Definition 1: A Hadamard matrix is an $n$ × $n$$n\times n$ matrix $H$ whose entries are either $1$ or $-1$ and whose rows are mutually orthogonal.
Definition 2: A matrix $A$ is half-skew-centrosymmetric if there exist two square matrices $B$ and $C$ of order $n$ such that
\begin{equation} A = \begin{bmatrix} B & R C R \\ C & -R B R \end{bmatrix}. \end{equation}
where $R$ is the reverse identity matrix.
One day I found that these two definitions can be considered simultaneously. For example, \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} is the simplest half-skew-centrosymmetric Hadamard matrix.
Here comes my questions:
First, can you give some references about half-skew-centrosymmetric Hadamard matrices? Are there other names for these matrices?
Second one is about construction methods. It is easy to figure out how to construct a half-skew-centrosymmetric Hadamard matrix of order $2^k*n$$2^k\cdot n$ based on a Hadamard matrix of order $n$ by using a variant of Sylvester's construction. Can you propose more methods to construct half-skew-centrosymmetric Hadamard matrices?
Third, can you prove the following conjecture or give a counter-example?
Conjecture: A half-skew-centrosymmetric Hadamard matrix exists for $n=2$ or $n$ is a multiple of $4$.
