The best possible upper bound on the probability of undetected error for linear codes of full support

There is a known best possible upper bound on the probability of undetected error for linear codes. The $[n,k;q]$ codes with probability of undetected error meeting the bound have support of size $k$ only. In this note, linear codes of full support ($=n$) are studied. A best possible upper bound on the probability of undetected error for such codes is given, and the codes with probability of undetected error meeting this bound are characterized.