An improved sphere-packing bound for finite-length codes over symmetric memoryless channels

We present an improved sphere-packing (ISP) bound for finite-length error-correcting codes whose transmission takes place over symmetric memoryless channels, and the codes are decoded by an arbitrary list decoder. Some applications of the ISP bound are also exemplified. Its tightness under maximum-likelihood (ML) decoding is studied by comparing the ISP bound to previously reported upper and lower bounds on the ML decoding error probability, and also to computer simulations of iteratively decoded turbo-like codes.