I am currently looking into how to construct a **skew-Hadamard matrix** of order 292. Where can I find such construction?


According to multiple papers (e.g. [Koukouvinos and Stylianou - On skew-Hadamard matrices](https://doi.org/10.1016/j.disc.2006.06.037) and [Seberry and Yamada - Amicable Hadamard matrices and amicable orthogonal designs](https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=d42da0293057242ecdd9ce5654c7e4d6fbf5a10e)) this matrix has been constructed. However, none of the papers I have found gives a reference to the actual construction that is being used to obtain it.


From what I have seen, it looks like the first paper which mentions its existence was published in 1978 ([Seberry - On skew Hadamard matrices](https://ro.uow.edu.au/infopapers/988)), with the first (*non-skew*) construction of an Hadamard matrix of order 292 being published in 1975 ([Spence - Skew-Hadamard Matrices of the Goethals–Seidel Type](https://doi.org/10.4153/CJM-1975-066-9)).




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The idea is to make constructions for various types of Hadamard matrices available in [SageMath](https://sagemath.org). We currently have quite a few already (and discovered gaps in literature along the way).



Construction of skew-Hadamard matrix of order 292
The skew Hadamard matrix of order 292 can be constructed from skew supplementary difference sets (kindly supplied to us by Prof. Djokovic). This same construction is used, for example, in [Djokovic - Skew-Hadamard Matrices of Orders 436,580, and 988 Exist](https://arxiv.org/abs/0706.1973).


An implementation of this can now be found in [SageMath](https://trac.sagemath.org/ticket/34848).


Note that $H=\{1,2,4,8,16,32,64,55,37\}$ is the order 9 subgroup of $Z\_{73}$, and let
$$
J\_1 = \{5,9,11,25\}\\
J\_2 = \{11,13,17,25\}\\
J\_3 = \{5,9,13,17\}\\
J\_4 = \{1,3,13\}\\
$$


Then, we can obtain supplementary difference sets $(X\_1,X\_2,X\_3,X\_4)$ with parameters $(73; 36,36,36,28; 63)$ and with block $X\_1$ skew as follows:
$$
X\_1 = \bigcup\_{x\in J\_1} xH\\
X\_2 = \bigcup\_{x\in J\_2} xH\\
X\_3 = \bigcup\_{x\in J\_3} xH\\
X\_4 = \{0\} \cup \bigcup\_{x\in J\_4} xH\\
$$


From each set $X\_i$, let $a\_i = (a\_{i,0}, a\_{i,1},...,a\_{i,72})$ be the {±1}-row vector such that $a\_{i,j} = −1$ *iff* $j \in X\_i$.


Then, we can construct four circulant matrices $A\_1, A\_2, A\_3, A\_4$ where $a\_i$ is the first row of $A\_i$. We have that $A\_1$ is skew, and by plugging them in the Goethals-Seidel array we obtain a skew Hadamard matrix of order **292**:
$$
H = \left(\begin{array}{rrrr}
 A\_1 & A\_2R & A\_3R & A\_4R \\
 -A\_2R & A\_1 & -A\_4^TR & A\_3^TR \\
 -A\_3R & A\_4^TR & A\_1 & -A\_2^TR \\
 -A\_4R & -A\_3^TR & A\_2^TR & A\_1
 \end{array}\right)
$$


Here $R$ denotes the matrix having ones on the back-diagonal and all other entries zero.


