Construction of symmetric Hadamard matrices of order $4v$ for $v=47,73,113$

We continue our systematic search for symmetric Hadamard matrices based on the so called propus construction. In a previous paper this search covered the orders $4v$ with odd $v\le41$. In this paper we cover the cases $v=43,45,47,49,51$. The odd integers $v<120$ for which no symmetric Hadamard matrices of order $4v$ are known are the following: $$47,59,65,67,73,81,89,93,101,103,107,109,113,119.$$ By using the propus construction, we found several symmetric Hadamard matrices of order $4v$ for $v=47,73,113$.


Introduction
In this paper we continue the systematic investigation, begun in [1], of the propus construction of symmetric Hadamard matrices.
Let us recall that a Hadamard matrix is a {1, −1}-matrix H of order m whose rows are mutually orthogonal, i.e. HH T = mI m , where I m is the identity matrix of order m. We say that H is skew-Hadamard matrix if also H + H T = 2I m . The famous Hadamard conjecture asserts that Hadamard matrices exist for all orders m which are multiples of 4. (They also exist for m = 1, 2.) Similar conjectures have been proposed for symmetric Hadamard matrices and skew-Hadamard matrices, see e.g. [2, V.1.4]. The smallest orders 4v for which such matrices have not been constructed are 668 for Hadamard matrices, 276 for skew-Hadamard matrices, and 188 for symmetric Hadamard matrices. Let us also mention that symmetric Hadamard matrices of orders 116, 156, 172 have been constructed only very recently, see [3,1].
Since the size of a Hadamard matrix or a skew or symmetric Hadamard matrix can always be doubled, while preserving its type, we are interested mostly in the case where these matrices have order 4v with v odd.
The propus construction is based on the so called Propus array (1) In this paper, except in section 4, the matrices C i will be circulants of order v and the matrix R will be the back-circulant identity matrix of order v, The matrix H will be a Hadamard matrix if (Superscript T denotes transposition of matrices.) If also C T 1 = C 1 and C 2 = C 3 then H will be a symmetric Hadamard matrix.
To construct the circulants C i satisfying the above conditions we use the cyclic propus difference families (A 1 , A 2 , A 3 , A 4 ) with parameters (v; k 1 , k 2 , k 3 , k 4 ; λ) such that A 2 = A 3 and at least one of the base blocks A 1 , A 4 is symmetric. The parameters must satisfy the three equations We refer to such parameter sets as the propus parameter sets.
For the definitions of the terms that we use here and the facts we mention below, we refer the reader to [1]. Without any loss of generality, we impose the following additional restrictions: For convenience we say that the propus parameter sets satisfying these additional conditions are normalized. For a given odd v there exist at least one normalized propus parameter set, see [1, Theorem 1]. However, there exist even v for which this is not true, see [1, Theorem 2].
It is conjectured in [1] that for each odd v there exists at least one propus difference family in the cyclic group Z v of integers modulo v. But this may fail if we specify not only v (odd) but also the parameters k 1 , k 2 = k 3 , k 4 . Our computations suggest that these exceptional propus parameter sets must have all k i equal to each other. For instance, there is no cyclic propus difference family having the parameters (25; 10, 10, 10, 10; 15). (This is also true for the propus difference families over the elementary abelian group Z 5 × Z 5 .) One of the authors developed a computer program to search for propus difference families. For the description of the algorithm used in the program we refer the reader to [1]. We used that program on PCs to construct many such families for odd (or even) v. The first version of the program was used in the range v < 43. The second, improved version, was capable of finding solutions for v ≤ 51. Some of the timings for these computations are given in section 5.
In section 2 we give several examples of symmetric Hadamard matrices of new orders 188, 292, and 452.
In section 3 we list the normalized propus parameter sets for odd v ∈ {43, 45, . . . , 59} and for each of them we indicate whether propus families with that parameter set exist and, if they do, which of the blocks A or D can be chosen to be symmetric. This list together with a similar list in [1] shows that there is a rich supply of propus type symmetric Hadamard matrices for orders 4v with odd v < 50. Sporadic examples are also known for v = 53, 55, 57. The first undecided case is v = 59.
In section 4 we focus on the case where v = s 2 is an odd square. We count the number of propus parameter sets (v; x, y, y, z; λ) with v = s 2 by dropping the normalization condition x ≥ z. This number, N s , is also the number of positive odd integer solutions of a simple quadratic Diophantine equation, namely (9). When s is an odd prime then we conjecture that N s − s − 1 ∈ {+1, −1}. We refer to the cases where v is odd and all k i are equal as exceptional cases. They occur only when v = s 2 . We also conjecture that every prime s ≡ 1 (mod 4) can be written uniquely as s = (a 2 + b 2 )/(a − b) where a and b are positive integers and 1 < a ≤ (s − 1)/2. Moreover, the denominator a − b is either a square or 2 times a square.
Finally in section 5, for each of the normalized propus parameter sets with odd v = 43, 45, . . . , 51, but excluding the exceptional parameter set (49; 21, 21, 21, 21; 35), we list one or two examples of propus difference families. with v = 47. In each case we constructed many such matrices, but here we record just two examples for each parameter set. In all four cases, A is symmetric in the first and D symmetric in the second example. As B = C we omit the block C. The examples are separated by semicolons. The circulant matrices C 1 , C 2 = C 3 , C 4 whose first rows are the sequences a, b = c, d are depicted on Figure 1. Our convention is that a white square represents +1 and the black square −1.

Symmetric Hadamard matrices of new orders
Note that C 1 is symmetric. By plugging these circulants into the propus array (1) we obtain the symmetric Hadamard matrix shown in Figure 2.
Next we give the symmetric Hadamard matrices of order 4v where v = 73, 113. No symmetric Hadamard matrices of these orders were known previously.
We For v = 73 we use the parameter set (73; 36, 36, 36, 28; 63). The base blocks of the two propus difference families are: The first three families share the same block A, and the second and third family differ only in block D. In spite of that, the four families are pairwise nonequivalent.

Normalized propus parameter sets
We list here all normalized propus parameter sets (v; x, y, y, z; λ) for odd v = 43, 45, . . . , 59. The cyclic propus families consisting of four base blocks A, B, C, D ⊆ Z v having sizes x, y, y, z, respectively, and such that B = C and A or D is symmetric give symmetric Hadamard matrices of order 4v. (If only D is symmetric we have to switch A and D before plugging the blocks into the propus array.) If x = z = y then the parameter set (v; y, x, x, y; λ) is also normalized and is included in our list. In the former case the two base blocks of size y have to be equal, while in the latter case the base blocks of size x have to be equal.
The four base blocks, subsets of Z v , are denoted by A, B, C, D. We require all propus difference families to have B = C. If we know such a family exists with symmetric block A, we indicate this by writing the symbol A after the parameter set, and similarly for the symbol D. If we know that there exists a propus family with both A and D symmetric, then we write the symbol AD. Finally, the question mark means that the existence of a cyclic propus difference familiy remains undecided.
The symbol T indicates that the parameter set belongs to the Turyn series of Williamson matrices. Since in that case all four base blocks are symmetric, the symbol T implies AD. Further, the symbol X indicates that the parameter set belongs to another infinite series (see [3,Theorem 5]) which is based on the paper [5] of Xia, Xia, Seberry, and Wu. In our list below the symbol X implies D. More precisely, for a difference family A, B, C, D in the X-series two blocks are equal, say B = C, and one of the remaining blocks is skew, block A in our list, and the last one is symmetric, block D.

Exceptional series of propus parameter sets
We say that a propus parameter set (v; k 1 , k 2 , k 3 , The exceptional parameter sets are parametrized by just one integer s > 1 and are given by the formula There exists a cyclic propus difference family with parameter set Π 3 . There exists also a propus difference family (A, B, C, D) over the group Z 3 × Z 3 with the same parameter set and such that A is symmetric and B = C = D. By using the finite field Z 3 [α] where α 2 = −1, we can take For s = 5, it is reported in [1] that there are no propus difference families in Z 25 having Π 5 as its parameter set. We performed another exhaustive search and found no such families in For s = 7, our non-exhaustive searches found no cyclic propus difference families having the parameter set Π 7 . However, we found a cyclic difference family with parameter set Π 7 and B = C with neither A nor D symmetric: 2,3,4,8,11,12,14,19,21,24,26,27,29,37,38,41,44,45,46], While computing the propus parameter sets (v; x, y, y, z; λ) in the case when v = s 2 is an odd square, we observed an interesting feature. Namely, if in the definition of normalized propus difference sets we drop only the condition that x ≥ z and if s is an odd prime then the number, N s , of such parmeter sets is either s or s + 2. It follows from the proof of [1, Theorem 1] that N s is equal to the number of odd positive integer solutions of the Diophantine equation After making additional computations, we decided to propose the following conjecture.
We have verified our conjecture for all odd primes less than 10000. There are 1228 such primes. For 606 of them we have N s = s and for the remaining 622 we have N s = s + 2. Thus the sequence N s − s − 1 is a {+1, −1}-sequence when s runs through odd primes < 10000. We have sketched the partial sums of this sequence on Figure 3.
If s is a prime congruent to 1 (mod 4), we observed that apart from Π s there is another normalized propus parameter set with v = s 2 and k 2 = k 3 = s 2 . Let us denote this new parameter set by The integers α and β are positive and satisfy the quadratic Diophantine equation We propose another conjecture.
Conjecture 2 For any odd prime s ≡ 1 (mod 4) the Diophantine equation (11), in the unknowns α and β, has a unique solution (a, b), where a and b are positive integers and 1 < a ≤ (s − 1)/2. Moreover, a − b is either a square or 2 times a square.
We have verified that this conjecture holds for s < 100000. If we drop the condition 1 < a ≤ (s − 1)/2, than there exists one more solution, namely (s − a, b). Note also that the two solutions share the same b, and so the integer b is uniquely determined by s.

Appendix
The cyclic propus difference families listed below, except some of the families that belong to one of the two infinite series T and X, have been constructed by using a computer program written by one of the authors. The program was run on two PCs, each with a single 64-bit processor. For v = 39 it takes about 5 minutes to obtain a solution, about 20 minutes for v = 41, about 1 hour for v = 43, about 3 or 4 hours for v = 45, about 12 hours for v = 47, about 2 days for v = 49, and 5 days for v = 51. In all families below the base block B = C, and to save space we omit the block C. The families are terminated by semicolons.