A characterization of codes with extreme parameters

Let C be an [n,k,d]-code over GP(q) with k⩾2. Let s=def(C)=n+1-k-d denote the defect of C. The Griesmer bound implies that d⩽q(s+1). If d>qs and s⩾2, then using a previous result of Faldum and Willems, k⩽q. Thus fixing s⩾2 the extreme parameters for a code with def(C)=s are d=q(s+1); k=q, and n=k+d+s-1=(q+1)(s+2)-3. In this correspondence we characterize the codes with such parameters