On nonlinear 1-quasi-perfect codes and their structural properties (Q6660008)

This publication is devoted to the nonlinear 1-quasi-perfect \(q\)-ary codes, which are quasi-perfect codes with packing radius 1 over a finite field of \(q\) elements. Quasi-perfect codes are defined as having a covering radius one unit more than the code's packing radius. The author has been able to prove that for any length \(n=q^m,\) there exists a nonlinear 1-quasi-perfect \(q\)-ary code of rank \(n-m-1+t,\) where \(1\leq t\leq m+1.\) This is done under the assumption that \(m\geq 5\) for ternary and quaternary codes, \(m\geq 4\) for \(5\leq q\leq 19,\) and \(m\geq 3\) for \(q\geq 23.\)\N\NAlso in this paper the existence of nonlinear 1-quasi-perfect \(q\)-ary codes of all possible ranks for code length \(n = q^m\), size \(q^{n-m-1}\), and minimum distance 3 is proven. In the case of \(n=q^m,\) the author proposes a new construction of 1-quasi-perfect \(q\)-ary codes and rank \(n-m.\) Denote \([m]_q=1+q+\dots+q^m.\) When \(q\geq 3,\) \(m\geq 2\) a switching construction of nonlinear 1-quasi-perfect codes with kernel dimension at least \(n-[m]_q-1\) is presented. The existence of nonlinear 1-quasi-perfect \(q\)-ary codes of length \(n\) with kernel dimension at least \(n-[m]_q-q-q^{m-1}\) is proven.