Over fields of characteristic unequal to $2$, we can identify symmetric matrices with homogeneous polynomials of degree $2$. This allows us to view symmetric rank-metric codes as living inside the space of such polynomials. In this paper, we generalize the construction of symmetric Gabidulin codes to polynomials of degree $d>2$ over field of characteristic $0$ or $>d$. To do so, we equip the space of homogeneous polynomials of degree $d\geq 2$ with the metric induced by the essential rank, which is the minimal number of linear forms needed to express a polynomial. We provide bounds on the minimal distance and dimension of the essential-rank metric codes we construct and provide an efficient decoding algorithm. Finally, we show how essential-rank metric codes can be seen as special instances of rank-metric codes and compare our construction to known rank-metric codes with the same parameters.

翻译：在特征不等于$2$的域上，我们可以将对称矩阵与二次齐次多项式等同。这使得对称秩度量码可视为生活在此类多项式空间中。本文我们将对称Gabidulin码的构造推广至特征为$0$或大于$d$的域上的$d>2$次多项式。为此，我们赋予$d\geq 2$次齐次多项式空间以本质秩诱导的度量，本质秩是表达一个多项式所需的最少线性形式个数。我们为所构造的本质秩度量码提供了最小距离与维数的界，并给出了一种高效译码算法。最后，我们展示了本质秩度量码如何被视为秩度量码的特例，并将我们的构造与具有相同参数的已知秩度量码进行了比较。