A characterization of certain Griesmer codes: MMD codes in a more general sense

Let C be an [n,k,d]_q linear code. The defect of C is the parameter s=s(C)=n-k+1-d. If k⩾m+1⩾2 then by the Griesmer bound d⩽(q^m(q-1)/q_m-1)(s+m). The author´s interest is in those linear codes having the maximum minimum distance, i.e., d=(q^m(q-1)/q^m-1)(s+m). For m=1 we have d=q(s+1) and the codes are maximum minimum distance (MMD) codes in the sense of Faldum and Willems (see ibid., vol.44, p.1555-58, 1998). Thus we consider MMD codes in a more general sense. We refer to them simply as MMD codes. All MMD codes with m=1 are known up to formal equivalence. Note that two codes are formally equivalent if they have the same weight distribution. The author classifies up to formal equivalence the MMD codes with m⩾2