Generalized Reed-Solomon Codes with Sparsest and Balanced Generator Matrices

We prove that for any positive integers $n$ and $k$ such that $n\geq k\geq 1$, there exists an $[n, k]$ generalized Reed-Solomon (GRS) code that has a sparsest and balanced generator matrix (SBGM) over any finite field of size $q \geq n+\lceil\frac{k(k-1)}{n}\rceil$, where sparsest means that each row of the generator matrix has the least possible number of nonzeros, while balanced means that the number of nonzeros in any two columns differ by at most one. Previous work by Dau et al (ISIT'13) showed that there always exists an MDS code that has an SBGM over any finite field of size $q\geq \binom{n-1}{k-1}$ I, and Halbawi et al (ISIT'16, ITW'16) showed that there exists a cyclic Reed-Solomon code (i.e., $n=q-1$) with an SBGM for any prime power q. Hence, this work extends both of the previous results.