Rank Generalized Subsapce subcode

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.