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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 thathas a sparsest and balanced generator matrix (SBGM) over any finite field ofsize $q\!\geq\! n\!+\!\lceil\frac{k(k-1)}{n}\rceil$, where sparsest means thateach 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 byat most one. Previous work by Dau et al (ISIT'13) showed that there alwaysexists an MDS code that has an SBGM over any finite field of size $q\geq{n-1\choose k-1}$, and Halbawi et al (ISIT'16, ITW'16) showed that there existsa 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.

Generalized Reed-Solomon Codes with Sparsest and Balanced Generator Matrices