Gorenstein matrices

Abstract

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 \(\{1, \ldots , n\}\) 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

\(\sigma({{\mathcal E}}_{m})=\begin{pmatrix} 1&2&3&\ldots &2^{m}-1&2^{m}\\ 2^{m}&2^{m}-1&2^{m}-2&\ldots & 2&1\end{pmatrix}.\)