Abstract:

We consider quasi-perfect codes with packing radius 1 over a finite field of $q$ elements. We call these codes 1-quasi-perfect $q$-ary codes. We study the structural properties of nonlinear 1-quasi-perfect $q$-ary codes, namely the rank and dimension of the kernel. In this paper, we propose a construction of 1-quasi-perfect $q$-ary codes with parameters of generalized Reed-Muller codes of order $r = (q -1)m - 2$, where  $m$ is a positive integer. For $q \geq 3$, $m \geq 2$, the proposed construction allows one to construct nonlinear 1-quasi-perfect $q$-ary codes  with different kernel dimensions. The dimensions of the kernel of nonlinear 1-quasi-perfect $q$-ary codes constructed using the proposed construction are calculated.