Gorenstein matrices

Let $A=(a_{ij})$ be an integral matrix. We say that $A$ is $(0, 1, 2)$-matrix if $a_{ij}\in\{0,1,2\}$. There exists the Gorenstein $(0, 1, 2)$-matrix for any permutation $\sigma$ on the set $\{1,\dots,n\}$ without fixed elements. For every positive integer $n$ there exists the Gorenstein cyclic $(0, 1, 2)$-matrix $A_{n}$ such that $inx\,A_{n}=2$.
If a Latin square ${\mathcal L}_{n}$ with a first row and first column $(0,1,\ldots,n-1)$ is an exponent matrix, then $n=2^{m}$ and ${\mathcal L}_{n}$ is the Cayley table of a direct product of $m$ copies of the cyclic group of order 2. Conversely, the Cayley table ${{\mathcal E}}_{m}$ of the elementary abelian group $G_{m}=(2)\times\ldots\times(2)$ of order $2^{m}$ is a Latin square and a Gorenstein symmetric matrix with first row $(0,1,\ldots,2^{m}-1)$ and
$$ \sigma({{\mathcal E}}_{m})=\begin{pmatrix}1&2&3&\ldots &2^{m}-1&2^{m}\\ 2^{m}&2^{m}-1&2^{m}-2&\ldots & 2&1\end{pmatrix}. $$