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 BBZ₄-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 BBZ₄-linear codes. The dimension of the kernel for these BBZ₄ -linear codes is established generalizing the known results about the dimension of the kernel for BBZ₄ -linear Hadamard and BBZ₄ -linear extended 1-perfect codes.