On subcodes of the generalized Reed-Solomon codes

In this paper, we study a class of subcodes of codimension $1$ in the $[n,k+1]_q$ generalized Reed-Solomon (GRS) codes, whose generator matrix is derived by removing the row of degree $k-r$ from the generator matrix of the $[n,k+1]_q$ GRS codes, where $1 \le r \le k-1$. We show equivalent characterizations for this class of subcodes of the GRS codes being self-dual or near-MDS, which extends the results for $r=1$ in the literature. Along with these characterizations, families of self-dual near-MDS subcodes of the GRS codes are also proposed. Finally, for $r = 1,2$, the dual codes of the subcodes of the GRS codes are found out. In some cases, the subcodes of the GRS codes can be closed under taking dual codes. In other cases, the dual codes turn out to be the twisted GRS codes.