An Improved Sphere-Packing Bound for Finite-Length Codes on Symmetric   Memoryless Channels
  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. 
Gil Wiechman and Igal Sason