Abstract
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 <inline-formula> <tex-math notation="LaTeX">$\sqrt {\frac {log_{2}e}{2\pi nC}}2^{-n(C/2-R)}$ </tex-math></inline-formula> where <inline-formula> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula> represents the capacity of the AWGN channel, <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> 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.
Highlights
T HE great progress on channel coding has been made in the recent three decades [1]–[5], toward achieving the capacity with affordable implementation complexity
We demonstrate that all the above coding structures considered exhibit a uniform bound of error probability on additive white Gaussian noise (AWGN) channels, which can be briefly expressed in advance as log2e 2−n(C/2−R). 2πnC
The above result exhibits that the block error probability of the folded Hadamard code is exponentially descending with the code length only if the rate is less than C/2
Summary
A new bound on the error probability of coding with limited code length over additive white Gaussian noise (AWGN) channels is proposed. We further extend the above folded structure to Gaussian random coding and to bound its error probability. We demonstrate that the bound of the above two constructions can be unified as 2loπgn2Ce 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
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