You need to check with the building inspector in your area to know for sure. When I was doing finish carpentry, the contractor made me install a 2 x 6 on each side of the stairs at the bottom, screwed into the side of the stair stringers and extending up to the floor joists of the 1st floor, where they were also screwed in. Then, 2 x 6's were angle cut to the angle of the stairs (39-40*), and attached to the under side of the 1st floor joists at the edge of the stair well and down to the vertical 2 x 6 at the bottom of the stairs. These 2 x 6's were 5" apart. The top one served as an attaching surface for handrail brackets and a wood handrail. That filled the requirements for that area.