BRST symmetry and $W$-algebra in higher derivative models

In this paper we discuss the (anti)BRST symmetries and $W$-algebra of higher derivative theories of relativistic particles satisfying general gauge conditions. Using this formalism, the connection between the (anti)BRST symmetries and $W$-algebra for the massless particle with rigidity is established. Incidentally, the full $W$-algebra emerges only when the anti-BRST transformations are considered in tandem with the BRST ones. Further, the BRST symmetry is made finite and coordinate dependent. We show that such finite coordinate-dependent BRST symmetry changes the BRST invariant gauge-fixing fermion within the functional integration. This is exploited to connect two different arbitrary gauge conditions.