The skew Hadamard matrix of order 292 can be constructed from skew supplementary difference sets. This same construction is used, for example, in Djokovic - Skew-Hadamard Matrices of Orders 436,580, and 988 Exist.

An implementation of this can be found in SageMath.

Let $H=\{1,2,4,8,16,32,64,55,37\}$ be a 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.

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.

An implementation of this can now be found in SageMath.

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.