1 and for any u = u ′ , h f u , f u ′ i = X r ∈R f u ( r )[ f u ′ ( r )] ∗ = X g ∈G χ u ( g )[ χ u ′ ( g )] ∗ − χ u ( g R )[ χ u ′ ( g R )] ∗ = − χ u ( g R )[ χ u ′ ( g R )] ∗ , and so |h f u , f u ′ i| = | χ u ( g R ) || χ u ′ ( g R ) | = 1 , as needed.As such, in order to show that harmonic ETFs constructedvia McFarland difference sets (16) can be constructed asKirkman ETFs (8), we need to show that the restricted additivecharacters exp( π i P tr q j /p ( vγ r s )) can be written in the form h s ( r,v ) ( s ) . Here, the key observation is that any element of V = F q j +1 can be decomposed in terms of the canonicalhyperplane (14). Indeed, since tr q j /q is a nontrivial linearfunctional of the vector space F q j +1 with respect to the field F q , there exists δ ∈ F q j +1 such that tr q j /q ( δ ) = 1 . Moreover,since δ lies outside of the hyperplane S , we can tack δ onto abasis for S to form a basis for F q j +1 . As such, any element of V can be uniquely written as s + tδ where s ∈ S and t ∈ F q .In particular, for any r ∈ R and v ∈ V , there exists a unique s ( r, v ) ∈ S and t ( r, v ) ∈ F q such that vγ r δ = s ( r, v ) + t ( r, v ) δ. (17) Note that for any s ∈ S , multiplying (17) by sδ − and thenapplying the linear functional tr q j /p yields: tr q j /p ( vγ r s ) = tr q j /p ( s ( r, v ) sδ − + t ( r, v ) s )= tr q j /p ( s ( r, v ) sδ − ) + t ( r, v ) tr q j /p ( s ) . (18)At this point, we introduce a third trace tr q/p : F q → F p whichcomplements the other two. Recalling F p ⊆ F q ⊆ F q j , it iswell known that these three traces satisfy the nice propertythat tr q j /p = tr q/p ◦ tr q j /q . In particular, recalling (14) andthe linearity of the trace, any s ∈ S satisfies tr q j /p ( s ) = tr q/p (tr q j /q ( s )) = tr q/p (0) = 0 . (19)As such, (18) reduces to tr q j /p ( vγ r s ) = tr q j /p ( s ( r, v ) sδ − ) ,implying that the formula (16) for our McFarland ETF can berewritten as ψ u,v ( r, s ) = D − χ u ( g r ) exp( π i P tr q j /p ( s ( r, v ) sδ − )) . Comparing this to (8), showing that this McFarland harmonicETF is a Kirkman ETF boils down to showing two claims:(i) that { h s ′ } s ′ ∈S , h s ′ ( s ) := exp( π i P tr q j /p ( s ′ sδ − )) is aunimodular orthogonal basis for C S and (ii) that the meansof identifying s ( r, v ) from a given r and v according to (17)corresponds to a resovable Steiner system.The truth of the first claim arises from the orthogonality ofthe additive characters of F q j +1 . Indeed, for any s ′ = s ′′ , thefact that ( s ′ − s ′′ ) δ − = 0 gives that X v ∈V exp( π i P tr q j /p (( s ′ − s ′′ ) vδ − )) . (20)Decomposing any v ∈ V as v = s + tδ where s ∈ S , t ∈ F q and then using (19) in the case where “ s ” is s ′ − s ′′ ∈ S gives tr q j /p (( s ′ − s ′′ ) vδ − ) = tr q j /p (( s ′ − s ′′ ) sδ − ) . Substituting this into (20) then gives our first claim: X s ∈S X t ∈ F q exp( π i P tr q j /p (( s ′ − s ′′ ) sδ − )= q h h s ′ , h s ′′ i . For the second claim, we let (17) define a block design.To be precise, let B = { b r,s } ( r,s ) ∈R×S be a set of subsets of V = F q j +1 where for any r = 0 , . . . , R − and s ∈ S wesay that v ∈ b r,s if and only if there exists a t ∈ F q such that vγ r δ = s + tδ ; solving for v reveals the ( r, s ) block to be b r,s = { sγ − r δ − + tγ − r : t ∈ F q } . (21)Recalling that for any element of F q j +1 there exists exactly one s ∈ S and t ∈ F q so that it can be written as s + tδ , we see thatfor any fixed r ∈ R , there exists exactly one s = s ( r, v ) suchthat v ∈ b r,s . As such, every v ∈ V is contained in exactly R blocks and moreover, for any fixed r ∈ R , B r = { b r,s } s ∈S forms a partition of V . Also, every block b r,s contains thesame number of points, namely the K = q points that arisefrom the various choices of t .Thus, in order to see that B is a resolvable (2 , K, V ) -Steiner system over V , all that remains to be shown is thatany two distinct v, v ′ ∈ V determine a unique block. Thisgets to the heart of an affine geometry over a finite field: the blocks are affine lines (21), which are determined by anonzero direction vector γ − r , which is only unique up tononzero scalar multiples, along with an initial point sγ − r δ − which lies in some hyperplane. Any two distinct points v, v ′ determine a direction v ′ − v = 0 ; since { γ r } R − r =0 representevery nonzero element of F q j +1 modulo scalar multiplication,we know there exists a unique r ∈ R and t ∈ F q suchthat v ′ − v = t γ − r . Moreover, for this particular r , weknow there exists unique s, s ′ ∈ S and t, t ′ ∈ F q suchthat v = sγ − r δ − + tγ − r and v ′ = s ′ γ − r δ − + t ′ γ − r ,respectively. Combining these facts gives v ′ = ( v ′ − v ) + v = t γ − r + ( sγ − r δ − + tγ − r )= sγ − r δ − + ( t + t ) γ − r , at which point the uniqueness gives s ′ = s and t ′ = t + t .Since s ′ = s , these two points v and v ′ are contained in thesame block b r,s = b r,s ′ . Also, this common block is unique:if v, v ′ ∈ b r,s are distinct, then v ′ − v uniquely determines r ; knowing v and r , s is always uniquely determined. Wesummarize these results in the following theorem, which isthe second main result of this paper. Theorem 2:
Let j ≥ and let q be a power of a prime p .Let R = { , . . . , R − } where R = ( q j +1 − / ( q − , andlet S be the hyperplane (14). Let { χ u } Ru =0 be the charactersof an Abelian group G = { g r } Rr =0 and let γ be a primitiveelement of F q j +1 , whose additive group is denoted V .Then the harmonic ETF generated by the McFarland dif-ference set (15), namely the vectors { ψ u,v } R − u =0 ,v ∈V ⊆ C R×S given in (16), is an example of a Kirkman ETF constructedby Theorem 1. To be precise, taking any δ ∈ F q j +1 suchthat tr q j /q ( δ ) = 1 , the blocks B = { b r,s } R − r =0 ,s ∈S definedin (21) are a resolvable (2 , q, q j +1 ) -Steiner system (affinegeometry) which generates this same ETF, provided we let f u ( r ) = χ u ( g r ) and h s ′ ( s ) := exp( π i P tr q j /p ( s ′ sδ − )) .We emphasize that this result tells us nothing new about theexistence of ETFs. Rather, the significance of Theorem 2 isthat it provides (sparse) Steiner ETF representations for one ofthe only two known flexible classes of harmonic ETFs; as wenow describe, this has ramifications on the use of such ETFsfor compressed sensing. A. Kirkman ETFs and the Restricted Isometry Property
Given L ≤ M ≤ N and δ < , the vectors { ϕ n } n ∈N in C M have the ( L, δ ) - Restricted Isometry Property (RIP) if forany L -element subset L of N , the eigenvalues of the Grammatrix of { ϕ n } n ∈L lie in [1 − δ, δ ] . That is, for all such L ,we want k Φ ∗L Φ L − I k ≤ δ where Φ L y := P n ∈L y ( n ) ϕ n is the corresponding restricted synthesis operator. In essence,an ( L, δ ) -RIP matrix Φ has the property that any L -elementsubset of its columns are nearly orthonormal.Though other paradigms exist, this property is undeniablycentral to compressed sensing. For a given M and N , thegoal is to design { ϕ n } n ∈N so that it is ( L, δ ) -RIP for L being as large as possible, subject to the constraint that δ issufficiently small compared to . To date, the most successfulexamples of such matrices are given by random matrix theory; such random constructions typically yield matrices Φ that,with high probability, are ( L, δ ) -RIP for L on the orderof M/ polylog( N ) . This is in stark contrast to nearly alldeterministic constructions of such matrices which, with theexception of [4], are only provably ( L, δ ) -RIP for L on theorder of M . In the compressed sensing literature, this isknown as the square-root bottleneck . These facts are commonknowledge, and are more thoroughly explained in [2].For most deterministic constructions, it is unknown whetherthis bottleneck is due to a lack of good proof techniques ormore seriously, is due to a fault in the construction itself. Tobe precise, the Gershgorin Circle Theorem gives k Φ ∗L Φ L − I k ≤ max n ∈L X n ′ ∈L n ′ = n |h ϕ n , ϕ n ′ i| ≤ ( L − µ, (22)where µ is the coherence (1) of { ϕ n } n ∈N . To use this fact toprove that { ϕ n } n ∈N is ( L, δ ) -RIP, we thus want to choose L such that ( L − µ ≤ δ < . In the case where { ϕ n } n ∈N isa sequence of unit vectors with redundancy ρ := N/M , theWelch bound (2) then yields the bottleneck: L − ≤ δµ < (cid:0) M ( N − N − M (cid:1) = (cid:0) ρM − ρ − (cid:1) ≤ (cid:0) ρρ − (cid:1) M . (23)As such, in order to push beyond this bottleneck, we need tofirst find vectors for which the bounds in (22) are too coarse,and then find a better way for estimating the eigenvaluesof the resulting submatrices. These are hard problems sincethe Gershgorin Circle Theorem, though easily proven, yieldsbounds which are surprisingly sharp.Indeed, the bounds in (22) are good in the case where { ϕ n } n ∈N is a Steiner ETF. To see this, recall from Section IIIthat the Welch bound of any such ETF is /R and so (23)becomes L − < R . That is, for any L ≤ R , there exists δ < such that { ϕ n } n ∈N is ( L, δ ) -RIP. Remarkably, the converseof this fact is also true. To elaborate, note that if { ϕ n } n ∈N is ( L, δ ) -RIP for some fixed L ≤ M and δ < , then atthe very least, any L -element subset of { ϕ n } n ∈N is linearlyindependent. This means its spark —the number of vectors inits smallest linearly dependent subcollection—is at least L +1 .However, as noted in [10], the spark of any Steiner ETF is atmost R + 1 . Indeed, in the special case where the underlyingSteiner system is resolvable, note that for any fixed v ∈ V ,the subcollection { ϕ u,v } Ru =0 of the Steiner ETF (7) defined inTheorem 1 is only supported over the indices ( r, s ) of thoseblocks b r,s which contain v . Since there are R +1 such vectorsbut only R such blocks, these vectors are necessarily linearlydependent. As such, if { ϕ n } n ∈N is ( L, δ ) -RIP for some δ < then L +1 ≤ spark( { ϕ n } n ∈N ) ≤ R +1 . In summary, a SteinerETF is ( L, δ ) -RIP for some δ < if and only if L ≤ R .We now combine this fact with the main results of thissection and the previous one to prove, for the first time,that some harmonic ETFs are not good RIP matrices. Indeed,Theorem 2 states that every ETF generated by a McFarlanddifference set—one of only two known flexible constructionsof harmonic ETFs—is, in fact, a Kirkman ETF. Moreover,Theorem 1 states that any Kirkman ETF can be obtained byapplying a unitary transformation to a Steiner ETF; it is wellknown that such transforms preserve RIP. Together, we have: Corollary 1: If { ϕ n } n ∈N is any Steiner or Kirkman ETFfor C M , then it is ( L, δ ) -RIP for some δ < if and only if L ≤ R = (cid:0) ρM − ρ − (cid:1) , where ρ = NM . In particular, it is impossible to surpass thesquare-root bottleneck using harmonic ETFs generated fromMcFarland difference sets.It remains an open problem whether or not there exists anETF which is a good RIP matrix for values of L which arelarger than numbers on the order of M . However, in light ofCorollary 1, there is only one known flexible class of ETFs leftto investigate, namely the harmonic ETFs generated by Singerdifference sets. These difference sets are cyclic, meaning theseETFs are obtained by extracting M = ( q j − / ( q − rowsfrom a standard DFT matrix of size N = ( q j +1 − / ( q − where j ≥ and q is some prime power. Here, there are somereasons for hope: when j = 2 , such ETFs are numericallyerasure-robust frames and as such, cannot be sparse in anybasis [9]. Moreover, Singer harmonic ETFs are full spark —their spark is M + 1 —when N is prime, such as when j = 2 and q = 2 . But even this can fail when N is but a primepower, such as when j = 4 and q = 3 [1].Also, there are inflexible families of non-Steiner ETFswhose RIP characteristics bear further study. One example ofthese are Paley ETFs which are constructed by modifying aquadratic-residue-based harmonic ETF into a redundancy-twoETF in the manner of [23]. There at least, spark is not theissue: any Paley ETF is ( L, δ ) -RIP for all L ≤ M for some δ < [2]. However, it is unknown how this δ behaves as afunction of L , M and N ; this is related to longstanding openproblems regarding the clique numbers of Paley graphs [2].These problems are nontrivial, and it is much easier to provethat a given set of vectors is not ( L, δ ) -RIP than to prove thatit is. Put simply, Corollary 1 does not tell you where to findgood RIP matrices but rather, where not to look.V. H ADAMARD
ETF
S AND THE G REY -R ANKIN B OUND
In this section, we apply the results of Sections III and IVto produce new examples of certain types of optimal binarycodes. An ( M, N ) -binary code is a set of N codewords(vectors) in Z M , that is, a sequence { c n } Nn =1 of M × vectorswhose entries lie in Z := { , } . The distance of such a codeis the minimum pairwise Hamming distance between any twocodewords, namely dist( { c n } Nn =1 ) := min n = n ′ d( c n , c n ′ ) where d( c , c ′ ) counts the number of entries of c , c ′ ∈ Z M that differ. The Grey-Rankin bound [13] is an upper boundon the number of codewords N one can have with a givendistance ∆ in a space with given dimension M . We show thatthe Grey-Rankin bound is equivalent to a special case of theWelch bound, and then exploit this equivalence, using codingtheory to prove new results in frame theory, and vice versa.To be precise, the Grey-Rankin bound only applies to self-complementary codes , that is, codes in which the complement ( c n + )( m ) := c n ( m ) + 1 mod 2 of any codeword c n alsolies in the code. In the work that follows, it is convenient forus to regard the second half of these ( M, N ) -binary codesas the complements of the first half, namely c n + N = c n + for all n = 1 , . . . , N . Denoting the distance of such a code as ∆ , the Grey-Rankin bound states: N ≤ M − ∆) M − ( M − (24)provided the right-hand side is positive; since the self-complementarity of the code guarantees that ≤ M , thispositivity is equivalent to having > M − M .We rederive (24) by applying the Welch bound (2) to frameswhose entries are all ± M − . To be clear, we can exponentiateany codeword c n ∈ Z M to form a corresponding unit normvector ϕ n ∈ R M , ϕ n ( m ) := M − ( − c n ( m ) . Under thisidentification, the Euclidean distance between any two of thesereal vectors can be written in terms of the Hamming distancebetween their corresponding codewords: k ϕ n − ϕ n ′ k = M M X m =1 | ( − c n ( m ) − ( − c n ′ ( m ) | = M M X m =1 (cid:26) , c n ( m ) = c n ′ ( m )0 , c n ( m ) = c n ′ ( m ) (cid:27) = M d( c n , c n ′ ) . We also have k ϕ n − ϕ n ′ k = 2(1 − h ϕ n , ϕ n ′ i ) , and solvingfor the inner product gives h ϕ n , ϕ n ′ i = M ( M − c n , c n ′ )) . (25)Grey himself used this identification in his derivation of (24).However, Grey’s argument [13] relies on prior work byRankin [20] concerning the packing of spherical caps, whereaswe instead make use of Welch’s bound (2).The mathematical novelty here is debatable: Rankin’s workis a forerunner to Welch’s bound. In fact, a little simplificationreveals Equation 26 of [20] to be equivalent to the real-variable version of the Welch bound; since it predates [29] bynearly two decades, one can argue that (2) should be calledthe “Rankin-Welch” bound. From this perspective, our workbelow serves to modernize Grey’s original argument. Thisitself has value: unlike Rankin’s work, the Welch bound iswidely studied. Also, as seen from (3), the Welch bound canbe quickly proven from basic principles. Most importantly,by using the Welch bound to streamline Grey’s approach, weallow the large body of existing ETF/WBE literature to bequickly and directly applied to open problems in coding theory.Returning to the argument itself, taking the maximums ofboth sides of (25) over all n, n ′ = 1 , . . . , N , n = n ′ gives max n,n ′ ∈{ ,..., N } n = n ′ h ϕ n , ϕ n ′ i = M − M . (26)Moreover, the self-complementarity of the code { c n } Nn =1 givesthat ϕ n + N = − ϕ n for all n = 1 , . . . , N . As such, the left-hand side of (26) can be rewritten as µ = max n,n ′ ∈{ ,...,N } n = n ′ |h ϕ n , ϕ n ′ i| = M − M . (27)At this point, the Welch bound (2) gives (cid:0) N − MM ( N − (cid:1) ≤ M − M . (28) Squaring both sides and then solving for N then gives theGrey-Rankin bound (24). Moreover, note that by this argu-ment, we obtain equality in (24) if and only if we haveequality in (28), which in light of (27), happens preciselywhen { ϕ n } Nn =1 is a real-valued constant-amplitude ETF. Thatis, every Grey-Rankin-bound-equality (GRBE) code generatessuch an ETF.Importantly, the converse is also true: if { ϕ n } Nn =1 is anyconstant-amplitude ETF for R M , then we may build code-words { c n } Nn =1 by letting c n ( m ) := log − (sgn( ϕ n ( m )) ∈ Z for all m = 1 , . . . , M and all n = 1 , . . . , N . We then extendthis to a self-complementary code by letting c n + N := c n + for all n = 1 , . . . , N . Since exponentiating these codewordsproduces our original ETF, we know that (28) holds withequality, meaning we also have equality in the Grey-Rankinbound (24). For example, the real × Kirkman ETF givenin Figure 2 yields the × self-complementary code given inFigure 3. We summarize these results as our third main result: Theorem 3:
Any ( M, N ) -binary self-complementary code { c n } Nn =1 with c n + N = c n + for all n = 1 , . . . , N satisfiesthe Grey-Rankin bound (24). Moreover, identifying { c n } Nn =1 with constant-amplitude vectors { ϕ n } Nn =1 ⊆ R M accordingto ϕ n ( m ) = M − ( − c n ( m ) , the code { c n } Nn =1 achieves theGrey-Rankin bound (24) if and only if { ϕ n } Nn =1 is an ETF.In light of Theorem 3, we turn our attention to the problemof constructing real-valued constant-amplitude ETFs. As notedin [8], harmonic examples of such ETFs can be constructedover the additive group of F j +2 by forming a difference set D as the support of a bent function . Such ETFs have parameters M = 2 j (2 j +1 ± , N = 2 j +2 (29)for some j ≥ . In the context of the previous sections, itis easier to understand ETFs with parameters (29) as specialcases of harmonic ETFs generated from McFarland differencesets. Indeed, M = 2 j (2 j +1 − and N = 2 j +2 is a specialcase of (13) where q = 2 . Here, the corresponding McFarlanddifference set (15) lies in the group H = G × V where V isthe additive group of F j +1 and G = Z j +12 . The resulting ETFis real since every element of H has order : for any h ∈ H and any character χ of H , [ χ ( h )] = χ ( h + h ) = χ (0) = 1 and so χ ( h ) = ± . Moreover, the set complement of anydifference set is another difference set [16]. Thus, there alsoexists a real-valued constant-amplitude ETF of N = 2 j +2 vectors in a space of dimension N − M = 2 j (2 j +1 + 1) ;this complementary ETF is a special case of the Naimarkcomplement of a tight frame { ϕ } n ∈N , which in general, isformed by finding a orthonormal basis for the orthogonalcomplement of the row space of Φ .Surveying the literature, we find that all known real-valuedharmonic ETFs are either regular simplices or have param-eters (29). As we now explain, these are the only possibledimensions for a real-valued harmonic ETF. To be precise,a code is linear if the codewords are the points in somesubspace of Z M . And, in the case where the underlyingreal ETF is harmonic, the corresponding code generated byTheorem 3 is necessarily linear: signed characters are closedunder multiplication, and so their corresponding codewords Fig. 3. A (6 , -binary code, the left half of which is obtained by converting the + ’s and − ’s of the ETF in Figure 3 into ’s and ’s, respectively. Thiscode is self-complementary, meaning its right half is obtained by adding to the left half, modulo . This code achieves the Grey-Rankin bound for M = 6 and ∆ = 2 , meaning it is the widest possible self-complementary matrix of height such that the Hamming distance of any two columns is at least . Weshow that such Grey-Rankin-bound-equality matrices are equivalent to real-valued constant-amplitude ETFs, and then exploit this result to prove new resultsabout ETFs using coding theory, and vice versa. are closed under addition. Moreover, it is known that a linearGRBE code with M ≥ must either have dimensions M = 2 j +1 − and N = 2 j +2 for some j ≥ —meaning itscorresponding ETF is a real-valued constant-amplitude regularsimplex—or alternatively, dimensions M = 2 j (2 j +1 ± and N = 2 j +3 for some j ≥ [19]. As such, we find that: Corollary 2:
If there exists a real-valued harmonic ETF of N vectors in an M -dimensional space with M ≥ , then either(i) the ETF is a regular simplex of N = 2 j +1 vectors for some j ≥ or (ii) the dimensions of the ETF are of the form (29).This corollary illustrates how coding theory can be used tofind new results in frame theory. However, from the point ofview of coding theory itself, this corollary is disappointing:GRBE codes with these parameters are already known toexist [19]. It is here that the not-necessarily-harmonic ETFsof Theorem 1 truly shine: we can find Kirkman ETFs thatlie outside the confines of Corollary 2; these ETFs are nec-essarily non-harmonic, and the resulting codes are necessarilynonlinear.To be precise, in order to construct real-valued constant-amplitude ETFs using Theorem 1, we want both our uni-modular regular simplex { f u } Ru =0 as well as our unimodularorthogonal basis { h s } s ∈S to be real-valued. Since such asimplex necessarily extends to a real unimodular orthogonalbasis, we in particular want Hadamard matrices of size R + 1 and S = B/R = V /K . It is well-known that this requires both R + 1 and V /K to either be or divisible by ; the Hadamardconjecture posits that these necessary conditions are sufficient.Also, recall that in order for a resolvable (2 , K, V ) -Steinersystem to exist, we necessarily have V ≡ K mod K ( K − .Writing V = W K ( K −
1) + K for some W ≥ , we want R + 1 = W K + 2 , VK = W ( K −
1) + 1 (30)to either be or divisible by . Note that since W ≥ and K ≥ , we cannot have R = 2 , and so this condition on R is equivalent to having W K + 2 ≡ . Meanwhile, if W ( K −
1) + 1 = 2 then we necessarily have K = 2 and W = 1 ; the resulting (2 , , -Steiner system (4) yields the × code of Figure 3; as discussed above, linear GRBE codeswith these parameters are well-known, and can be generatedas McFarland harmonic ETFs, letting j = 1 in (29). As such,we also assume W ( K −
1) + 1 ≡ . At this point,subtracting W ( K −
1) + 1 from
W K + 2 , we see that thesetwo necessary conditions are equivalent to having W ≡ and K ≡ . Combining the above discussion withTheorems 1 and 3 gives the following result: Corollary 3:
Given K ≡ and W ≡ ,let V = K [ W ( K −
1) + 1] . Then both parameters in (30)are divisible by , and if there exist Hadamard matrices ofthese sizes and there also exists a resolvable (2 , K, V ) -Steinersystem, then there exists a real-valued Kirkman ETF with M = ( W K + 1)[ W ( K −
1) + 1] ,N = K ( W K + 2)[ W ( K −
1) + 1] , meaning there exists a ( M, N ) -self-complementary code thatachieves the Grey-Rankin bound (24).To explore the consequences of this result, we first considerthe case where K = 2 . Recall from Section III that (2 , , V ) -Steiner systems are resolvable as a round-robin tournament forany even V ≥ . Here, V = K [ W ( K −
1) + 1] = 2 W + 2 for some W ≡ . To make the resulting Kirkman ETFreal-valued, we want Hadamard matrices of size R + 1 =2 W + 2 = V and V /K = W + 1 = V / . It thus suffices forthere to exist a Hadmard matrix of size V / , since we can takethe tensor product of it with the canonical Hadamard matrixof size to form one of size V . When V is a power of ,the ETFs produced by Corollary 3 have the same dimensionsas those produced by the real-valued harmonic ETFs of (29).However, when V is not a power of , Corollary 2 tells us thatthese Kirkman ETFs cannot be harmonic. For example, letting W = 11 yields V = 24 , and we know there exists a Hadamardmatrix of size V / . As such, there exists a real-valued × Kirkman ETF that cannot be harmonic, and theresulting × GRBE code is not linear. There are aninfinite number of nonharmonic Kirkman ETFs of this type:at the very least, Paley’s quadratic-residue based constructionof Hadamard matrices gives the existence of such an ETFwhenever W ≡ is a prime power.For K ≡ such that K > , the true implications ofCorollary 3 are harder to ascertain. We could not find any ex-plicit infinite families of resolvable (2 , K, V ) -Steiner systemsfor such values of K in the literature. For K = 6 in particular,we need V ≡ ; it is known [11] that a resolvable (2 , , V ) -Steiner system (i) does not exist for V = 36 , (ii)may or may not exist for V = 66 and V = 96 , (iii) does existfor V = 126 (unital design), V = 156 (projective geometry)and V = 186 . Unfortunately, the only one of these values of V that satisfies the hypotheses of Corollary 3 is V = 96 . If a resolvable (2 , , -Steiner system does exist it would, toour knowledge, give the first example of a GRBE code whoseredundancy N/M = 3840 / is not approximately . For K = 10 and larger, the minimum corresponding V whichcould satisfy the assumptions of Corollary 3 is V = 280 ,which lies beyond the range of the tables of known resolvabledesigns we encountered.At this point, we turn to asymptotic existence results. Recallthat for any K , there exists V ( K ) such that for all V ≥ V ( K ) with V ≡ K mod K ( K − , there exists a resolvable (2 , K, V ) -Steiner system [22]. As such, if the Hadamardconjecture is true, then for any K ≡ , there exists W ( K ) such that for all W ≥ W ( K ) with W ≡ ,there exists a real-valued Kirkman ETF whose parameters aregiven by Corollary 3. In particular, if the Hadamard conjectureis true, for any K ≡ there exists real-valued constant-amplitude ETFs and GRBE codes whose redundancies areapproximately K and K , respectively.VI. C ONCLUSIONS AND F UTURE W ORK
We now have a method for transforming certain SteinerETFs into constant-amplitude ETFs, as desired for certainwaveform design applications. We also now know that animportant class of previously discovered harmonic ETFs arisein this fashion, making them less attractive for deterministiccompressed sensing. Finally, we have seen how the problem ofconstructing a real-valued constant-amplitude ETF is equiva-lent to that of constructing a type of optimal binary code,allowing us to apply results from one area to the other. Severalimportant questions remain open: To what degree do Singerharmonic ETFs satisfy RIP? Are Denniston Kirkman ETFsharmonic? More generally, can we use these results alongwith ideas from resolvable designs to build new examplesof difference sets, or vice versa? Do there exist nontrivialreal-valued constant-amplitude ETFs whose redundancy is notessentially ? Equivalently, do there exist nontrivial GRBEcodes whose redundancy is not essentially ?A CKNOWLEDGMENT