Let F be a set of f points in a finite projective geometry PG( t, q) of t dimensions where t≥2, f≥1 and q is a prime power. If (a) | F∩ H|≥ m for any hyperplane H in PG( t, q) and (b) | F∩ H| = m for some hyperplane H in PG( t, q), then F is said to be an { f, m; t, q}-min.hyper (or an { f, m; t, q}-minihyper) (cf. Hamada and Tamari (1978) and Hamada (1985, 1987a)) where m≥0 and | A| denotes the number of elements in the set A. Using a characterization of { ν μ + 1 , ν μ ; k − 1, q}-min.hypers and { ν μ + 1 + 1, ν μ ; k − 1, q}-min.hypers in PG( k − 1, q), Tamari (1981a, 1984) characterized all ( n, k, d; q)-codes (i.e., all q-ary linear codes with length n, dimension k, and minimum distance d) meeting the Griesmer bound (1.1) for the case d = q k − 1 − q μ or d = q k − 1 − (1+ q μ ) where k≥3, 1 ≤ μ ≤ k − 2 and ν l = (q l − 1) (q − 1) for any integer l ≥ 0. The purpose of this paper is to extend the above results, i.e. to characterize all ( n, k, d; q)-codes meeting the Griesmer bound (1.1) for the case d = q k − 1 − ( ε + q μ ) using a characterization of { ν μ + 1 + ε, ν μ ; k − 1, q}-min.hypers where k≥3, 1≤ μ≤ k−2, 2<− ε < √ q and q > 4.