Code Performance as a Function of Block Size

The recent development of turbo codes has stirred new interest in trying to understand how closely practical codes can approach the theoretical limits on code performance. This article reformulates Shannon’s sphere-packing bound in a normalized form conducive to analyzing the relative performance of families of turbo codes. Even though this bound is achievable only by a (generally mythical) perfect spherical code for the continuous-input additive white Gaussian noise (AWGN) channel, it still can serve as a useful performance benchmark against which all codes of a given rate r and information block size k can be compared. A measure of imperfectness relative to this bound is defined and evaluated for a family of turbo codes and several other well-known codes. Simulations show that these turbo codes approach perfectness closely (within 0.7 dB) and almost uniformly over a wide range of code rates and block sizes. The implication is that turbo codes are near-optimal codes even for block sizes much smaller than previously believed. Finally, the sphere-packing lower bound also is compared to the average performance of randomly selected codes for the continuous-input AWGN channel and to an upper bound on the average performance of randomly selected codes for the binary-input AWGN channel.