On the number of $q$-ary quasi-perfect codes with covering radius 2

Abstract

In this paper we present a family of $q$-ary nonlinear quasi-perfect codes with covering radius 2. The codes have length $n=q^m$ and size $M=q^{n-m-1}$ where $q$ is a prime power, $q\ge 3$, $m$ is an integer, $m\ge 2$. We prove that there are more than $q^{q^{cn}}$ nonequivalent such codes of length $n$, for all sufficiently large $n$ and a constant $c=\frac{1}{q}-\epsilon$.