Quasi-Perfect Linear Codes With Minimum Distance $4$

Some new infinite families of short quasi-perfect linear codes are described. Such codes provide improvements on the currently known upper bounds on the minimal length of a quasi-perfect [n,n-m,4]
_q
-code when either 1) q=16, m ges 5, m odd, or 2) q=2
ⁱ
, 7 les i les 15, m ges 4, or 3) q=2
^2lscr
, lscr ges 8, m ges 5, m odd. As quasi-perfect [n,n-m,4]
_q
-codes and complete n-caps in projective spaces PG(m-1,q) are equivalent objects, new upper bounds on the size of the smallest complete cap in PG(m-1,q) are obtained