An Improved Sphere-Packing Bound Targeting Codes of Short to Moderate   Block Lengths and Applications
  This paper derives an improved sphere-packing (ISP) bound targeting codes of short to moderate block lengths. We first review the 1967 sphere-packing (SP67) bound for discrete memoryless channels, and a recent improvement by Valembois and Fossorier. 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. Under a mild condition, the ISP bound is applicable to general memoryless channels, and some of its applications are exemplified. Its tightness is studied by comparing it with bounds on the ML decoding error probability. It is exemplified that the ISP bound suggests an interesting alternative to the 1959 sphere-packing (SP59) bound of Shannon for the Gaussian channel, especially for digital modulations of high spectral efficiency. 
Gil Wiechman and Igal Sason