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**1 - 2**of**2**### 1On List-decodability of Random Rank Metric Codes

"... In the present paper, we consider list decoding for both random rank metric codes and random linear rank metric codes. Firstly, we show that, for arbitrary 0 < R < 1 and > 0 ( and R are independent), if 0 < n m ≤ , then with high probability a random rank metric code in Fm×nq of rate R ..."

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In the present paper, we consider list decoding for both random rank metric codes and random linear rank metric codes. Firstly, we show that, for arbitrary 0 < R < 1 and > 0 ( and R are independent), if 0 < n m ≤ , then with high probability a random rank metric code in Fm×nq of rate R can be list-decoded up to a fraction (1−R−) of rank errors with constant list size L satisfying L ≤ O(1/). Moreover, if n m ≥ ΘR(), any rank metric code in Fm×nq with rate R and decoding radius ρ = 1−R − can not be list decoded in poly(n) time. Secondly, we show that if n m tends to a constant b ≤ 1, then every Fq-linear rank metric code in Fm×nq with rate R and list decoding radius ρ satisfies the Gilbert-Varsharmov bound, i.e., R ≤ (1 − ρ)(1 − bρ). Furthermore, for arbitrary > 0 and any 0 < ρ < 1, with high probability a random Fq-linear rank metric codes with rate R = (1−ρ)(1−bρ) − can be list decoded up to a fraction ρ of rank errors with constant list size L satisfying L ≤ O(exp(1/)).