Wachter-Zeh in [42], and later together with Raviv [31], proved that Gabidulin codes cannot be efficiently list decoded for any radius $\tau$, providing that $\tau$ is large enough. Also, they proved that there are infinitely many choices of the parameters for which Gabidulin codes cannot be efficiently list decoded at all. Subsequently, in [41] these results have been extended to the family of generalized Gabidulin codes and to further family of MRD-codes. In this paper, we provide bounds on the list size of rank-metric codes containing generalized Gabidulin codes in order to determine whether or not a polynomial-time list decoding algorithm exists. We detect several families of rank-metric codes containing a generalized Gabidulin code as subcode which cannot be efficiently list decoded for any radius large enough and families of rank-metric codes which cannot be efficiently list decoded. These results suggest that rank-metric codes which are $\mathbb{F}_{q^m}$-linear or that contains a (power of) generalized Gabidulin code cannot be efficiently list decoded for large values of the radius.
翻译:在 [42] 和后来与Raviv [31] 一起,证明加比杜林编码无法有效地为任何半径解码,只要$\tau$足够大。此外,它们证明对加比杜林编码无法有效解码的参数有无限的选择。随后,在[41] 中,这些结果扩大到了通用加比杜林编码的家属,以及MRD-code的大家庭。在本文中,我们提供了包含通用加比杜林编码的等分编码列表大小的界限,以确定是否存在多数值-时间解码算法。我们发现了若干包含通用加比杜林编码的等级编码组,作为无法有效解码的子编码组,任何范围足够大的半径和无法有效解码的等分码组都无法有效解码。这些结果表明,$\mathb{F ⁇ q ⁇ m} 美元线式编码的等分数代码组,或含有(有效)通用加比勒码的大值列表。