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.