Quasiperfect Linear Binary Codes with Distance 4 and Complete Caps in Projective Geometry

We prove that if a linear binary code with distance $d=4$ is quasiperfect (i.e., has a covering radius 2) and the code length is $N\ge 2^{r-2}+2$, where $r$ is the number of check symbols, then the check matrix is symmetric in the following sense: the matrix columns may be partitioned into $N/2$ pairs so that the sum of the columns in each pair is constant. As a corollary, we derive all possible values of the length $N$ of a binary linear quasiperfect code with $d=4$ for $N\ge 2^{r-2}+1$ and construct all such nonequivalent codes for $N>2^{r-2}+2^{r-6}$. The results are extended to complete caps in the projective geometry $PG(r-1,2)$.