We prove the non--existence of [gqβ(4,d),4,d]qβ codes for d=2q3βrq2β2q+1 for 3β€rβ€(q+1)/2, qβ₯5; d=2q3β3q2β3q+1 for qβ₯9; d=2q3β4q2β3q+1 for qβ₯9; and d=q3βq2βrqβ2 with r=4,5 or 6 for qβ₯9, where gqβ(4,d)=βi=03ββd/qiβ. This yields that nqβ(4,d)=gqβ(4,d)+1 for 2q3β3q2β3q+1β€dβ€2q3β3q2, 2q3β5q2β2q+1β€dβ€2q3β5q2 and q3βq2βrqβ2β€dβ€q3βq2βrq with 4β€rβ€6 for qβ₯9 and that nqβ(4,d)β₯gqβ(4,d)+1 for 2q3βrq2β2q+1β€dβ€2q3βrq2βq for 3β€rβ€(q+1)/2, qβ₯5 and 2q3β4q2β3q+1β€dβ€2q3β4q2β2q for qβ₯9, where nqβ(4,d) denotes the minimum length n for which an [n,4,d]qβ code exists