Twin bent functions, strongly regular Cayley graphs, and Hurwitz-Radon theory

The real monomial representations of Clifford algebras give rise to two sequences of bent functions. For each of these sequences, the corresponding Cayley graphs are strongly regular graphs, and the corresponding sequences of strongly regular graph parameters coincide. Even so, the corresponding graphs in the two sequences are not isomorphic, except in the first 3 cases. The proof of this non-isomorphism is a simple consequence of a theorem of Radon.


Introduction
A recent paper of the author [3] describes a generalization of Williamson's construction for Hadamard matrices [6] using the real monomial representation of the basis elements of the Clifford algebras R m,m . In that paper, the following three conjectures appear: Conjecture 1. For all m 0 there is a permutation π of the set of 4 m canonical basis matrices, that sends an amicable pair of basis matrices with disjoint support to an anti-amicable pair, and vice-versa.

Conjecture 2.
For all m 0, for the Clifford algebra R m,m , the subset of transversal graphs that are not self-edge-colour complementary can be arranged into a set of pairs of graphs with each member of the pair being edgecolour complementary to the other member. The author's subsequent paper on bent functions [4] refines Conjecture 1 into the following question. In this paper it is proved that Conjecture 3 fails for m 4, and as a consequence, so do Conjectures 1 and 2. This also resolves Question 1: the only values for which such an automorphism exists are 1, 2, and 3.
The paper is organized as follows. Section 2 repeats the relevant definitions from the previous two papers [3,4], and includes further definitions as necessary. Section 3 states and proves the main results.

Preliminaries
This section sets out the main definitions and properties used in this paper. It is mostly based on the previous papers [3,4] with a few additions.
Clifford algebras and their real monomial representations.
The following definitions and results appear in the paper on Hadamard matrices and [3], and are presented here for completeness, since they are used below. Further details and proofs can be found in that paper, and in the paper on bent functions and Clifford algebras, unless otherwise noted.
The signed group G p,q of order 2 1+p+q is extension of Z 2 by Z p+q 2 , defined by the signed group presentation The following construction of the real monomial representation P (G m,m ) of the group G m,m is used in [3].
The 2 × 2 orthogonal matrices generate P (G 1,1 ), the real monomial representation of group G 1,1 . The cosets of {±I} ≡ Z 2 in P (G 1,1 ) are ordered using a pair of bits, as follows.
For m > 1, the real monomial representation P (G m,m ) of the group G m,m consists of matrices of the form are ordered by concatenation of pairs of bits, where each pair of bits uses the ordering as per P (G 1,1 ), and the pairs are ordered as follows.
This ordering is called the Kronecker product ordering of the cosets of {±I} in P (G m,m ).
Recall that the group G m,m and its real monomial representation P (G m,m ) satisfy the following properties.
1. Pairs of elements of G m,m (and therefore P (G m,m )) either commute or anticommute: for g, h ∈ G m,m , either hg = gh or hg = −gh.
2. The matrices E ∈ P (G m,m ) are orthogonal: 3. The matrices E ∈ P (G m,m ) are either symmetric and square to give I or skew and square to give −I: either E T = E and E 2 = I or E T = −E and E 2 = −I.
Taking the positive signed element of each of the 2 2m cosets listed above defines a transversal of {±I} in P (G m,m ) which is also a monomial basis for the real representation of the Clifford algebra R m,m in Kronecker product order, called this ordered monomial basis the positive signed basis of P (R m,m ).
Definition. We define the function γ m : Z 2 2m → P (G m,m ) to choose the corresponding basis matrix from the positive signed basis of P (R m,m ), using the Kronecker product ordering. This ordering also defines a corresponding function on Z 2m 2 , which we also call γ m .
The graphs used in Conjectures 2 and 3 and Question 1.
The following definitions appear in the previous two papers [3,4], and are repeated here for completeness, because Conjectures 2 and 3 and Question 1 depend on these definitions. Definition 1. Let ∆ m be the graph whose vertices are the n 2 = 4 m canonical basis matrices of the real representation of the Clifford algebra R m,m , with each edge having one of two labels, −1 or 1: • Matrices A j and A k are connected by an edge labelled by −1 if they have disjoint support and are anti-amicable, that is, A j A −1 k is skew. As noted in the paper on Clifford algebras and Hadamard matrices [3], a transversal graph for the Clifford algebra R m,m is any induced subgraph of ∆ m that is a complete graph on 2 m vertices. That is, each pair of vertices in the transversal graph represents a pair of matrices, A j and A k with disjoint support.

Two bent functions and their strongly regular Cayley graphs.
The previous two papers [3,4] define two bent functions on Z 2m 2 , σ m and τ m , respectively. (For the definition and properties of bent functions, see the previous two papers [3,4] and references therein.) In the paper on bent functions [4], the following two properties of σ m are noted.  As stated by Geramita and Pullman [1, Theorem A], Radon [5] proved the following result, which is used as a lemma in this paper. Lemma 1. Any Hurwitz-Radon family of order n has at most ρ(n) − 1 members.

Main Results
Here we state and prove the main results of this paper.
Theorem 2. For m 4 the following hold.
1. There exist transversal graphs that do not have an edge-colour complement, and therefore Conjecture 3 does not hold.
2. As a consequence, Conjectures 1 and 2 also do not hold. Proof. If we label the vertices of the subgraph ∆ m [−1] with the elements of Z 2m 2 , then any clique in this subgraph is mapped to another clique if a constant is added to all of the vertices. Thus without loss of generality we can assume that one of the vertices is labelled 0. If we then use γ m to label the vertices with elements of R m,m , we have one vertex labelled with the identity matrix I of order 2 m , and (since we have a clique in ∆ m [−1]) the other vertices A 1 to A s (say) must necessarily be skew matrices that are pairwise anti-amicable:

But then
and therefore {A 1 , . . . , A s } is a Hurwitz-Radon family. By Lemma 1, s is at most ρ(2 m ) − 1 and therefore the size of the clique is at most ρ(2 m ).
The second lemma puts a lower bound on the size of the maximum clique in ∆ m [1]. This set is closed under addition in Z 2m 2 , and therefore forms a clique of order 2 m in ∆ m [1].
With these two lemmas in hand, the proof of Theorem 2 follows easily. Proof of Theorem 2. Assume that m 4. A transversal graph is a subgraph of ∆ m which is a complete graph of order 2 m . The edges of a transversal graph are labelled with -1 (red) or 1 (blue). By Lemma 2, the largest clique of ∆ m [−1] is of order ρ(2 m ) < 2 m , and by Lemma 3, the largest clique of ∆ m [1] is of order 2 m . If we take a blue clique of order 2 m as a transversal graph, this cannot have an edge-colour complement in ∆ m , because no red clique can be this large. More generally, we need only take a transversal graph containing a blue clique with order larger than ρ(2 m ). This falsifies Conjecture 3.
Since Conjecture 3 fails for m 4, the pairing of graphs described in Conjecture 2 is impossible for m 4. Thus Conjecture 2 is also false.
Finally, Conjecture 1 fails as a direct consequence of Lemmas 2 and 3, since, for m 4, the difference between ∆ m [−1] and ∆ m [1] in the order of the largest clique means that these two subgraphs of ∆ m cannot be isomorphic. Therefore, for m 4, there can be no automorphism of ∆ m that swaps the edge colours.