On perfect and Reed–Muller codes over finite fields

We consider error-correcting codes over a finite field with $q$ elements ($q$-ary codes). We study relations between single-error-correcting $q$-ary perfect codes and $q$-ary Reed–Muller codes. For $q\ge 3$ we find parameters of affine Reed–Muller codes of order $(q-1)m-2$. We show that affine Reed–Muller codes of order $(q-1)m-2$ are quasi-perfect codes. We propose a construction which allows to construct single-error-correcting $q$-ary perfect codes from codes with parameters of affine Reed–Muller codes. A modification of this construction allows to construct $q$-ary quasi-perfect codes with parameters of affine Reed–Muller codes.