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arXiv · 2303.06745

Higher-degree symmetric rank-metric codes

Arthur Bik·Alessandro Neri
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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 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.

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Arthur Bik, Alessandro Neri. 2023-03-12. Higher-degree symmetric rank-metric codes. https://arxiv.org/abs/2303.06745

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