I. Introduction
MDS codes with constrained generator matrices have been attracting much attention recently due to their applications in weakly secure cooperative data exchange [1]–[3], multiple access networks [4], [5], wireless sensor networks [6], and so on. The relations among them are well explained in [7], [8]. An interesting problem of this topic is to construct an MDS code with a sparse and balanced generator matrix (SBGM) , where ‘sparse’ means that each row of has the least possible number of nonzeros, i.e., nonzeros, and ‘balanced’ means that the number of nonzeros in any two columns differs by at most one, i.e., or . This problem was first considered in [6] motivated by the benefits from such a matrix during the encoding process [9], [10]. On the one hand, since the time required to compute each code symbol is a function of the number of nonzeros in a specified column of , each code symbol is computed in roughly the same amount of time due to the balanced property of . This ensures that the computational load is balanced, which is required in scenarios such as the storage system. On the other hand, when is sparse, then updating a single message symbol impacts exactly storage nodes in the storage system, which has a minimal impact.