Mutually Unbiased Bush-type Hadamard Matrices and Association Schemes

It was shown by LeCompte, Martin, and Oweans in 2010 that the existence of mutually unbiased Hadamard matrices and the identity matrix, which coincide with mutually unbiased bases, is equivalent to that of a $Q$-polynomial association scheme of class four which is both $Q$-antipodal and $Q$-bipartite. We prove that the existence of a set of mutually unbiased Bush-type Hadamard matrices is equivalent to that of an association scheme of class five. As an application of this equivalence, we obtain the upper bound of the number of mutually unbiased Bush-type Hadamard matrices of order $4n^2$ to be $2n-1$. This is in contrast to the fact that the upper bound of mutually unbiased Hadamard matrices of order $4n^2$ is $2n^2$. We also discuss a relation of our scheme to some fusion schemes which are $Q$-antipodal and $Q$-bipartite $Q$-polynomial of class $4$.


INTRODUCTION
A Hadamard matrix is a matrix H of order n with entries in (−1, 1) and orthogonal rows in the usual inner product on R n . Two Hadamard matrices H and K of order n are called unbiased if HK t = √ nL for some Hadamard matrix L, where K t denotes the transpose of K. In this case, it follows that n must be a perfect square. A Hadamard matrix of order n for which the row sums and column sums are all the same, necessarily √ n, is called regular, see [11].

Definition 1.1. A Bush-type Hadamard matrix is a block matrix H = [H i j ]
of order 4n 2 with block size 2n, H ii = J 2n and H i j J 2n = J 2n H i j = 0, i = j, 1 ≤ i ≤ 2n, 1 ≤ j ≤ 2n where J 2n is the 2n by 2n matrix of all 1 entries.
It was shown by LeCompte, Martin, and Oweans in [9] that the existence of mutually unbiased Hadamard matrices and the identity matrix, which coincide with mutually unbiased bases, is equivalent to that of a Q-polynomial association scheme of class four which is both Q-antipodal and Q-bipartite.
Our aim in this paper is to show that the existence of unbiased Bush-type Hadamard matrices is equivalent to the existence of a certain association scheme of class five. As an application of this equivalence, we obtain the upper bound of the number of mutually unbiased Bush-type Hadamard matrices of order 4n 2 is 2n − 1, whereas the upper bound of mutually unbiased Hadamard matrices of order 4n 2 is 2n 2 [8,Theorem 2]. Also we discuss a relation of our scheme to such association schemes of class four.

ASSOCIATION SCHEMES
A symmetric d-class association scheme, see [1], with vertex set X of size n and d classes is a set of symmetric (0, 1)-matrices A 0 , . . ., A d , which are not equal to zero matrix, with rows and columns indexed by X , such that: (1) A 0 = I n , the identity matrix of order n.
Since disjoint (0, 1)-matrices A i 's form a basis of A, the algebra A is closed under the entrywise multiplication denoted by •. The Krein parameters . Each of the matrices A i 's can be considered as the adjacency matrix of some graph without multiedges. The scheme is imprimitive if, on viewing the A i 's as adjacency matrices of graphs G i on vertex set X , at least one of the G i 's, i = 0, is disconnected. Then there exists a set I of indices such that 0 and such i are elements of I and ∑ j∈I A j = I p ⊗ J q for some p, q with p < n. Thus the set of n vertices X are partitioned into p subsets called fibers, each of which has size q. The set I defines an equivalence relation on {0, 1, . . ., d} by j ∼ k if and only if p k i j = 0 for some i ∈ I . Let I 0 = I , I 1 , . . . , I t be the equivalent classes on {0, 1, . . ., d} by ∼. Then by [1,Theorem 9.4] and the matrices A j (0 ≤ j ≤ t) define an association scheme on the set of fibers. This is called the quotient association scheme with respect to I For fibers U and V , let I (U,V ) denote the set of indices of adjacency matrices that has an edge between U and V . We define a (0, where B 1 and B 2 are disjoint (0, 1)-matrices. By reworking a result of Mathon, see [3], we have the following: Proof. The intersection numbers are: We now impose a further structure on the regular Hadamard matrices H i s and assume that they are all of Bush-type. First we need the following.

Lemma 3.2. Let H and K be two unbiased Bush-type Hadamard matrices
Similarly, we have X L = 2nX . Thus L is a Bush-type Hadamard matrix.
This enables us to add two more classes and we have the following.
Proof. We work out the intersection numbers, using some of the relations in Lemma 3.1. Note that A 0 + A 1 , A 3 , (A 0 + A 1 + A 2 ) are block matrices of block size 2n, (4n 2 , respectively), where each block is either the zero or the all one's matrix. On the other hand A 4 and A 5 are block matrices of block size 2n, where the blocks are either the zero matrix or of row and column sum n. So, it is straightforward computation to see the following: Using these, the facts that A 3 + A 4 = B 1 , A 5 (A 3 + A 4 ) = B 2 B 1 , and the intersection numbers in Lemma 3.1 we have: Finally, noting that A 4 − A 5 is a block matrix of block size 2n, where the blocks are either the zero matrix or of row and column sum zero, it follows that The existence and a construction method for MUBH matrices to use mutually suitable Latin squares were given in [8,Theorem 13]. However, in order to obtain Bush-type Hadamard matrices as defined here, an additional assumption on the MSLS is needed as follows. The construction is exactly same as [8,Theorem 13]. The resulting mutually unbiased Hadamard matrices are all of Bush-type. Indeed, each Latin square has the entries 1 on diagonal, thus the resulting Hadamard matrix has the all ones matrices on diagonal blocks.
The equivalence of MOLS and MSLS was given in [8, Lemma 9]. The assumption on MOLS corresponding to MSLS with all ones entries on diagonal is that each Latin square has (1, 2, · · · , n) as the first row. The MOLS having this property is constructed by the use of finite fields as follows. For each α ∈ F q \ {0}, define L α as (i, j)-entry equal to αi + j, where i, j ∈ F q . By switching rows so that the first row corresponds to 0 ∈ F q and mapping F q to {1, 2, . . ., n} such that each first row becomes (1, 2, . . ., n), we obtain the desired MOLS. Thus we have the same conclusion as [8, Corollary 15].
(1) Rewriting A 4 A 4 = A 5 A 5 as: It is seen that, for m = 2n − 1, A 5 is the adjacency matrix of a strongly regular graph and A 4 is the adjacency matrix of a Deza Graph, see [5,6]. This is true for n = 2 k , for each integer k ≥ 1. (2) The association scheme of class 5 is uniform. Any two fibers define a coherent configuration, which is a strongly regular design of the second kind, see [7]. The first and second eigenmatrices and B * 5 are as follows: Thus the association scheme certainly satisfies [4,Proposition 4.7]. Since the Krein number q 1 1,2 = 2n−m−1 m+1 must be positive, we obtain m ≤ 2n − 1 holds. This means that the number of MUBH matrices of order 4n 2 is at most 2n − 1. The example attaining the upper bound is given in [8,Corollary 15]. (3) The first, second eigenmatrices and the Krein matrix B * 1 of the class 3 association scheme are as follows: This association scheme is a Q-antipodal Q-polynomial scheme of class 3. By [3, Theorem 5.8], this scheme comes from a linked systems of symmetric designs.
Next we show the converse implication as follows. Proof. Let A 0 , . . ., A 5 be the adjacency matrices of an association scheme with the same eigenmatrices in Remark 3.5.

8 CLASS ASSOCIATION SCHEMES
Linked systems of symmetric designs with specific parameters have the extended Q-bipartite double which yields an association scheme of mutually unbiased bases [10, Theorem 3.6]. Next we show an association scheme from our association schemes of class 5 has a double cover and show a relation to an association scheme of class 4 as a fusion scheme.
Proof. Follows from the calculation in Theorem 3.3.