  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.
