Title :
Sphere-packing bounds revisited for moderate block lengths
Author :
Valembois, Antoine ; Fossorier, Marc P C
Author_Institution :
Dept. of Electr. Eng., Hawaii Univ., Honolulu, HI, USA
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.
Keywords :
AWGN channels; block codes; channel coding; error statistics; iterative decoding; AWGN channel; Shannon theory; additive white Gaussian noise channel; block code; block length; channel decoding; error probability; information theory; iteratively decodable code; moderate code length; modulation scheme; sphere-packing bound; telecommunication system; AWGN; Additive white noise; Engineering profession; Error analysis; Error probability; Information theory; Iterative decoding; Modulation coding; Parity check codes; Turbo codes; 65; AWGN; Additive white Gaussian noise; block codes; lower bound on error probability;
Journal_Title :
Information Theory, IEEE Transactions on
DOI :
10.1109/TIT.2004.838090