We prove the non--existence of [gq(4,d),4,d]q codes for d=2q3−rq2−2q+1 for 3≤r≤(q+1)/2, q≥5; d=2q3−3q2−3q+1 for q≥9; d=2q3−4q2−3q+1 for q≥9; and d=q3−q2−rq−2 with r=4,5 or 6 for q≥9, where gq(4,d)=∑i=03⌈d/qi⌉. This yields that nq(4,d)=gq(4,d)+1 for 2q3−3q2−3q+1≤d≤2q3−3q2, 2q3−5q2−2q+1≤d≤2q3−5q2 and q3−q2−rq−2≤d≤q3−q2−rq with 4≤r≤6 for q≥9 and that nq(4,d)≥gq(4,d)+1 for 2q3−rq2−2q+1≤d≤2q3−rq2−q for 3≤r≤(q+1)/2, q≥5 and 2q3−4q2−3q+1≤d≤2q3−4q2−2q for q≥9, where nq(4,d) denotes the minimum length n for which an [n,4,d]q code exists