Classification of Some Families of Quaternary Reed–Muller Codes

Recently, new families of quaternary linear Reed–Muller codes have been introduced. They satisfy that, after the Gray map, the corresponding
${\BBZ}_{4}$
-linear codes have the same parameters and properties as the codes of the binary linear Reed–Muller family. A structural invariant, the dimension of the kernel, for binary codes is used to classify completely these
${\BBZ}_{4}$
-linear codes. The dimension of the kernel for these
${\BBZ}_{4}$
-linear codes is established generalizing the known results about the dimension of the kernel for
${\BBZ}_{4}$
-linear Hadamard and
${\BBZ}_{4}$
-linear extended 1-perfect codes.