On q-ary Grey-Rankin bound and codes meeting this bound

We consider the q-ary analog of the binary Grey-Rankin bound, recently suggested by Fu, Kloeve and Shen. For any prime power q/spl ges/2, we give an infinite family of codes which reach this bound with equality. If the outer and inner codes are chosen as linear, a linear resulting code is obtained by the concatenation construction.