  We demonstrate the existence of the first-class constraints on the massless
Abelian 3-form theory which generate the classical gauge symmetry
transformations for this theory in any arbitrary D-dimension of spacetime. We
write down the explicit expression for the generator in terms of these
first-class constraints. Using the celebrated Noether theorem, corresponding to
the gauge symmetry transformations, we derive the Noether conserved current and
conserved charge. The latter is connected with the first-class constraints of
the theory in a subtle manner as we demonstrate clearly in our present
investigation. We comment on the first-class constraints within the framework
of Becchi-Rouet-Stora-Tyutin (BRST) formalism where the conserved (anti-)BRST
charges are the generalizations of the above generator for the classical gauge
symmetry transformation. The standard Noether conserved (anti-)BRST charges are
found to be non-nilpotent. We derive the nilpotent versions of the (anti-)BRST
charges. One of the interesting observations of our present endeavor is the
result that only the nilpotent versions of the conserved (anti-)BRST charges
lead to the annihilation of the physical states by the operator form of the
first-class constraints at the quantum level which is consistent with the Dirac
quantization condition for the systems that are endowed with any kind of
constraints. We comment on the existence of the Curci-Ferrari (CF) type
restrictions from different theoretical angles, too.
