Higher-degree symmetric rank-metric codes
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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≥ 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.
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