  This paper derives an improved sphere-packing (ISP) bound for finite-length
codes whose transmission takes place over symmetric memoryless channels. We
first review classical results, i.e., the 1959 sphere-packing (SP59) bound of
Shannon for the Gaussian channel, and the 1967 sphere-packing (SP67) bound of
Shannon et al. for discrete memoryless channels. A recent improvement on the
SP67 bound, as suggested by Valembois and Fossorier, is also discussed. These
concepts are used for the derivation of a new lower bound on the decoding error
probability (referred to as the ISP bound) which is uniformly tighter than the
SP67 bound and its recent improved version. The ISP bound is applicable to
symmetric memoryless channels, and some of its applications are exemplified.
Its tightness is studied by comparing it with bounds on the ML decoding error
probability, and computer simulations of iteratively decoded turbo-like codes.
The paper also presents a technique which performs the entire calculation of
the SP59 bound in the logarithmic domain, thus facilitating the exact
calculation of this bound for moderate to large block lengths without the need
for the asymptotic approximations provided by Shannon.
