# Antonia Wachter-Zeh – Bounds on list decoding of rank-metric codes

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So far, there is no polynomial-time list decoding algorithm for Gabidulin codes. These codes can be seen as the rank-metric equivalent of Reed–Solomon codes. In this talk, we provide bounds on the list size of rank-metric codes in order to understand whether polynomial-time list decoding is possible.
Three bounds on the list size are proven. The first one is a lower exponential bound for Gabidulin codes and shows that for these codes no polynomial-time list decoding beyond the Johnson radius exists. Second, an exponential upper bound is derived, which holds for any rank-metric code of length $n$ and minimum rank distance $d$. The third bound proves that there exists a rank-metric code over $\Fqm$ of length $n \leq m$ such that the list size is exponential in the length for any radius greater than half the minimum rank distance. This implies that there cannot exist a polynomial upper bound depending only on $n$ and $d$ similar to the Johnson bound in Hamming metric. All three rank-metric bounds reveal significant differences to bounds for codes in Hamming metric.