Sphere-Packing Bounds Revisited for Moderate Block Lengths

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Abstract
The main reference of this paper is the sphere-packing bound of 1967 (SP67) derived by Shannon, Gallager, and Berlekamp. It offers a lower bound on the decoding error probability over a very large variety of channels. If it has failed so far to provide any usable material in practical implementation of telecommunication systems, it is due to an original focus on asymptotic results (making it inapplicable for moderate code lengths) and to the difficulty of the involved methods (which makes the derivation of SP67 quite hermetic and uninspiring for further research). The purpose of this paper is two-fold: 1) to stir up some renewed interest in the topic on which Shannon concluded his career in information theory thanks to a qualitative (rather than technical) review of the derivation of SP67, introduced by a review of the simpler sphere-packing bound derived by Shannon in 1959; 2) to prove the practical interest of SP67 by extending its field of application to continuous output channels and particularly the additive white Gaussian noise (AWGN) channel used with any particular modulation scheme, and by improving its lower bound for the moderate code length case so that it becomes the best lower bound for most iteratively decodable codes (turbo codes, low-density parity-check (LDPC) codes, repeat-accumulate (RA) codes, etc.) of usual lengths.

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