We consider nonlinear quasi-perfect codes with packing radius 1 over a finite field of $q$ elements. We call these codes nonlinear 1-quasi-perfect $q$ -ary codes. We study the structural properties of nonlinear 1-quasi-perfect $q$ -ary codes, namely the rank and the dimension of the kernel. We prove that for $n=q^m$ and any $t\in\{1,2,\ldots,m+1\}$ there exist nonlinear 1-quasi-perfect $q$ -ary codes of length $n$ and rank $n-m-1+t$ . Here, $m\ge 5$ for $q=3$ and $4$ , $m\ge 4$ for $5\le q\le 19$ , and $m\ge 3$ for $q\ge 23$ . In particular, we prove that there exist full-rank nonlinear 1-quasi-perfect $q$ -ary codes. Also, for some nonlinear 1-quasi-perfect $q$ -ary codes we calculate the kernel dimension.
