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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 rigid relativistic particle 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.