  The performance of maximum-likelihood (ML) decoded binary linear block codes
over the AWGN channel is addressed via the tangential-sphere bound (TSB) and
two of its recent improved versions. The paper is focused on the derivation of
the error exponents of these bounds. Although it was exemplified that some
recent improvements of the TSB tighten this bound for finite-length codes, it
is demonstrated in this paper that their error exponents coincide. For an
arbitrary ensemble of binary linear block codes, the common value of these
error exponents is explicitly expressed in terms of the asymptotic growth rate
of the average distance spectrum.
