# Achievable Information Rates for Coded Modulation with Hard Decision Decoding for Coherent Fiber-Optic Systems

## TL;DR

This paper evaluates the achievable information rates for coded modulation with hard decision decoding in fiber-optic systems, demonstrating that bit-wise decoders outperform symbol-wise decoders under standard hard decision decoding.

## Contribution

It provides a detailed analysis of AIRs for hard decision decoders, clarifies the superiority of binary decoders with BICM in practical fiber-optic systems, and designs codes confirming the theoretical findings.

## Key findings

- Bit-wise decoders yield higher AIRs than symbol-wise decoders with hard decision decoding.
- Binary codes outperform nonbinary codes in the studied fiber-optic scenarios.
- Standard hard decision decoding is preferable for complexity and performance in fiber-optic communications.

## Abstract

We analyze the achievable information rates (AIRs) for coded modulation schemes with QAM constellations with both bit-wise and symbol-wise decoders, corresponding to the case where a binary code is used in combination with a higher-order modulation using the bit-interleaved coded modulation (BICM) paradigm and to the case where a nonbinary code over a field matched to the constellation size is used, respectively. In particular, we consider hard decision decoding, which is the preferable option for fiber-optic communication systems where decoding complexity is a concern. Recently, Liga \emph{et al.} analyzed the AIRs for bit-wise and symbol-wise decoders considering what the authors called \emph{hard decision decoder} which, however, exploits \emph{soft information} of the transition probabilities of discrete-input discrete-output channel resulting from the hard detection. As such, the complexity of the decoder is essentially the same as the complexity of a soft decision decoder. In this paper, we analyze instead the AIRs for the standard hard decision decoder, commonly used in practice, where the decoding is based on the Hamming distance metric. We show that if standard hard decision decoding is used, bit-wise decoders yield significantly higher AIRs than symbol-wise decoders. As a result, contrary to the conclusion by Liga \emph{et al.}, binary decoders together with the BICM paradigm are preferable for spectrally-efficient fiber-optic systems. We also design binary and nonbinary staircase codes and show that, in agreement with the AIRs, binary codes yield better performance.