Cubes of symmetric designs

We study $n$-dimensional matrices with $\{0,1\}$-entries ($n$-cubes) such that all their $2$-dimensional slices are incidence matrices of symmetric designs. A known construction of these objects obtained from difference sets is generalized so that the resulting $n$-cubes may have inequivalent slices. For suitable parameters, they can be transformed into $n$-dimensional Hadamard matrices with this property. In contrast, previously known constructions of $n$-dimensional designs all give examples with equivalent slices.

In this paper we study such objects under the name cubes of symmetric designs.These are n-dimensional {0, 1}-matrices of order v such that all 2-dimensional slices are incidence matrices of (v, k, λ) designs.Given an ordinary Hadamard matrix of order v, a proper n-dimensional Hadamard matrix of the same order is obtained by the product contruction of Yang [34].Thus, the spectrum of orders v such that proper Hadamard matrices exist is the same for all dimensions n ≥ 2. According to the famous Hadamard conjecture, it includes all orders divisible by 4. No such construction is known for symmetric designs.There are parameters such that symmetric designs exist, but n-dimensional cubes are not known for n ≥ 3, for example (25,9,3).However, cubes of arbitrary dimension arise from (v, k, λ) difference sets, in analogy with the group development construction of Hammer and Seberry [16].We generalise this construction to commence from symmetric designs that have difference sets as blocks, but are not necessarily developments.The ensuing group cubes have the interesting property that different slices may be inequivalent (v, k, λ) designs.In contrast, all known constructions of proper n-dimensional Hadamard matrices and other types of n-dimensional designs give examples with equivalent slices.
The outline of our paper is as follows.In Section 2 we recall the relevant definitions and fix notation.Equivalence of cubes and their autotopy and autoparatopy groups are defined.Invariants are introduced to be used later on to distinguish inequivalent cubes.Representations of cubes as orthogonal arrays and transversal designs are explained, enabling the application of computational tools for incidence structures to cubes of symmetric designs.
Section 3 is devoted to the construction of n-cubes from difference sets.Properties of such difference cubes and their autotopy groups are proved.Equivalence of difference sets and the corresponding difference cubes is compared.Distinctions are found between dimension n = 2 and dimensions n ≥ 3.
Applications of n-dimensional Hadamard matrices include spectroscopy, error-correcting codes, cryptography, and signal processing; see [18,Section 5.3] and [37].Cubes of symmetric designs could have similar applications, but this will be left for future work.

Definitions
Let V be a set of v points.A (v, k, λ) design over V is a collection D of k-subsets of V called blocks, such that every pair of points is contained in exactly λ blocks.The design is symmetric if the number of blocks is also v.All designs in this paper are symmetric.Given an ordering of the points p 1 , . . ., p v and blocks B 1 , . . ., B v , the incidence matrix A = (a ij ) of the design is defined by Here and in the sequel, we use the Iverson bracket [P ] which takes the value 1 if P is true and 0 otherwise; see [21].It is known that a v × v matrix A with {0, 1}-entries is the incidence matrix of a (v, k, λ) design if and only if A • A t = (k − λ)I + λJ holds, where I is the identity matrix, and J is the all-ones matrix.Furthermore, the transposed matrix A t satisfies the same equation and is an incidence matrix of the dual design D t .We refer to [24] for these and other results about symmetric designs.
Let {1, . . ., v} n be the Cartesian product of n copies of {1, . . ., v}.An n-dimensional incidence cube of order v is a function C : {1, . . ., v} n → {0, 1}.We will talk about n-cubes of order v for short.Thus, a 2-cube is a v × v matrix with {0, 1}-entries, a 3-cube is a 3-dimensional array of zeros and ones, etc.Given a pair of distinct integers (x, y) ∈ {1, . . ., n} 2 , a slice of the n-cube C is the matrix obtained by varying the coordinates in positions x and y, and taking some fixed values i 1 , . . ., i x−1 , i x+1 , . . ., i y−1 , i y+1 , . . ., i n ∈ {1, . . ., v} for the remaining coordinates.In other words, it is the restriction of C to the set We allow x > y, in which case the order of the factors indexed by x and y should be reversed.The slices corresponding to (x, y) and (y, x) are transposed matrices.Definition 2.1.An n-dimensional cube of symmetric (v, k, λ) designs is an n-cube of order v such that all of its slices are incidence matrices of (v, k, λ) designs.The set of all such n-cubes will be denoted by C n (v, k, λ).This is a special case of de Launey's proper n-dimensional transposable designs (v, Π R , Π C , β, S) n , see [ Let A 2 be the matrix obtained by a cyclic shift upwards of rows of A 1 .This is another incidence matrix of the Fano plane.Continue in the same way to get incidence matrices A 3 to A 7 : . Now put all of these matrices on top of each other: first A 1 , then A 2 and end up with A 7 as the top layer.We get a 3-cube of order 7 such that the horizontal slices are incidence matrices of the (7, 3, 1) design.Because of the cyclic nature of the construction, this also holds for slices in the other two directions.If we look at a particular row of A 1 , . . ., A 7 , say row j, we get the matrix A j .The matrices are symmetric and the conclusion also holds if we look at a particular column.
We now define equivalence of cubes of symmetric designs; cf.[25,13] for the corresponding concepts for n-dimensional Hadamard matrices.The direct product of symmetric groups (S v ) n = S v × . . .× S v acts on the set C n (v, k, λ) by permuting indices: ).The orbits of this action are the isotopy classes of cubes.The stabiliser of a cube C ∈ C n (v, k, λ) is its autotopy group Atop(C).Furthermore, permutations γ ∈ S n act by conjugation, i.e. changing the order of the indices: . A cube such that C γ = C for all γ ∈ S n is totally symmetric.The cube from Example 2.2 has this property.The combination of isotopy and conjugation is called paratopy and is the natural action of the wreath product S v ≀ S n on C n (v, k, λ).Two cubes are considered equivalent if they can be mapped onto each other by paratopy.The corresponding classes are the main classes and the stabiliser of a cube C in S v ≀ S n is its autoparatopy group Apar(C).
The terminology is borrowed from Latin squares [20].Indeed, a Latin square , since incidence matrices of symmetric designs with k = 1 are just permutation matrices.The equivalence is easily generalised to higher dimensions, but notice that different authors have considered different concepts under the name "Latin hypercubes".The appropriate definition for our purposes is the one used in [28]; see the discussion and references therein for other related concepts.In the case k = 2, symmetric designs are possible only for parameters (3, 2, 1).There is a unique (3, 2, 1) design up to isomorphism and it is cyclic.By Theorem 3.1, cubes of (3, 2, 1) designs exist for all dimensions n ≥ 2.
In the sequel we consider only designs with k ≥ 3.
To distinguish inequivalent cubes, we use invariants based on parallel slices.For dimension n = 2, cubes are simply incidence matrices of symmetric designs and equivalence means that the designs are isomorphic or dually isomorphic.For n > 2, we can choose the varying coordinates in n 2 ways and substitute the fixed coordinates in v n−2 ways.Slices are parallel if all but one of the fixed coordinates agree.We consider v mutually parallel slices as a multiset of (dual) isomorphism types of (v, k, λ) designs.The multiset of n 2 (n − 2)v n−3 such multisets must agree for equivalent cubes, i.e. this is a paratopy invariant.We may also look at weaker invariants, for example orders of the full automorphism groups of parallel slices.In either case, the multiset of such multisets agreeing is only a necessary, but not sufficient condition for equivalence of cubes; see the comment after Example 4.2.
An incidence cube C : {1, . . ., v} n → {0, 1} is the characteristic function of a set of n-tuples This is a one-to-one correspondence.If C ∈ C n (v, k, λ), then C is an orthogonal array with parameters OA(kv n−1 , n, v, n − 1) of index k; see [17] or [8,Section III.6] for the definition.It has the additional property that for any choice of coordinate positions x, y ∈ {1, . . ., n}, x = y, and values i Every orthogonal array with this property and the parameters above corresponds to an n-dimensional cube of (v, k, λ) designs.Thus, we can represent cubes of symmetric designs by orthogonal arrays.In practice we use an equivalent representation by transversal designs: C is represented as a collection of n-subsets {i 1 , v + i 2 , 2v + i 3 , . . ., (n − 1)v + i n } instead of n-tuples (i 1 , . . ., i n ).Now we have an incidence structure of nv points and kv n−1 blocks such that the usual notion of isomorphism agrees with paratopy of cubes.This has the advantage that we can check equivalence of cubes and compute autoparatopy groups by tools available in the computer algebra system GAP [14], notably the graph isomorphism programs nauty and Traces [27].By coloring the n classes of points {1, . . ., v}, . . ., {(n−1)v+1, . . ., nv} we can also check isotopy of cubes, compute autotopy groups, and use the Kramer-Mesner approach [22] to construct cubes with prescribed autotopy groups.These and other tools for handling cubes of symmetric designs are available in our GAP package Prescribed Automorphism Groups [23].

Difference cubes
Let G be a multiplicatively written group of order v.
is an n-dimensional cube of (v, k, λ) designs.
Proof.First consider the case n = 2. Notice that the design dev D with points ordered as g 1 , . . ., g v and blocks ordered as ] is an incidence matrix of the dual design (dev D) t with a different ordering of columns, since g i g j ∈ D ⇐⇒ g j ∈ g −1 i D holds.In the general case, a slice of the n-cube (1) is a matrix of the form This is just a permutation of rows and columns of the incidence matrix of (dev D) t or dev D, and therefore (1) is a (v, k, λ) cube of dimension n.
The general construction of Theorem 2.9 in [9] relies on collapsable functions, which were later specialised to abelian extension functions and 2-cocycles [12].Cocyclic Hadamard matrices are an important and widely studied class [18], more general than group developed Hadamard matrices.The latter are necessarily regular, and therefore of order v = 4u 2 for some u ∈ N.There is no such restriction for cocyclic Hadamard matrices, which are conjectured to exist for all orders divisible by 4 [12,Conjecture 3.6].Regarding symmetric designs, the symbols 0 and 1 in their incidence matrices cannot be exchanged and there are no nontrivial cocycles.In this case Theorem 2.9 of [9] reduces to the difference set construction of Theorem 3.1.
We shall call cubes arising from Theorem 3.1 difference cubes.A non-difference cube is a cube of symmetric designs not equivalent to any difference cube.The following properties of difference cubes are easy consequences of the definition.
) n and its image is an autotopy group of the difference cube (1): The set of all multipliers of D is a subgroup Mult(D) ≤ Aut(G) acting on the design dev D as automorphism group.This group also acts on the difference cube (1) as autotopy group: In the last line we see how to permute the indices to leave the difference cube (1) invariant.The combined actions of G n−1 and Mult(D) give an autotopy group of (1) isomorphic to their semidirect product.This proves the following theorem.
Equivalent difference sets clearly give rise to isomorphic designs and isotopic difference cubes.For designs, the converse does not hold: dev D 1 and dev D 2 can be isomorphic even if D 1 and D 2 are not equivalent.For example, according to the GAP package DifSets [29] there are 27 inequivalent (16, 6, 2) difference sets in 12 of the 14 groups of order 16 (see Table 1).On the other hand, there are only three (16,6,2) designs up to isomorphism and duality [8].By a computation in PAG [23], the 27 difference 3-cubes (16,6,2) are not equivalent (paratopic).The conclusion holds for all dimensions n ≥ 3 because any sub-3-cube of a difference n-cube is equivalent with the corresponding difference 3-cube.
The smallest examples of equivalent difference n-cubes (n ≥ 3) obtained from inequivalent difference sets have parameters (27,13,6).Difference sets exist in two of the five groups of order 27.There is a unique (27,13,6) difference set in Z 3 × Z 3 × Z 3 and two inequivalent difference sets in Z 9 ⋊ Z 3 [29].Another computation in PAG [23] shows that the difference 3-and 4-cubes constructed from the two difference sets in Z 9 ⋊ Z 3 are equivalent, but not isotopic.See Table 3 for other examples of equivalent difference cubes coming from inequivalent difference sets.For dimensions n ≥ 3, we did not find a single example of isotopic difference cubes coming from inequivalent difference sets, nor of equivalent difference cubes coming from difference sets in nonisomorphic groups.Both situations are possible for n = 2.

Group cubes
In this section we generalise the construction of difference cubes of Theorem 3.1.Let G = {g 1 , . . ., g v } be a group of order v and D = {B 1 , . . ., B v } a (v, k, λ) design over G such that all blocks B i are (v, k, λ) difference sets.The design D could be the development of a difference set, say B i = g −1 i B 1 , i = 1, . . ., v.In this case the corresponding difference cube can be written as is an n-dimensional cube of (v, k, λ) designs.
Proof.If the index i 1 is fixed, then (2) is just the (n − 1)-dimensional difference cube constructed from the difference set B i 1 , so any such slice is an incidence matrix of its development.On the other hand, if i 1 and one of the remaining indices are varied, then the corresponding slice is an incidence matrix of the design D t with reordered columns.
We shall call cubes constructed from Theorem 4.1 group cubes.The main question is whether there exist designs that are not developments, but all of their blocks are difference sets.In this case the construction may give non-difference cubes.Our first example is for parameters (21,5,1) of the projective plane of order 4.Here is a (21, 5, 1) design over F 21 with all blocks being difference sets, which is not the development of any of its blocks: As before, by considering sub-3-cubes, we see that the conclusion holds for all dimensions: C n (21, 5, 1) contains a non-difference group cube for every n ≥ 3.
The inequivalence of the cubes C 1 , C 2 , and C 3 was shown by calculating their full autotopy groups.There is only one (21, 5, 1) design D 0 up to isomorphism and duality, so these three cubes have the same slice invariant {{D 21 0 } 3 } defined in Section 2. For parameters (16,6,2), there are three inequivalent designs D 1 , D 2 , and D 3 [8]  i } e } for e = n 2 (n−2)16 n−3 and i ∈ {1, 2, 3}.Our next goal is to construct group cubes with different slice invariants, which will then clearly be non-difference cubes.
The design D 1 belongs to a family of symmetric designs with the symmetric difference property (SDP designs) studied by Kantor [19].They can be constructed from difference sets in elementary abelian groups G m = Z 2m 2 of orders 4 m as follows.Any singleton, e.g.D 1 = {00}, is a (4, 1, 0) difference set in the Klein four-group G 1 = {00, 01, 10, 11}.Here we use additive notation and 00 is the neutral element.By the product construction of Mann [26,Lemma 6.3.1],[19], and the design D 1 coincides with S −1 (4).
The other two (16, 6, 2) designs D 2 and D 3 can be obtained by switching submatrices in the incidence matrix of D 1 : Clearly any such switch leaves the row and column sums invariant.It can be checked that dot products of rows and columns are also not changed.To be specific, number the group elements of G 2 = Z 4 2 lexicographically: g 0 = 0000, g 1 = 0001, g 2 = 0010, g 3 = 0011, . .., g 15 = 1111.Let D 2 = {g 1 , g 2 , g 3 , g 4 , g 8 , g 12 } be the difference set for D 1 .Denote the blocks B i = g i + D 2 , so that D 1 = {B 0 , . . ., B 15 }.Then the blocks B 0 = D 2 and B 1 = {g 0 , g 2 , g 3 , g 5 , g 9 , g 13 } contain the elements g 2 , g 3 and do not contain the elements g 14 , g 15 .On the other hand, the blocks B 12 = {g 0 , g 4 , g 8 , g 13 , g 14 , g 15 } and B 13 = {g 1 , g 5 , g 9 , g 12 , g 14 , g 15 } do not contain g 2 , g 3 and do contain g 14 , g 15 .By replacing these four blocks by their symmetric difference with S 1 = {g 2 , g 3 , g 14 , g 15 } and leaving the other blocks unchanged, the design D 1 is transformed into D 2 .Denote the new blocks B ′ i = B i ∆S 1 for i = 0, 1, 12, 13 and B ′ i = B i for i = 2, . . ., 11, 14, 15.The design D 3 is obtained by applying another switch to D 2 .Take S 2 = {g 6 , g 7 , g 14 , g 15 } and replace the blocks Now notice that the changed blocks B ′ i and B ′′ i are also (16,6, 2) difference sets in the group G 2 , equivalent with the original difference set D 2 .Therefore, D 2 and D 3 are (16, 6, 2) designs with all blocks being difference sets in G 2 .The development of any one of these blocks is isomorphic to D 1 , thus D 2 and D 3 are not developments of their blocks.By applying Theorem 4.1 to these designs, we get group cubes of dimension n ≥ 3 with slice invariants {{D 16  1 } e 1 , {D 16  2 } e 2 }} and {{D 16  1 } e 1 , {D 16  3 } e 2 }}, respectively, for e 1 = n−1 2 (n − 2)16 n−3 and e 2 = (n − 1)(n − 2)16 n−3 .
We now generalise this construction to obtain non-difference cubes in C n (4 m , 2 m−1 (2 m − 1), 2 m−1 (2 m−1 − 1)).Let A be an incidence matrix of a symmetric design with these parameters and J the all-ones matrix of order 4 m .Then J − A is the incidence matrix of the complementary design with parameters (4 m , is an incidence matrix of a (4 m+1 , 2 m (2 m+1 −1), 2 m (2 n −1)) design.This recursive construction was described in [6] and [19] by using {−1, 1}matrices and the Kronecker product.When applied to the incidence matrix of D 1 , the construction gives the series of SDP designs S −1 (2m).Applied to D 2 and D 3 , the construction gives two other series of nonisomorphic designs S −1 2 (2m) and S −1 3 (2m) with the same parameters.The blocks of these designs are (4 m , 2 m−1 (2 m − 1), 2 m−1 (2 m−1 − 1)) difference sets in G m , because the first block row of ( 4) is obtained by the product construction (3) of rows of A with the singleton {00}, the second block row with the singleton {01}, etc.The developments of these difference sets are all isomorphic to S −1 (2m), so S −1 2 (2m) and S −1 3 (2m) are not developments of their blocks.Theorem 4.1 gives rise to two group n-cubes in G m .Slices with i 1 fixed are incidence matrices of S −1 (2m), and slices with i 1 varying are incidence matrices of S −1 2 (2m) or S −1 3 (2m).Therefore, these two n-cubes are not difference cubes and the following theorem holds.[33].The series of designs S −1 (2m), S −1 2 (2m), S −1 3 (2m) are transformed into Hadamard matrices which are Kronecker products of H 1 , H 2 , H 3 with the cyclic Hadamard matrix of order 4, and are also not equivalent.In conclusion, the two non-difference cubes of Theorem 4.3 can be transformed into proper n-dimensional Hadamard matrices with inequivalent slices, which cannot be obtained by [9, Theorem 2.9] and other known constructions.

Small examples
The smallest parameters of symmetric designs are (7, 3, 1), (11,5,2), and (13, 4, 1).There is a single design up to isomorphism and duality for each of these parameters, coming from a difference set in the cyclic group Z v .The next parameters are (15,7,3) with four designs up to isomorphism and duality, but only one coming from a difference set, namely P G 2 (3,2).In all four cases the only designs with difference sets as blocks are developments.Hence, there is a unique group cube in C n (7, 3, 1), C n (11, 5, 2), C n (13, 4, 1), and C n (15, 7, 3), equivalent with the difference cube.
The fifth smallest parameters are (16,6,2), and in this case there are many non-difference group cubes.The results of an exhaustive computer search for designs with (16,6,2)  order 16 in the GAP library of small groups [14] and a description of their structure.The third and fourth column (Nds and Ndc) contain numbers of inequivalent difference sets according to [29] and numbers of inequivalent difference cubes, respectively.These numbers coincide by the computation from Section 3. The fifth column contains designs arising as developments of difference sets in each of the groups.The total number of difference sets, including equivalent ones, is given in the sixth column (Tds).2, along with the corresponding numbers of difference cubes (Ndc) and non-difference group cubes (Ngc).
We have also found cubes of symmetric designs not equivalent with any group cube, which we shall call non-group cubes.Using our package PAG [23], we computed the full autotopy groups of available cubes C ∈ C 3 (16, 6, 2).We then chose subgroups G ≤ Atop(C) and used the Kramer-Mesner method to construct all 3-cubes of (16, 6, 2) designs with G as prescribed autotopy group.Clearly we must always get the cube C we started from, but often we also get other inequivalent cubes, some of which are non-group cubes.Here is an example.This transversal design represents a 3-cube of (16, 6, 2) designs with slice invariant {{D 4 1 , D 12 2 } 3 }.It is not equivalent with any group cube, because parallel slices in all three directions comprise both designs D 1 and D 2 .A 3-cube constructed from Theorem 4.1 must have slices isomorphic to D in two directions.
We found many non-group cubes of (16,6,2) designs by this method.
Proposition 5.3.The set C 3 (16, 6, 2) contains at least 1423 inequiva- lent non-group cubes.We did not find any other non-group cubes for smaller parameters (v, k, λ), and for larger parameters the Kramer-Mesner approach was too inefficient.However, we did find larger non-difference group cubes by the construction of Theorem 4. For parameters (27,13,6) we did an incomplete search and found 7 other group cubes besides the two difference cubes mentioned at the end of Section 3.For even larger parameters we searched for designs with difference sets as blocks and a prescribed automorphism group.Table 3.An on-line version of the table with links to files containing the cubes is available on the above-mentioned web page.The column Nds contains numbers of inequivalent difference sets according to [29].For parameters (64, 28, 12) some of the corresponding difference 3-cubes are equivalent, but we did not perform a complete enumeration.

The last column of
be written as a left difference g = d −1 1 d 2 for exactly λ pairs (d 1 , d 2 ) ∈ D × D. This is equivalent with the corresponding property for right differences g = d 1 d −1 2 [7].The development of D, i.e. the set of left translates dev D = {gD | g ∈ G} is a (v, k, λ) design over G.The right translates {Dg | g ∈ G} also form a design isomorphic to the dual (dev D) t .Difference sets give rise to cubes of symmetric designs of arbitrary dimension.The following theorem is a special case of [9, Theorem 2.9].An analogous theorem was originally proved for group developed n-dimensional proper Hadamard matrices [16, Theorem 4].Theorem 3.1.Let D be a (v, k, λ) difference set in the group G. Order the group elements as g 1 , . . ., g v .Then the function

Proposition 3 . 2 .
All slices of an n-dimensional difference cube are incidence matrices of the same (v, k, λ) design up to isomorphism and duality.Proposition 3.3.A difference cube constructed from an abelian group is totally symmetric, i.e. invariant under any conjugation.A design dev D constructed from a difference set D ⊆ G has G as an automorphism group acting regularly on the points and blocks.An ndimensional difference cube has the direct product G n−1 = G × . . .× G as autotopy group acting as follows.For a, b ∈ G, let α a b ∈ S v be the permutation defined by α a b

Example 4 . 2 .
There are two groups of order 21: the Frobenius group F 21 = a, b | a 3 = b 7 = 1, ba = ab 2 and the cyclic group Z 21 .Both groups allow one (21, 5, 1) difference set up to equivalence [29].We denote the corresponding 3-cubes C 1 and C 2 , respectively.Using PAG [23], we computed | Atop(C 1 )| = 1323 and | Atop(C 2 )| = 2646.Thus, C 1 and C 2 are not equivalent, and neither are the corresponding n-cubes for n > 3 because all of their sub-3-cubes are equivalent either with C 1 or with C 2 .This shows that there are exactly two inequivalent difference cubes in C n (21, 5, 1) for every n ≥ 3.

Theorem 4 . 3 .
For every m ≥ 2 and n ≥ 3, the set C n (4 m , 2 m−1 (2 m − 1), 2 m−1 (2 m−1 − 1)) contains at least two inequivalent group cubes that are not difference cubes.The total number of inequivalent group cubes inC n (4 m , 2 m−1 (2 m − 1), 2 m−1 (2 m−1 − 1)) ismuch larger; see Proposition 5.1 for n = 3 and m = 2. Notice that the parameters are of Menon type and therefore by exchanging 0 → −1 the cubes are transformed into n-dimensional proper Hadamard matrices.These matrices are totally regular, meaning that any slice is a regular Hadamard matrix.The designs D 1 , D 2 , D 3 are transformed into inequivalent Hadamard matrices H 1 , H 2 , H 3 .Up to equivalence, there are only two other Hadamard matrices H 4 and H 5 of order 16

Table 1 .
difference sets as blocks are given in The first two columns contain IDs of the groups of

Table 2 (
Nng) contains their distribution by slice invariants.The constructed examples are available on our web page: https://web.math.pmf.unizg.hr/~krcko/results/cubes.html The total number of non-group cubes in C 3 (16, 6, 2) is probably much larger.We have attempted construction only for groups of order |G| ≥ 512.

Table 3 .
The parameters for which we found non-difference group cubes are given in Difference and group cubes for n = 3.