All Reed-Muller codes are linearly representable over the ring of dual numbers over Z2

The statement given in the title is proved. Linear codes over chain rings (commutative and noncommutative) are a natural generalization of linear codes over finite fields and of linear codes over integer residue class rings of prime power order. In matters of linear representability there is no obvious reason why we should prefer one chain ring to the other. Yet, apart from Z₄, there is one further nontrivial chain ring with four elements: the ring Z₂[x]/(x²) of dual numbers over Z₂. It is natural to ask about the linear representability of the Reed-Muller codes over this ring. For the sake of completeness, we reformulate here in an obvious way the definition of a linearly representable code