  The punctured binary Reed-Muller code is cyclic and was generalized into the
punctured generalized Reed-Muller code over $\gf(q)$ in the literature. The
major objective of this paper is to present another generalization of the
punctured binary Reed-Muller code. Another objective is to construct a family
of reversible cyclic codes that are related to the newly generalized
Reed-Muller codes.
