In this letter, we confirm that, with overwhelming probability, random $\mathbb {F}_{{q}}$ -linear subcodes of Gabidulin codes can be list decoded with a decoding radius beyond half of the minimum distance and constant upper bound of the list size. Furthermore, our results reveal the existence of an $\mathbb {F}_{{q}}$ -linear subcode of a square Gabidulin code that is list decodable with radius approaching the Gilbert-Varshamov bound, which is shown to be the optimal list decoding radius. On the other hand, when we are considering a Gabidulin code with the ratio of the number of its rows and columns being $\epsilon $ , we can also find an $\mathbb {F}_{{q}}$ -linear subcode that is list decodable up to the Singleton bound. Moreover, we investigate the list decodability of $\mathbb {F}_{{q}^{{m}}}$ -linear subcodes of Gabidulin codes.

中文翻译：

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Gabidulin码的线性子码的列表可解码性

在这封信中，我们确认以极大的概率随机 $ \ mathbb {F} _ {{q}} $ 可以用超过最小距离的一半和列表大小恒定上限的解码半径对Gabidulin代码的-linear子代码进行列表解码。此外，我们的结果表明存在 $ \ mathbb {F} _ {{q}} $ -方形Gabidulin码的-linear子码，其列表可解码，半径接近吉尔伯特-瓦尔沙莫夫边界，被证明是最佳列表解码半径。另一方面，当我们考虑Gabidulin码时，其行和列数之比为 $ \ epsilon $ ，我们还可以找到一个 $ \ mathbb {F} _ {{q}} $ -线性子代码，可对列表进行解码，直到Singleton界限。此外，我们调查了清单的可解码性 $ \ mathbb {F} _ {{q} ^ {{m}}} $ -Gabidulin代码的线性子代码。