Construction of a Class of Real Array Rank Distance Codes

Abstract

Rank distance codes are known to be applicable in various applications such as distributed data storage, cryptography, space time coding, and mainly in network coding. Rank distance codes defined over finite fields have attracted considerable attention in recent years. However, in some scenarios where codes over finite fields are not sufficient, it is demonstrated that codes defined over the real number field are preferred. In this paper, we proposed a new class of rank distance codes over the real number field $\mathbb{R}$ . The real array rank distance (RARD) codes we constructed here can be used for all the applications mentioned above whenever the code alphabet is the real field $\mathbb{R}$ . From the class of RARD codes, we extract a subclass of equidistant constant rank codes which is applicable in network coding. Also, we determined an upper bound for the dimension of RARD codes leading the way to obtain some optimal RARD codes. Moreover, we established examples of some RARD codes and optimal RARD codes.