Codes approaching the Shannon limit with polynomial complexity per information bit

TL;DR

This paper introduces simple LDPC-type codes and a combined polar-LDPC scheme that approach the Shannon limit with polynomial complexity, improving low-noise channel performance and achieving vanishing error rates.

Contribution

It presents a novel coding scheme combining LDPC and polar codes that nearly reach the Shannon limit with polynomial complexity per bit.

Findings

LDPC-type codes improve channel output for SNR > -6 dB

Bounds on BER match simulation results

Combined scheme achieves vanishing BER near Shannon limit

Abstract

We consider codes for channels with extreme noise that emerge in various low-power applications. Simple LDPC-type codes with parity checks of weight 3 are first studied for any dimension $m\rightarrow\infty.$ These codes form modulation schemes: they improve the original channel output for any $SNR>$ $-6$ dB (per information bit) and gain $3$ dB over uncoded modulation as $SNR$ grows. However, they also have a floor on the output bit error rate (BER) irrespective of their length. Tight lower and upper bounds, which are virtually identical to simulation results, are then obtained for BER at any SNR. We also study a combined scheme that splits $m$ information bits into $b$ blocks and protects each with some polar code. Decoding moves back and forth between polar and LDPC codes, every time using a polar code of a higher rate. For a sufficiently large constant $b$ and $m\rightarrow\infty$, this design yields a vanishing BER at any SNR that is arbitrarily close to the Shannon limit of -1.59 dB. Unlike other existing designs, this scheme has polynomial complexity of order $m\ln m$ per information bit.