Dickson Matrix Based Construction of Linear Maximum Rank Distance Codes

Maximum rank distance (MRD for short) codes lately attract more attention due to their various applications in storage systems, network coding, cryptography and space time coding. Similar to Reed-Solomon codes in classical coding theory, Gabidulin codes are the most prominent family of MRD codes. Due to their poor performance in list decoding or in constructing McEliece-type cryptosystems, the focus moves from Gabidulin codes to other non-Gabidulin codes. A natural following challenge is then to see if we can construct an infinite family of MRD codes that are not equivalent to Gabidulin codes. In this paper, we utilize Dickson matrices to construct an infinite family of Fq-linear MRD codes. Our codes are characterized by each of their codewords corresponding to a linearized polynomial with leading coefficient determined by one of any other coefficients. The family of codes corresponding to the set of linearized polynomials with leading coefficients dependent on the linear terms provides an extension to both Twisted Gabidulin codes and generalized Twisted Gabidulin codes for dimensions 1 and n-1. Lastly, we also provide some analysis on the equivalence between our proposed codes with some known families of MRD codes.