  In this paper, nested Gallager regions with a single parameter is introduced
to exploit Gallager's first bounding technique (GFBT). We present a necessary
and sufficient condition on the optimal parameter. We also present a sufficient
condition (with a simple geometrical explanation) under which the optimal
parameter does not depend on the signal-to-noise ratio (SNR). With this general
framework, three existing upper bounds are revisited, including the tangential
bound (TB) of Berlekamp, the sphere bound (SB) of Herzberg and Poltyrev, and
the tangential-sphere bound (TSB) of Poltyrev. This paper also reveals that the
SB of Herzberg and Poltyrev is equivalent to the SB of Kasami et al., which was
rarely cited in literature.
