A proof from the book, perchance

Abstract

The fact that the groups ${\mathbb {Z}}_{2^m} \times {\mathbb {Z}}_{2^m}$ contain difference sets was first established by induction by Jim Davis in his Virginia dissertation of 1987. Later that year we gave a direct construction for a very large family of highly structured inequivalent difference sets in these groups. In this paper, we give a proof of our result which we presented long ago in a colloquium at Wright State University, but which has never been published. While the proof lays bare the rich structure of the difference sets, its utter simplicity suggests that it might just be from the book. We also provide some historical background and elaborate on some of the structural properties of these difference sets which have motivated some recent progress toward the goal of classifying all hadamard 2-groups.

1 Introduction

The k-subset D of the group G of order v is a difference set with parameters (v, k, $\lambda$) if, for all nonidentity elements g in G, the equation

$$\begin{aligned} xy^{-1} = g \end{aligned}$$

has exactly $\lambda$ solutions (x,y) with x and y in D; the related parameter n is defined to be equal to $k-\lambda$. The complement of such a difference set is itself a difference set with parameters (v, $v-k$, $v-2k+\lambda$) and the same parameter n. If we identify the subset D with the element of the group ring ${\mathbb {Z}}G$ which is its $\{0,1\}$-valued characteristic function, then we have that D is such a difference set precisely when it satisfies in ${\mathbb {Z}}G$ the equation

$$\begin{aligned} D D^{(-1)} = n + \lambda G, \end{aligned}$$

where “n” stands for “n times the identity element of G” and, for any element $A:=\sum _{g \in G} a_g g$ in ${\mathbb {Z}}G$, $A^{(-1)}$ denotes the element $\sum _{g \in G} a_g g^{-1}$. The associated $\{\pm 1\}$-valued function $D^{*}:=G - 2D$ then satisfies

$$\begin{aligned} D^{*} {D^{*}}^{(-1)} = 4n + (v - 4n)G; \end{aligned}$$

so that if either $v=1$ or $v=4n$ we have

$$\begin{aligned} D^{*} {D^{*}}^{(-1)} = |G|. \end{aligned}$$

(1)

In such an event, the difference set D is called a hadamard difference set since the $v \times v$ $\{\pm 1\}$-incidence matrix $[D^{*}]$, whose rows and columns are indexed by the elements of G and whose (g,h)$^{th}$ entry is the coefficient of $g^{-1}h$ in $D^{*}$, is hadamard. If the group G is abelian, then the condition (1) is equivalent to the condition

$$\begin{aligned} |\chi (D^{*})|^2 = |G| \quad \forall \chi \in \widehat{G}, \end{aligned}$$

(2)

where $\widehat{G}$ is the dual group of (irreducible) characters of G. If D is a difference set in G, then the elements of G and the translates of D in G constitute the points and blocks of a symmetric balanced incomplete block design called the development of D. Moreover, if an SBIBD has a group of automorphisms G which is sharply transitive on its points (and blocks), then we may identify the points with the group elements whereupon the blocks become difference sets in G. We therefore sometimes say that the notion of difference set is equivalent to the notion of SBIBD with a regular automorphism group. Two difference sets in a group G are equivalent if they are in the same orbit under the action induced on the subsets of G by the holomorph of G, i.e., one can be obtained from the other by applying some combination of automorphism and translation of G. A popular exercise is to prove that, up to equivalence,

$$\begin{aligned} D:=\{[0,0,0,0],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,1,1,1]\} \end{aligned}$$

is the unique (16, 6, 2) difference set in G = ${\mathbb {Z}}_2^4$. In 1955, R. H. Bruck [8] used this example to illustrate his new notion of noncyclic difference set. While James Singer [35] had formally defined the notion of difference set in his pioneering work on finite geometries in 1938, all of the considerable research on the subject was confined to cyclic groups until Bruck extended it. An outstanding problem in combinatorics is that of determining the class ${\mathcal H}$ of groups which contain a hadamard difference set. We shall call ${\mathcal H}$ the class of hadamard groups. In the next section, we give some background on hadamard groups, mainly to set the stage back to the time that Jim Davis achieved his breakthrough. We speak mainly about 2-groups, although some of the theorems we recall are more general and there certainly have been a number of more recent results on hadamard groups of other orders. Good general references on difference sets and designs include [3, 4, 22] and [26]; and more complete references for hadamard difference sets include [1, 12, 18] and [26]. Section 3 reviews some Fourier analysis for finite abelian groups, mainly to establish the terminology and notation we use in the sequel and to draw attention to some useful ways of evaluating certain linear transformations by manipulating appropriate matrix products. The penultimate section presents the novel heretofore unpublished proof which we developed on the happy occasion of a visit to K. T. Arasu and his colleagues at Wright State University and which, we pray, might engender warm memories on this happy occasion of his $65^{th}$ birthday.

2 Some historical background

P. Kesava Menon [32] was the first to recognize that the special parameter relation $v = 4n$ forced the parameters to be

$$\begin{aligned} (v, k, \lambda ) = (4N^2, 2N^2-N, N^2-N) \end{aligned}$$

(3)

for some integer N. Here N can be positive or negative; and, for fixed N, the two values $\pm N$ give the parameters of complementary designs. Every group G of order v has trivial difference sets of cardinalities k in $\{0, 1, v-1, v\}$. In particular, if G is a group of order 4, then the singleton set $D=\{Id(G)\}$ is a trivial hadamard difference set whose corresponding matrix $[D^{*}]$ is

$$\begin{aligned}{}[D^{*}] = \left[ \begin{array}{llll} {-}&{}1&{}1&{}1\\ 1&{}{-}&{}1&{}1\\ 1&{}1&{}{-}&{}1\\ 1&{}1&{}1&{}{-} \end{array}\right] . \end{aligned}$$

(4)

A long-standing conjecture that there can be no larger circulant hadamard matrix is still open today. While it is natural to interpret this matrix as a circulix whose $(i,j)^{th}$ entry depends only on the difference $j-i$ mod 4, it can just as well be interpreted as the image $\left[ D^{*}\right]$ under the regular matrix representation of the group ring element $D^{*}=-Id(G) + g_1 + g_2 + g_3$ in ${\mathbb {Z}}G$, where $G:=\{g_0=Id(G), g_1, g_2, g_3\}$ is either group of order 4. Consideration of trivial difference sets may be avoided if we simply require that the parameter n be greater than 1. However, since the tensor (Kronecker) product of hadamard matrices is again hadamard, these trivial difference sets in groups of order 4 automatically spawn nontrivial difference sets in groups which are products of groups of order 4. P. Kesava Menon [32] and R. J. Turyn [37] both observed that the class of hadamard groups ${\mathcal H}$ is closed under direct products. More generally we have

Theorem 1

(The Product Theorem [15,16,17]) Let $G_1$ and $G_2$ be hadamard groups and suppose that G is a group containing subgroups $G_1$ and $G_2$ satisfying $G=G_1G_2$ and $G_1 \cap G_2$=1. Then G is a hadamard group.

Proof

The hypothesis implies that every element g of G may be written uniquely as $g=g_1g_2$ for some $g_1 \in G_1$ and $g_2 \in G_2$.

Let $D_1$ and $D_2$ be difference sets in $G_1$ and $G_2$, respectively; and let D be the subset of G defined by $D^{*}=D_1^{*}D_2^{*}$. Then

$$\begin{aligned} D^{*}{D^{*}}^{(-1)}= & {} (D_1^{*}D_2^{*})(D_1^{*}D_2^{*})^{(-1)}\\= & {} (D_1^{*}D_2^{*})({D_2^{*}}^{(-1)}{D_1^{*}}^{(-1)})\\= & {} D_1^{*}(D_2^{*}{D_2^{*}}^{(-1)}){D_1^{*}}^{(-1)}\\= & {} D_1^{*}(|G_2|){D_1^{*}}^{(-1)}\\= & {} D_1^{*}{D_1^{*}}^{(-1)}(|G_2|)\\= & {} |G_1||G_2|\\= & {} |G|. \end{aligned}$$

$\blacksquare$

Thus, not only are both the cyclic group ${\mathbb {Z}}_4$ and the elementary abelian group $E_4:= {\mathbb {Z}}_2 \times {\mathbb {Z}}_2$ hadamard groups, but so is any group which is a product of such groups in the sense of unique representation. It is sometimes convenient to use the notation $E_q$ to denote an elementary abelian group of order q, where q can be any prime power. Thus, for q a power of 2, $E_q$ is a hadamard group if and only if q is a square, i.e., $q=1$ or $q=2^{2s+2}$ for s any nonnegative integer. Such difference sets are precisely the sets whose characteristic function is a bent function [13, 34]. Notice that, for any group G which is such a product of subgroups $G_1$ and $G_2$, we may write the truth-table of any function f on G as a matrix whose rows and columns are indexed by the elements of $G_1$ and $G_2$, respectively. In particular, for any groups $G_1$ and $G_2$ of order 4, we can write the truth-table of the function

$$\begin{aligned} D^{*}=D_1^{*}D_2^{*}=(G_1-2Id(G_1))(G_2-2Id(G_2) \end{aligned}$$

as

$$\begin{aligned} \left[ \begin{array}{llll} 1&{}-&{}-&{}-\\ {-}&{}1&{}1&{}1\\ {-}&{}1&{}1&{}1\\ {-}&{}1&{}1&{}1 \end{array}\right] . \end{aligned}$$

This is NOT the hadamard matrix associated with D; it is rather the first column of the hadamard matrix $\left[ D^{*}\right]$ written as a short fat $4 \times 4$ vector instead of a long skinny $16 \times 1$ vector. In this form, we can see at a glance that the difference set D, which indexes the $-1's$ in the matrix, corresponds to the six cells of the $4 \times 4$ array $\left[ (i,j)\right] _{0 \le i,j \le 3}$ which share exactly one coordinate with (0, 0). Furthermore, for every cell (i, j), the six other cells in the same row or column correspond to a block in the (16, 6, 2)-design which is the development of the difference set D. Thus, we may describe the design succinctly as follows. Points are the 16 cells in a $4 \times 4$ array. Blocks are in 1-1 correspondence with points; for each point, the corresponding block consists of the six other cells appearing in the same row or column as that point.

This description was given by Camille Jordan [24, 25] who studied its doubly transitive automorphism group. Meanwhile, observe how the $4 \times 4$ rendition of the product difference set in $G=G_1G_2$ above makes it clear that this difference set is obtained from the partial spread $\{G_1, G_2 \}$ according to the following more general

Theorem 2

(Partial Spreads [13]) Let G of order $4N^2$, $N>0$, have subgroups $H_1$, $H_2$, $\dots$, $H_N$ satisfying $|H_i| = 2N \; \forall i$ and $H_i \cap H_j = 1 \;\forall i \ne j$.

Then $D:=(\cup _{i=1}^N H_i) \backslash \{1\}$ is a hadamard difference set in G.

We now restrict our attention to groups of order a power of 2. A theorem of H. B. Mann [29] says that any nontrivial difference set in a 2-group must be hadamard. Indeed, Mann proved that a symmetric block design with v a power of 2 must satisfy the Menon condition $v=4n$; a slightly more general result is proved in [13]. Thus, our groups of interest now will have orders $2^{2s+2}$; and difference sets in such a group will have parameters (up to complementation)

$$\begin{aligned} (v, k, \lambda ) = (2^{2s+2}, 2^s(2^{s+1} - 1), 2^s(2^s - 1)). \end{aligned}$$

P. Kesava Menon [32] proved a multiplier theorem for difference sets in general abelian groups by introducing character theory into that setting; and R. J. Turyn [37] was able to exploit such character theory to give algebraic restrictions on hadamard groups. A special case of his results which is relevant here is the

Theorem 3

(R. J. Turyn 1965) Let G be a hadamard group of order $2^{2s+2}$ and suppose that G has a normal subgroup K such that G/K is cyclic. Then G/K has order at most $2^{s+2}$.

Corollary 1

(Turyn’s Exponent Bound) If G is an abelian hadamard group of order $2^{2s+2}$, then G has exponent at most $2^{s+2}$.

We later showed how to twist Turyn’s proof of his above theorem into a proof of the following

Theorem 4

(The Dihedral Trick [15, 17]) The above theorem remains true if cyclic is replaced by dihedral.

We combine these two nonexistence criteria into the single

Theorem 5

(Turyn-Dillon 1985) If a group G of order $2^{2s+2}$ has a normal subgroup K such that G/K is either cyclic or dihedral of order greater than $2^{s+2}$, then G is not a hadamard group, i.e., G does not have a nontrivial difference set.

There are fourteen isomorphism classes of groups of order 16; and this theorem says that the cyclic and dihedral groups are not in ${\mathcal H}$. It is a remarkable fact that all twelve of the other groups of order 16 act regularly on the (16, 6, 2) Jordan design and are therefore hadamard groups. In 1978, R. E. Kibler [27] found by computer search all inequivalent difference sets in all of these groups.

Of all the abelian hadamard groups of order $2^{2s+2}$ produced by investigators in the early sixties, ${\mathbb {Z}}_4^{s+1}$ had the smallest rank, which is $s+1$. Twenty-five years later the known hadamard groups of least rank were ${\mathbb {Z}}_2^s \times {\mathbb {Z}}_{2^{s+2}}$, which were still of rank $s+1$; and this caused some to think that it was necessary that the rank be this large. Curiously, R. J. Turyn [37] had given an example of a difference set in ${\mathbb {Z}}_2 \times Z_8$ which has rank 2; but even this is equal to $s+1$ for groups of order $2^{2s+2} = 16$. A natural question was whether the rank could be as small as 2 for larger groups. According to Turyn’s exponent bound given by Corollary 1, the only abelian groups of order $2^{2s+2}$ which could possibly be hadamard are ${\mathbb {Z}}_{2^{s+1}} \times {\mathbb {Z}}_{2^{s+1}}$ and ${\mathbb {Z}}_{2^s} \times {\mathbb {Z}}_{2^{s+2}}$.

In 1987, Jim Davis surprised the world by proving, in his Virginia dissertation [11], that all of these groups are hadamard. That same year R. J. Turyn [38] found by computer all inequivalent difference sets in the group ${\mathbb {Z}}_8 \times {\mathbb {Z}}_8$, one of which is the beautiful Fig. 1.

Fig. 1

Turyn DS in ${\mathbb {Z}}_8 \times {\mathbb {Z}}_8$

Again, this is not the hadamard matrix that comes from the development of the difference set; but it is (the characteristic function of) the difference set itself, where we think of the group as naturally coordinatizing the cells of an $8 \times 8$ array.

Later Robert Kraemer [28] used Davis’ inductive techniques to prove that any abelian group of order $2^{2s+2}$ and exponent at most $2^{s+2}$ does indeed have a difference set, i.e.,

Theorem 6

(Turyn-Kraemer 1993) An abelian group of order $2^{2s+2}$ has a nontrivial difference set if and only if it has exponent at most $2^{s+2}$.

Thus, the existence question was completely settled for abelian 2-groups G. For such a group G then, of interest would be questions such as explicit constructions of difference sets in G, perhaps a number of inequivalent ones or perhaps some with particularly interesting properties. The next theorem achieves all of these goals. The Davis and Kraemer results were very important advances in the theory of hadamard difference sets and groups; but, because of the inductive nature of the constructions, it was not at all clear what these difference sets looked like. Turyn’s beautiful example in ${\mathbb {Z}}_8 \times {\mathbb {Z}}_8$ suggested the possibility of more general beautiful difference sets. Later in 1987 [16], we gave the following simple direct construction of difference sets in ${\mathbb {Z}}_{2^{s+1}} \times {\mathbb {Z}}_{2^{s+1}}$.

Theorem 7

Let $G={\mathbb {Z}}_{2^{s+1}} \times {\mathbb {Z}}_{2^{s+1}}$, where we regard ${\mathbb {Z}}_{2^{s+1}}$ as the additive group of integers mod $2^{s+1}$. Let $f:{\mathbb {Z}}_{2^{s+1}} \rightarrow \{\pm 1\}$ be any function satisfying the property that $f(x+2^s)=-f(x)$ for all x in ${\mathbb {Z}}_{2^{s+1}}$; and let $\pi :{\mathbb {Z}}_{2^{s+1}} \rightarrow {\mathbb {Z}}_{2^{s+1}}$ be the permutation which maps each residue $2^r t$, t odd, to the residue $2^r \overline{t}$, where $t \overline{t} \equiv 1 \pmod {2^{s+1}}$. Define $F:G \rightarrow \{\pm 1\}$ by $F(x,y):=f(\pi (x) y)$, where $\pi (x) y$ is the product of $\pi (x)$ and y, taken mod $2^{s+1}$. Then

• (i) F is the $\{\pm 1\}$-valued characteristic function of a hadamard difference set D in G;

• (ii) D is fixed by inversion, i.e., $-1$ is a multiplier of D;

• (iii) If $\eta (G)$ is the number of pairwise inequivalent difference sets with the parameters of D produced by this construction, then $\eta (G)$ is exponential in the sense that $\log {\eta (G)} \sim 2^s-1 = \frac{\sqrt{\vert G \vert }}{2} - 1$;

• (iv) Any one of these difference sets D in G = ${\mathbb {Z}}_{2^{s+1}} \times {\mathbb {Z}}_{2^{s+1}}$ may be transferred to a difference set ${\mathcal {D}}$ in ${\mathcal G}$ = ${\mathbb {Z}}_{2^s} \times {\mathbb {Z}}_{2^{s+2}}$.

Of the $2^{2^s}$ such functions f, those with $f(0)=1$ produce difference sets having parameters (3) with $N=2^s$; and those with $f(0)=-1$ give the complementary parameters with $N=-2^s$. One natural choice for the function f is $f(x):=(-1)^{\lfloor \frac{x}{2^s} \rfloor }$, i.e., $(-1)^{x_s}$, where $x_s$ is the high-order bit in the binary expansion of x.

Corollary 2

Let $G={\mathbb {Z}}_{2^{s+1}} \times {\mathbb {Z}}_{2^{s+1}}$, where ${\mathbb {Z}}_{2^{s+1}}=\{0,1,2,\dots ,2^{s+1}-1\}$ is the additive group of integers mod $2^{s+1}$. Let $\tau :{\mathbb {Z}}_{2^{s+1}} \rightarrow \{0,1\}$ be the characteristic function of the set of residues greater than or equal to $2^s$; and let $\pi$ be the permutation on ${\mathbb {Z}}_{2^{s+1}}$ which maps the residue $2^rt$, t odd, to the residue $2^r \overline{t}$, where $t \overline{t} = 1 \bmod 2^{s+1}$. Let D be the subset of G given by

$$\begin{aligned} D = \{(x,y):\tau (\pi (x) y) = 1\}. \end{aligned}$$

Then D is a difference set in G. Moreover, this difference set may be transferred to a difference set in the group ${\mathbb {Z}}_{2^{s}} \times {\mathbb {Z}}_{2^{s+2}}$.

Notice that when $s \le 2$, the permutation $\pi$ is the identity and a difference set may be defined by

$$\begin{aligned} D:=\{(x,y):0 \le x,y < 2^{s+1} \; \mathrm{and} \; \lfloor \frac{xy}{2^s} \rfloor = 1\}. \end{aligned}$$

For $s=2$ this is precisely Turyn’s example given in Figure 1.

We note that this construction also provides 2-dimensional synchronization patterns of size $2^{s+1} \times 2^{s+1}$ for all s; see [9, 18] for more on this idea. In fact, Jonathan Jedwab [23] independently developed much of the theory of abelian hadamard groups from the point of view of perfect binary arrays. Early proponents of this approach were Calabro and Wolf [9], Chan, Siu and Tong [10] and, especially, Bömer and Antweiler [6], who gave examples for the groups of order 64. Arasu, himself, together with his student James Reis [2], had independently conducted successful computer searches for difference sets in these same groups of order 64.

It is the main purpose of this paper to give a new proof of Part (i) of this theorem. We put that off to Section 4 and here give quick verification of the other parts of the theorem.

Proof of Parts (ii), (iii), (iv) of Theorem7.

Part (ii) is immediate since

$$\begin{aligned} F(-x,-y) = f(\pi (-x)(-y)) = f((-1)\pi (x)(-y)) = f(\pi (x)y) = F(x,y). \end{aligned}$$

More generally, for all odd t,

$$\begin{aligned} F(tx,ty) = f(\pi (tx)(ty)) = f(t^{-1}\pi (x)(ty)) = f(\pi (x)y) = F(x,y); \end{aligned}$$

so D is fixed by every numerical automorphism of G, confirming a theorem of Schur.

Part (iii) is clear because there are $2^{2^s-1}$ functions f for which f(0) = 1 and which therefore produce difference sets of size k = $2^s(2^{s+1}-1)$; but the order of the holomorph of G is less than $(2^{s+1})^6$.

To verify Part (iv), we work in the group ring ${\mathbb {Z}}G$ where it is most convenient to use multiplicative notation for the group operation. We assume, for now, that Part (i) is true, i.e., D is a difference set in G. If G = $\langle a \rangle \times \langle b \rangle \cong {\mathbb {Z}}_{2^{s+1}} \times {\mathbb {Z}}_{2^{s+1}}$, we let H be the subgroup $\langle a^2 \rangle \times \langle b \rangle$ and write $G = H + aH$ and $D^{*}$ = $A^{*} + aB^{*}$. The hypothesized condition on f that $f(x+2^s)$ = $-f(x)$ implies that $A^{*}$ = $(1+b^{2^s})\varDelta _0$ and $B^{*}$ = $(1-b^{2^s})\varDelta _1$ for some $\pm 1$-valued functions $\varDelta _0$ and $\varDelta _1$ on some set of representatives of the cosets of $\langle b^{2^s} \rangle$ in H, and therefore that $B^{*}{A^{*}}^{(-1)}$ = 0 = $A^{*}{B^{*}}^{(-1)}$ and

$$\begin{aligned} 2|H| = |G|&= D^{*}{D^{*}}^{(-1)}\end{aligned}$$

(5)

$$\begin{aligned}&= (A^{*} + a B^{*})(A^{*} + aB^{*})^{(-1)}\end{aligned}$$

(6)

$$\begin{aligned}&= (A^{*}{A^{*}}^{(-1)} + B^{*}{B^{*}}^{(-1)}) + a(B^{*}{A^{*}}^{(-1)} + a^{-2}A^{*}{B^{*}}^{(-1)})\end{aligned}$$

(7)

$$\begin{aligned}&= A^{*}{A^{*}}^{(-1)} + B^{*}{B^{*}}^{(-1)}. \end{aligned}$$

(8)

Now let ${\mathcal {G}}$ = $\langle a^2 \rangle \times \langle c \rangle \cong {\mathbb {Z}}_{2^s} \times {\mathbb {Z}}_{2^{s+2}}$, where $c^2 = b$. We write ${\mathcal {G}}$ = $H + cH$; and define a subset ${\mathcal {D}}$ of ${\mathcal {G}}$ by ${\mathcal {D}}^{*}$ = $A^{*} + c B^{*}$ in ${\mathbb {Z}}{\mathcal {G}}$. Then, computing just as above, we have

$$\begin{aligned} {\mathcal {D}}^{*}{{\mathcal {D}}^{*}}^{(-1)}&= (A^{*} + c B^{*})(A^{*} + c B^{*})^{(-1)} \end{aligned}$$

(9)

$$\begin{aligned}&= A^{*}{A^{*}}^{(-1)} + B^{*}{B^{*}}^{(-1)} \end{aligned}$$

(10)

$$\begin{aligned}&= |G| \end{aligned}$$

(11)

$$\begin{aligned}&= 2|H| \end{aligned}$$

(12)

$$\begin{aligned}&= |{\mathcal {G}}|. \end{aligned}$$

(13)

Thus, $\mathcal {D}$ is a difference set in ${\mathcal {G}}$.$\blacksquare$

Note that the factors $\chi _0$ := $1+b^{2^s}$ and $\chi _1$ := $1-b^{2^s}$ of $A^{*}$ and $B^{*}$, respectively, are the characters of the subgroup $\langle b^{2^s} \rangle$ of H which are orthogonal, i.e., $\chi _0 \chi _1$ = 0 in ${\mathbb {Z}}H$. This technique of using orthogonal pieces in a group to construct a difference set in an extension of that group has been generalized in [14, 16,17,18] and most recently in [1]. We pointed out in [17, 18] how Turyn’s difference set in ${\mathbb {Z}}_8 \times {\mathbb {Z}}_8$ could be transferred to any group of order 64 which contains ${\mathbb {Z}}_4 \times {\mathbb {Z}}_4$ as a normal subgroup. In this case, $D^{*}$ is decomposed into four pieces which are multiples of the four characters of the ${\mathbb {Z}}_2 \times {\mathbb {Z}}_2$ subgroup and are therefore pairwise orthogonal. The same technique was used in [16] to prove our generalizations of McFarland’s Theorem 8. Reference [1] shows how to construct difference sets in a wide class of groups G which contain an elementary abelian subgroup E and a subgroup K which satisfy $E< K < G$ and [G : K] = |E|. This new construction has helped to complete the project begun in [19] to classify the hadamard groups of order 256. It is intriguing to note that, for 2-groups of square order up to and including 256, the only groups which are not hadamard are those ruled out by the long-known Theorem 5. The interested reader is invited to consult [18] and [1] for all the details on how this classification was accomplished.

3 Fourier analysis

While most of what we have reviewed so far in this paper has been on the existence question and has therefore focused mainly on nonabelian groups, in which case (1) was the condition of choice, we now want to restrict our attention to a class of abelian groups for which we can exploit the condition (2) in a novel way. The group algebra ${\mathbb C}G$ is an inner product space under the Hermitian inner product $\langle A, B \rangle := \sum _{g \in G}A(g)\overline{B(g)}$. For any function A in ${\mathbb C}G$ the character sums

$$\begin{aligned} \chi (A) := \sum _{g \in G}A(g)\chi (g) = \langle A, \overline{\chi } \rangle \end{aligned}$$

are the unnormalized Fourier coefficients which allow us to express A in terms of the orthonormal basis of normalized characters $\frac{1}{\sqrt{\vert G \vert }} \chi$. We have

$$\begin{aligned} A = \frac{1}{\sqrt{|G|}}{\sum _{\chi } \widehat{A}(\chi ) \chi }, \end{aligned}$$

where $\widehat{A}(\chi ) := \frac{1}{\sqrt{\vert G \vert }}\langle A, \chi \rangle$. It will be convenient for us to take a cyclic group of order m, $C_m$, to be the additive group of integers mod m whose elements we will list in the usual way

$$\begin{aligned} 0, 1, 2, \dots , m-1 \end{aligned}$$

and use to also index the elements of the character group $\widehat{C_m}$ as

$$\begin{aligned} \chi _0, \chi _1, \chi _2, \dots , \chi _{m-1}. \end{aligned}$$

We usually define $\chi _a$ by $\chi _a: x \mapsto \zeta _m^{ax}$ for all x in $C_m$, where $\zeta _m$ := $\exp {\frac{2\pi i}{m}}$. Then we may take the character table ${\mathscr {C}}_m$ of $C_m$ to be the $m \times m$ matrix whose rows and columns are indexed by $C_m$ and whose $(a,b)^{th}$ entry is $\chi _a(b)$ = $\zeta _m^{ab}$ for all a and b in $C_m$, i.e., ${\mathscr {C}}_m = \left[ \zeta _m^{ab}\right]$. Now an arbitrary finite abelian group G, being a direct product of cyclic groups, has a natural induced order on its elements which we may also use to index its characters as well. For example, the Sylvester hadamard matrix

$$\begin{aligned} H_m := \otimes ^m {{\mathscr {C}}}_2 = \otimes ^m \left[ \begin{matrix} 1 &{} 1\\ 1 &{} - \end{matrix}\right] = \left[ (-1)^{a \cdot b}\right] \end{aligned}$$

is the character table of the elementary abelian group $E_{2^m} := C_2^m$. We have belabored a bit the idea of indexing the characters of G by the elements of G because we want to consider the Fourier transform $A \mapsto \widehat{A}$ as a unitary linear operator on ${\mathbb C}G$; we will say that $\widehat{A}$ is the Fourier transform of A. We restate the fundamental equivalences relating to the notion of hadamard difference sets.

Proposition 1

For a $\{\pm 1\}$-valued function F on the finite abelian group G

the following are equivalent:

• Combinatorial Criterion: F is (the ${\pm 1}$-valued characteristic function of) a hadamard difference set in G;

• Group Ring Criterion: $FF^{(-1)} = |G|$;

• Fourier Transform Criterion: $|\widehat{F}(g)| = 1 \quad \forall g \in G$, i.e., $\widehat{F}$ has constant magnitude 1.

We now want to express the Fourier transform operator in terms of matrices. For a certain class of groups G we will then be able to obtain difference sets by defining F in such a way that the Fourier Transform Criterion of Proposition 1 is graphically obvious. For A any function in ${\mathbb C}G$, we denote by $\left[ A \right]$ the truth table of A, i.e., the vector indexed by G whose $g^{th}$ coordinate is A(g). If ${{\mathscr {C}}}_G$ is the character table for G whose $(a,b)^{th}$ entry is $\chi _a(b)$, then the Fourier transform $\widehat{A}$ of A is

$$\begin{aligned}{}[\widehat{A}] = {{\mathfrak F}}_G [A], \end{aligned}$$

(14)

where ${{\mathfrak F}}_G := {\vert G \vert }^{-\frac{1}{2}} \overline{{{\mathscr {C}}}_G}$. We will call ${{\mathfrak F}}_G$ the Fourier matrix of G.

We now want to restrict our attention to groups G of the form G = $G_1 \times G_2$. Then ${\mathfrak F}_G$ = ${\mathfrak F}_{G_1} \otimes {\mathfrak F}_{G_2}$ = $({\mathfrak F}_{G_1} \otimes I_{|G_2|})(I_{|G_1|} \otimes {\mathfrak F}_{G_2})$; and so we may express (14) as

$$\begin{aligned}{}[\widehat{A}]^\Box = {\mathfrak F}_{G_1} [A]^\Box {\mathfrak F}_{G_2}^\intercal . \end{aligned}$$

(15)

where, for any function X on $G_1 \times G_2$, $X^\Box$ denotes the $|G_1| \times |G_2|$ matrix whose rows (resp. columns) are indexed by $G_1$ (resp. $G_2$) and whose $(a,b)^{th}$ entry is X(a, b). For example, suppose that $G :=G_1 \times E_{2^m}$, where $G_1$ is any abelian group of order $2^m$; and suppose that we define A by

$$\begin{aligned} A^\Box := \varDelta \Pi H_m, \end{aligned}$$

where $\varDelta$ is a diagonal matrix with diagonal entries $\pm 1$ and $\Pi$ is a permutation matrix. Then Equation (15) becomes

$$\begin{aligned}{}[\widehat{A}]^\Box&= {\mathfrak F}_{G_1} \left[ A\right] ^\Box {\mathfrak F}_{G_2}^\intercal \end{aligned}$$

(16)

$$\begin{aligned}&= {\mathfrak F}_{G_1}\left( \varDelta \Pi H_m\right) {\mathfrak F}_{E_{2^m}}\end{aligned}$$

(17)

$$\begin{aligned}&= 2^{-m} \overline{{\mathscr {C}}_{G_1}}\left( \varDelta \Pi H_m\right) H_m\end{aligned}$$

(18)

$$\begin{aligned}&= \overline{{\mathscr {C}}_{G_1}} \varDelta \Pi . \end{aligned}$$

(19)

Since the entries of this matrix are roots of unity, it follows by the Fourier Criterion of Proposition 1 that A is a difference set and that therefore $G = G_1 \times E_{2^m}$ is a hadamard group. We have here an exceedingly transparent proof of the following theorem of McFarland [30], in the case that $G_1$ is abelian.

Theorem 8

(R. L. McFarland 1973) $G_1 \times E_{2^m} \in {\mathcal H}$ for all groups $G_1$ of order $2^m$.

When $G_1 = E_{2^m}$ the difference sets A given in the above proof are given by

$$\begin{aligned} A^\Box = \varDelta \Pi H_m = \varDelta \Pi \left[ (-1)^{a \cdot b}\right] , \end{aligned}$$

so we have $A(x,y) = (-1)^{f(x,y)}$, where f(x, y) is the Boolean function on $GF(2)^{2m} \cong GF(2)^m \oplus GF(2)^m$ given by

$$\begin{aligned} f(x,y) = \pi (x) \cdot y + g(x), \end{aligned}$$

where $(-1)^{g(x)} := \varDelta (x,x)$. These are the bent functions which were said to be in Family ${\mathcal M}$ in [13] and are nowadays usually called Maiorana-McFarland bent functions. The proof given here which exploits Equation (15) in the case of elementary abelian groups is due to D. P. Cargo (private communication). The partial spread construction of Theorem 2 produces many bent functions which are not equivalent to any in Family ${\mathcal M}$. The interested reader is referred to the encyclopedic books [33] and [36] for much more on bent functions.

In 1985 [15], we generalized McFarland’s Theorem 8 to

Theorem 9

A group G of order $2^{2m}$ is hadamard if it has a central subgroup which is elementary abelian of order $2^m$.

This result shows that the groups ${\mathbb {Z}}_2^s \times {\mathbb {Z}}_{2^{s+2}}$ are hadamard and therefore Turyn’s exponent bound (Corollary 1) is sharp. We conjectured that the theorem would remain true if the hypothesis central were replaced by normal; and this was proved by Art Drisko [20].

4 The new proof

Here is a new general idea which we introduced in a Colloquium talk given at Wright State University while visiting our esteemed colleague and friend K. T. Arasu.

Theorem 10

Let C denote the additive group ${\mathbb {Z}}_{2^m}$ of integers mod $2^m$. Let G =$C \times C$ and let F be a ${\pm 1}$-valued function on G. Let ${\mathfrak F}$ be the Fourier matrix of C and let $F^{\Box }$ be the $2^m \times 2^m$ matrix whose rows and columns are indexed by C and whose $(x,y)^{th}$ entry is F(x, y). Suppose that $F^{\Box }{\mathfrak F}$ is symmetric. Then F is a difference set in G.

Proof

Note that the character table ${\mathscr {C}}$ = $[\zeta ^{ab}]$ of C and hence the Fourier matrix ${\mathfrak F}$ are symmetric and, since ${{\mathscr {C}}}^2(a,b)$ = $\sum _{c \in C}\zeta ^{c(a+b)}$ = $2^m\delta _{-a,b} \forall a,b$, ${{\mathfrak F}}^2$ is the permutation matrix $P:=[\delta _{-a,b}]$. Therefore,

$$\begin{aligned} {\widehat{F}}^{\Box } = {\mathfrak F}F^{\Box } {\mathfrak F}^\intercal = {\mathfrak F}^2{F^{\Box }}^\intercal = P{F^{\Box }}^{\intercal }, \end{aligned}$$

which, graphically, has entries $\pm 1$; and the result follows by the Fourier Transform Criterion of Proposition 1.$\blacksquare$

Proof of Part (i) of Theorem 7. According to the previous Theorem we need only prove that the matrix product $F^{\Box } {\mathfrak F}^{\intercal }$ is symmetric; so consider its $(a,b)^{th}$ entry

$$\begin{aligned} F^{\Box } {\mathfrak F}^{\intercal }(a,b) = 2^{-\frac{m}{2}}\sum _{c \in C}f(\pi (a)c)\zeta ^{-cb}, \end{aligned}$$

which we claim is equal to its $(b,a)^{th}$ entry. We first suppose that a and b have the same order in C; say a = rb for some odd r. The claim is certainly true if b = 0. Otherwise we have

$$\begin{aligned} 2^{\frac{m}{2}}F^\Box {\mathfrak F}^\intercal (a,b)= & {} \sum _{c} f(\pi (a)c)\zeta ^{-cb}\\= & {} \sum _{c} f(\pi (rb)c)\zeta ^{-cb}\\= & {} \sum _{c} f(r^{-1}\pi (b)c)\zeta ^{-cb}\\= & {} \sum _{c} f(\pi (b)c)\zeta ^{-rcb}\\= & {} \sum _{c} f(\pi (b)c)\zeta ^{-ca}\\= & {} 2^{\frac{m}{2}}F^\Box {\mathfrak F}^\intercal (b,a). \end{aligned}$$

We now show that all other entries in the matrix are 0. If a = 0, then

$$\begin{aligned} 2^{\frac{m}{2}}F^\Box {\mathfrak F}^\intercal (0,b) = \sum _{c} f(0)\zeta ^{-cb} = f(0)\sum _{c}\zeta ^{-cb} = f(0) 2^m \delta _{b,0}, \end{aligned}$$

which is 0 for all nonzero b. Similarly, if b = 0, then

$$\begin{aligned} 2^{\frac{m}{2}}F^\Box {\mathfrak F}^\intercal (a,0) = \sum _{c} f(\pi (a)c) = 2^m f(0)\delta _{a,0}, \end{aligned}$$

which is 0 for all nonzero a because f sums to 0 on the subgroup of C generated by $\pi (a)$. Now suppose that a and b are nonzero elements having different orders; say

$$\begin{aligned} 1< o(a) < o(b) = 2^\beta . \end{aligned}$$

Then

$$\begin{aligned} 2^{\frac{m}{2}} \cdot 2F^\Box {\mathfrak F}^\intercal (a,b)= & {} \sum _{c} f(\pi (a)c)\zeta ^{-cb} + \sum _{c} f(\pi (a)(c+2^{\beta -1}))\zeta ^{-(c+2^{\beta -1})b}\\= & {} \sum _{c} f(\pi (a)c)[\zeta ^{-cb} + \zeta ^{-cb+2^s} ]\\= & {} \sum _{c} f(\pi (a)c)[\zeta ^{-cb} + \zeta ^{-cb}(-1) ]\\= & {} 0. \end{aligned}$$

Similarly, if

$$\begin{aligned} 1< o(b) < o(a) = 2^\alpha , \end{aligned}$$

then

$$\begin{aligned} 2^{\frac{m}{2}} \cdot 2F^\Box {\mathfrak F}^\intercal (a,b)= & {} \sum _{c} f(\pi (a)c)\zeta ^{-cb} + \sum _{c} f(\pi (a)(c+2^{\alpha -1}))\zeta ^{-(c+2^{\alpha -1})b}\\= & {} \sum _{c} [f(\pi (a)c) +f(\pi (a)c +2^s)]\zeta ^{-cb}\\= & {} \sum _{c} [f(\pi (a)c) - f(\pi (a)c)]\zeta ^{-cb}\\= & {} 0. \end{aligned}$$

We have shown that $F^\Box {\mathfrak F}^\intercal$ is symmetric. Theorem 10 is now the coup de grâce that despatches Theorem 7(i).$\blacksquare$

We have completed the proof of the entire Theorem 7 which establishes not only the simple direct construction of these difference sets, but also a number of interesting properties that they possess. The fact that any one of these ${\pm 1}$-valued functions F corresponds to a difference set came from showing that its Fourier transform $\widehat{F}$ also took values ${\pm 1}$. Now notice that the Fourier transform of $\widehat{F}$ is given by

$$\begin{aligned} {\mathfrak F}\widehat{F}^{\Box } {\mathfrak F}= {\mathfrak F}({\mathfrak F}F^{\Box } {\mathfrak F}) {\mathfrak F}= PF^{\Box }P, \end{aligned}$$

so that $\widehat{\widehat{F}}(x,y) = F(-x,-y) = F(x,y)$ and the Fourier transform of $\widehat{F}$ coincides with F. Thus, the Fourier transform interchanges the difference set functions F and $\widehat{F}$, which is exactly what happens if the group is elementary abelian in which case the Fourier transform interchanges every bent function F with its so-called dual bent function $\widehat{F}$. In the present case, since $\widehat{F}(x, y) = F(y, -x)$, the difference set given by $\widehat{F}$ is equivalent to the one given by F under the automorphism of G given by $(x, y) \mapsto (y, -x)$.

5 An appreciation

We would like to acknowledge our debt of gratitude to a number of people who have contributed to the development of the theory of hadamard groups and from whose work we have benefitted. P. Kesava Menon and Dick Turyn started the ball rolling in the early sixties. Bob McFarland’s work in the seventies fueled our imagination about the wide variety of such groups. Jim Davis’ 1987 breakthrough on the question of the rank of an abelian hadamard 2-group and Robert Kraemer’s subsequent application of Jim’s inductive techniques to finish off the complete characterization of abelian hadamard 2-groups marked the beginning of a golden age. Jim, Jonathan Jedwab, Ben Meisner, K. T. Arasu, Surinder Sehgal, Bob Liebler, Ken Smith and Joel IIams were some of the players. Research on the projects to classify the hadamard groups of orders 64 and 256 led to many new constructions which led to a better understanding of hadamard groups in general. Jim and Ken have been involved in all of this work; and Jonathan has been a frequent collaborator. We are especially indebted to John Cannon and his Cayley/Magma team for providing the computer algebra system which makes it possible to test groups for hadamardicity and to our colleague Al Schwartz, who did much of the computer testing and wrote the computer program which displays difference sets and designs in such a beautiful way as Fig. 1. Joe Bohannon and Alexander Hulpke developed some search algorithms in GAP. Most of the testing required by these classifications have been done independently by different teams using Magma and GAP, both of which use the same SmallGroups database. We note that the groups of order 1024 have recently been enumerated; they number 49,487,365,422. Ok, who’s up for that project? :)