A New Bound to Error Probability on AWGN Channels: From Folded Coding to Gaussian Random Coding

A new bound on the error probability of coding with limited code length over additive white Gaussian noise (AWGN) channels is proposed. The developed bound is proved to be universal for two connected encoding ways. On the one hand, we conceive folding the conventional codes, such as Hadamard and binary random ones, in order to adapt shorter code length. On the other hand, we further extend the above folded structure to Gaussian random coding and hence to bound its error probability. Finally, we demonstrate that the bound of the above two constructions can be unified as $\sqrt {\frac {log_{2}e}{2\pi nC}}2^{-n(C/2-R)}$ where $C$ represents the capacity of the AWGN channel, $n$ and $R$ stand for the code length and rate, respectively. This theoretical contribution confirms that, in the context of short code length and low rate, the developed two constructions exhibit excellent performance even close to the Shannon bound on AWGN channels.