@MISC{Ding_1onlist-decodability, author = {Yang Ding}, title = {1On List-decodability of Random Rank Metric Codes}, year = {} }

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Abstract

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/)).