Rank metric codes were study by E. Gabidulin in 1985 after a brief introduction by Delaste in 1978 as an alternative to Reed-Solomon codes based on linear polynomials. They have found applications in many area including linear network coding and space-time coding. They are also used in cryptography to reduce the size of the keys compared to Hamming metric codes at the same level of security. Despite this prowess, these codes suffer from structural attacks due to the strong algebraic structure from which they are defined. It therefore becomes interesting to find new families in order to address these questions. This explains their elimination from the NIST post-quantum cryptography competition. \par In this paper we provide a generalisation of subspace subcodes in rank metric introduced by Gabidulin and Loidreau. we also characterize this family by giving an algorithm which allows to have its generator and parity-check matrices based on the associated extended codes. We also have bounded the cardinal of these codes both in the general case and in the case of Gabidulin codes. We have also studied the specific case of Gabidulin codes whose the underlined Gabidulin decoding algorithms are known.

翻译：秩度量码由E. Gabidulin在1985年研究提出，此前Delaste于1978年将其作为基于线性多项式的里德-所罗门码的替代方案进行了简要介绍。这类码在包括线性网络编码和空时编码在内的多个领域得到应用，同时在密码学中用于在相同安全等级下缩减密钥尺寸（相较于汉明度量码）。然而尽管具有这些优势，由于定义时所依赖的强代数结构，此类码易遭受结构性攻击。因此寻找新型码族以应对这些问题变得颇具研究价值，这也解释了它们为何在NIST后量子密码竞赛中被淘汰。\par 本文提出了Gabidulin与Loidreau引入的秩度量子空间子码的推广形式。我们通过给出一种基于关联扩展码生成其生成矩阵与校验矩阵的算法来刻画该族码的特性，同时在一般情形与Gabidulin码情形下界定了此类码的基数。我们还专门研究了其底层Gabidulin解码算法已知的Gabidulin码的特例。