Linear codes with covering radius R=2, 3 and codimension tR

Let [n,n-r]_qR denote a linear code over F_q with length n, codimension r, and covering radius R. We use a modification of constructions of [2q+1, 2q-3]_q2 and [3q+1, 3q-5]_q3 codes (q⩾5) to produce infinite families of good codes with covering radius 2 and 3 and codimension tR