Higher-Degree Symmetric Rank-Metric Codes
Arthur Bik, Alessandro Neri · SIAM Journal on Applied Algebra and Geometry · 2024
Abstract. 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 Delsarte–Gabidulin codes to polynomials of degree [Formula: see text] over fields of characteristic 0 or [Formula: see text]. To do so, we equip the space of homogeneous polynomials of degree [Formula: see text] 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 minimum 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.