Perfect mixed codes from generalized Reed-Muller codes

In this paper, we propose a new method for constructing 1-perfect mixed codes in the Cartesian product F(n)xF(q)(n), where F-n and F-q are finite fields of orders n = q(m) and q. We consider generalized Reed-Muller codes of length n=qm and order(q-1)m-2. Codes whose parameters are the same as the parameters of generalized Reed-Muller codes are called Reed-Muller-like codes. The construction we propose is based on partitions of distance-2MDS codes into Reed-Muller-like codes of order(q-1)m-2. We construct a set of q(qcn) nonequivalent 1-perfect mixed codes in the Cartesian product FxF(q)(n), where the constant c satisfies c<1,n=q(m) and m is a sufficiently large positive integer. We also prove that each1-perfect mixed code in the Cartesian product F(n)xF(q)(n) corresponds to a certain partition ofa distance-2 MDS code into Reed-Muller-like codes of order(q-1)m-2.