Hadamard Matrices Related to Projective Planes

Let n be the order of a quaternary Hadamard matrix. It is shown that the existence of a projective plane of order n is equivalent to the existence of a balancedly multi-splittable quaternary Hadamard matrix of order n 2


Introduction
K. A. Bush [2] was the first to establish a link between projective planes of even order and specific Hadamard matrices, that was later labeled as Bush-type in 1971. H. J. Ryser [11] found the same connection as an application of factors of design matrix in 1977. Eric Verheiden [12] provided a direct construction for the matrices using the incidence matrices of the corresponding projective planes.
Frans C. Bussemaker, Willem Haemers and Ted Spence [3] used an exhaustive search and found no strongly regular graph with parameters (36,15,6,6) and chromatic number six or, equivalently, there is no symmetric Bush-type Hadamard matrix of order 36. Many Bush-type Hadamard matrices of order 100 are constructed, but none is known to be symmetric. The proof of the nonexistence of a symmetric Bush-type Hadamard matrix of order 100 would be exciting and is an alternative to the proof of the nonexistence of projective plane of order 10, however, there has been no attempt at showing it so far. The nonexistence of the projective plane of order 10 was finally established by a long computational method by C. W. H. Lam et al. in [9,10], and an alternate approach is still highly desirable.
The connection between projective planes and Hadamard matrices shown in [2,11,12] are all one-sided results in which from a projective plane of even order symmetric Bushtype Hadamard matrices are constructed.
Balancedly splittable Hadamard matrices were introduced by the authors in 2018 in [8], and the results were widely expanded in a recent paper by Jonathan Jedwab et al. in [5]. It is known [7] that the existence of a Hadamard matrix of order 4n would lead to a balancedly splittable Hadamard matrix of order 64n 2 . However, there is no balancedly splittable Hadamard matrix of order 4n 2 , n odd, see [8]. The case of Hadamard matrices of order 16n 2 , n > 1 odd, remains open, and no balancedly splittable Hadamard matrix of order 144 is known.
Concentrating on the order 144, the authors were led to some exotic classes of balancedly splittable Hadamard matrices, which is dubbed as balancedly multi-splittable Hadamard matrices. There is a balancedly multi-splittable Hadamard matrix of order 4 m for every positive integer m, and it seems that these are probably the only Hadamard matrices with this property.
It will be shown in this paper that the existence of a projective plane of order 4n is equivalent to the existence of a balancedly multi-splittable Hadamard matrix of order 16n 2 provided that 4n is the order of a Hadamard matrix. In doing so we use the fact that the existence of projective planes are equivalent to the existence of orthogonal arrays, see [1,Theorems 3.18 and 3.20], and the latter is equivalent to the balancedly multi-splittable Hadamard matrices.
There is also a similar equivalence between the projective plane of order 2n, n odd, and balancedly multi-splittable quaternary Hadamard matrices will be presented too.
The establishment of the nonexistence of a balancedly multi-splittable (quaternary) Hadamard matrix of order 144 (100) would be significant.

Codes
Let n, q be positive integers n, q 2, and let Q = {0, 1, . . . , q − 1}. A subset C of Q n is called to be a q-ary code of length n. For x, y ∈ Q n with x = x 1 x 2 · · · x n and y = y 1 y 2 · · · y n , the Hamming distance between codewords x and y is given by dist(x, y) = |{i : x i = y i }|. A code C is said to be an equi-distance code or a 1-distance set if the Hamming distance d(x, y) does not depend on x, y ∈ C with x = y.

Hadamard matrices
An n × n matrix H is a Hadamard matrix of order n if its entries are 1, −1 and it satisfies HH = I n , where I n denotes the identity matrix of order n. A Hadamard matrix H of order n is said to be balancedly splittable if there is an × n submatrix H 1 of H such that inner products for any two distinct column vectors of H 1 take at most two values.
More precisely, there exist integers a, b and the adjacency matrix A of a graph such that H 1 H 1 = I n + aA + b(J n − A − I n ), where J n denotes the all-ones matrix of order n. We say the quadruple (v, , a, b) the parameter. In this case we say that H is balancedly splittable with respect H 1 . Only the special case of (v, , a, b) = (4n 2 , 2n 2 , n, −n) will be used in this note.
The same concept can be extended to orthogonal designs [7]. Here, we adopt the following definition for quaternary Hadamard matrices. An n×n matrix H is a quaternary Hadamard matrix of order n if its entries are ±1, ±i and it satisfies HH * = nI n . A quaternary Hadamard matrix H of order n is said to be balancedly splittable if there is an × n submatrix H 1 of H such that the off-diagonal entries of H * 1 H 1 are in the set where α, β are some complex numbers. In this paper, we restrict to the case α = β and we say that a quaternary Hadamard matrix H of order n is balancedly splittable if H * 1 H 1 = I + αS where α is some positive real number and S is a (0, ±1, ±i)-matrix with zero diagonal entries and nonzero off-diagonal entries.

Orthogonal arrays
An orthogonal array of strength t and index λ is an N × k matrix over the set {1, . . . , q} such that in every N × t subarray, each t-tuple in {1, . . . , q} t appears λ times. We denote this property as OA λ (N, k, q, t). Note that N = λq t and (N, k, q, t) is the parameter of the orthogonal array. For t = 2e, the following lower bound on N was shown by Rao (see i . An orthogonal array with parameters (N, k, q, 2e) is said to be complete if the equality holds in above.
When t = 2 and λ = 1, the complete orthogonal array has the parameters OA 1 (q 2 , q + 1, q, 2), and it is known that its existence is equivalent to that of a projective plane of order q. For the orthogonal version of a projective plane is used in the next section.
The following lemmas will be used later.
Let D be the distance matrix, ie., D is an N × N matrix whose rows and columns indexed by the rows of A with (i, j)entry defined by the Hamming distance between the i-th row and the j-th row of A. Then Lemma 2. Assume that there exists an orthogonal array A with parameters (q 2 , q+1, q, 2). Proof. The proof for (i) and (ii) are exactly the same as [6, Lemma 2.5].
The assumed orthogonal array is a 2-design and 1-distance set with Hamming distance q in the Hamming association scheme. The case (iii) follows from the fact that C is a 1-distance set with Hamming distance q. holds. Equality holds if and only if the matrix whose rows consists of the codewords of C is an orthogonal array OA 1 (q 2 , q + 1, q, 2).

Balancedly multi-splittable Hadamard matrices
We consider the following property of a Hadamard matrix. Let H be a Hadamard matrix of order 4n 2 . Assume that H is normalized so that the first column of H is the all-ones vector. A Hadamard matrix H is said to be balancedly multi-splittable if there is a block The main results of this paper are as follows: Theorem 4. Let n be a positive integer. The following are equivalent.
Theorem 5. Let n be a positive integer. The following are equivalent.

Proof of Theorem 4
The proof of (ii) ⇒ (i where r i is a 1 × (4n − 1) matrix for any i.
(ii) For any distinct i, j, r i r j = −1.
(ii)D is a Hadamard matrix of order 16n 2 .
Let A be a submatrix of A obtained by restricting the columns to a 2n element set.
. . , 4n}) are disjoint 16n 2 × 2n (0, 1)-matrices. Lemma 8. There exists a symmetric (0, 1)-matrix B with diagonal entries 0 such that Proof. The rows of the matrix A is a 1-distance set with Hamming distance 4n and A is obtained from A by restricting some 2n coordinates. Therefore by Lemma 1(iii), the Hamming distances between the rows of A are 2n or 2n − 1. Thus, the distance matrix of the code of rows of A is 2nB + (2n − 1)(J 16n 2 − I 16n 2 − B) for some symmetric matrix (0, 1) B with zero diagonals. Since This with (i) shows (ii).
The proof of (i) ⇒ (ii). Assume that H is a balancedly multi-splittable Hadamard matrix of order 16n 2 with respect to the following block form: Proof. We show the case i = 1. Since H is a Hadamard matrix of order 16n 2 , HH = 16n 2 I 16n 2 , that is, By the assumption of balanced multi-splittability, we have that all inner products of distinct rows in both H 2 · · · H 2n+1 and H 2n+2 · · · H 4n+1 are ±2n. Thus, where S and S are (0, 1, −1)-matrices with diagonal entries 0 and off-diagonal entries ±1. Then Since both S and S are (0, ±1)-matrix, by inspecting the equation involving H 1 H 1 it can be seen that S + S is a (0, ±2)-matrix with diagonal entries 0. However, the off-diagonal entries of H 1 H 1 cannot be −4n − 1, S + S is (0, −2)-matrix. Therefore, H 1 H 1 is a (4n − 1, −1)-matrix.
For each i, consider the matrixH i = 1 H i . Then, by Lemma 9,H iH i is a (4n, 0)matrix. Thus some of rows ofH i coincide. SinceH iH i = 16n 2 I 4n , the rank ofH i is 4n. Therefore there exist exactly 4n distinct rows ofH i that correspond to the rows of a Hadamard matrix, sayK i , of order 4n. WriteK i = 1 K i and fix i. Some rows of H i also coincide and any row of H i coincides with some row of K i . In the matrix 1 H 1 · · · H 4n+1 , we then assign a symbol j to any row in H i , which equals the j-th row of K i . Let A be the resulting 16n 2 × (4n + 1) matrix over the symbol set {1, . . . , 4n}.
Lemma 10. The code C with codewords consisting of the rows of A is an equidistance code with the number of codewords 16n 2 , equidistance 4n, of length 4n + 1.
Proof. It is enough to see the case for the first row and second row. Let the first and second rows of H be the following forms: 1 r 1,1 · · · r 1,4n+1 , 1 r 2,1 · · · r 2,4n+1 .
Consider the inner product between them: By Lemma 9, r 1,i r 2,i ∈ {4n − 1, −1} for any i. Then there exists i 0 such that r 1,i 0 r 2,i 0 = 4n − 1 and r 1,i r 2,i = −1 for any i = i 0 . Therefore the distance between the first row and second row is 4n.

Proof of Theorem 5
The proof of (ii) ⇒ (i). Assume that there exists a quaternary Hadamard matrix H of order 2n. Write H as where r i is a 1 × (2n − 1) matrix for any i.
(ii)D is a quaternary Hadamard matrix of order 4n 2 .
Let A be a submatrix of A obtained by restricting the columns to an n element set.
. . , 2n}) are disjoint 4n 2 × n (0, 1)-matrices. Lemma 13. There exists a symmetric (0, 1)-matrix B with diagonal entries 0 such that Proof. The rows of the matrix A is a 1-distance set with Hamming distance 2n and A is obtained from A by restricting some n coordinates. Therefore by Lemma 1(iii), the Hamming distances between the rows of A are n or n − 1. Thus, the distance matrix of the code of rows of A is nB + (n − 1)(J 4n 2 − I 4n 2 − B) for some symmetric matrix (0, 1) B with zero diagonals. Since This with (i) shows (ii).
Then, by Lemma 13, Therefore the quaternary Hadamard matrix D is balancedly multi-splittable.
The proof of (i) ⇒ (ii). Assume that H is a balancedly multi-splittable quaternary Hadamard matrix of order 4n 2 with respect to the following block form: We show the case i = 1. Since H is a quaternary Hadamard matrix of order 4n 2 , HH * = 4n 2 I 4n 2 , that is, By the assumption of balanced multi-splittability, we have that the inner product of distinct rows of matrices H 2 · · · H n+1 or H n+2 · · · H 2n+1 are ±2n, ±2i. Thus, where S and S are (0, ±1, ±i)-matrix with diagonal entries 0 and off-diagonal entries ±1, ±i. Then Since both S and S are (0, ±1, ±i)-matrix, by inspecting the equation involving H 1 H * 1 it can be seen that S + S is a (0, ±2, ±2i)-matrix with diagonal entries 0. However, the absolute values of off-diagonal entries of H 1 H * 1 cannot exceed 2n − 1, S + S is (0, −2)matrix. Therefore, H 1 H * 1 is a (2n − 1, −1)-matrix. For each i, consider the matrixH i = 1 H i . Then, by Lemma 14,H iH * i is a (2n, 0)matrix. Thus some of rows ofH i coincide. SinceH * iH i = 4n 2 I 2n , the rank ofH i is 2n. Therefore there exist exactly 2n distinct rows ofH i that correspond to the rows of a Hadamard matrix, sayK i , of order 2n. WriteK i = 1 K i and fix i. Some rows of H i also coincide and any row of H i coincides with some row of K i . In the matrix H 1 · · · H 2n+1 , we then assign a symbol j to any row in H i , which equals the j-th row of K i . Let A be the resulting 4n 2 × (2n + 1) matrix over the symbol set {1, . . . , 2n}.
Lemma 15. The code C with codewords consisting of the rows of A is an equidistance code with the number of codewords 4n 2 , equidistance 2n, of length 2n + 1.
Proof. It is enough to see the case for the first row and second row. Let the first and second rows of H be the following forms: 1 r 1,1 · · · r 1,2n+1 , 1 r 2,1 · · · r 2,2n+1 .
Consider the inner product between them: By Lemma 14, r 1,i r * 2,i ∈ {2n − 1, −1} for any i. Then there exists i 0 such that r 1,i 0 r * 2,i 0 = 2n − 1 and r 1,i r * 2,i = −1 for any i = i 0 . Therefore the distance between the first row and second row is 2n.

Example
In this section, we present an example of balancedly multi-splittable Hadamard matrices following the construction in Theorem 4.
Example 16. Take an OA 1 (16, 5, 4, 2) A and a Hadamard matrix H of order 4 as: Then the matrix D constructed in Theorem 4 is a balancedly multi-splittable Hadamard the electronic journal of combinatorics 30(2) (2023), #P2.49 matrix of order 16: Conversely, we demonstrate how an orthogonal array and a Hadamard matrix can be constructed from a balancedly multi-splittable Hadamard matrix D. for i ∈ {2, 3, 4, 5}. In the matrix H 1 · · · H 5 , we assign a symbol j ∈ {1, 2, 3, 4} to any row in H i , which equals the j-th row of K i . The resulting matrix A is reconstructed as aforementioned.