High-Rate Fair-Density Parity-Check Codes
Hessam Mahdavifar
TL;DR
FDPC codes provide high-rate, low-latency forward error correction by constructing a base matrix $H_b$ with dimension $2\sqrt{n}\times n$ and expanding it via random permutations to order-$s$ codes. The approach enables explicit weight-distribution characterization, enabling ML-type bounds that closely follow the random coding bound on the BEC and competitive performance on B-AWGN, while maintaining manageable decoding complexity through the MP-PL decoder. Key contributions include a deterministic $H_b$ construction for $n=4t^2$, the order-$s$ stacking framework, exact irreducible-codeword counts, ML bounds with shortening techniques, and a practical MP-PL decoding scheme that outperforms 5G-LDPC and polar-CRC in high-rate scenarios. The results indicate FDPC’s potential for ultra-high-throughput communications and optical channels, offering near-ML performance with low-complexity decoding and room for further decoding enhancements.
Abstract
We introduce fair-density parity-check (FDPC) codes targeting high-rate applications. In particular, we start with a base parity-check matrix $H_b$ of dimension $2 \sqrt{n} \times n$, where $n$ is the code block length, and the number of ones in each row and column of $H_b$ is equal to $\sqrt{n}$ and $2$, respectively. We propose a deterministic combinatorial method for picking the base matrix $H_b$, assuming $n=4t^2$ for some integer $t \geq 2$. We then extend this by obtaining permuted versions of $H_b$ (e.g., via random permutations of its columns) and stacking them on top of each other leading to codes of dimension $k \geq n-2s\sqrt{n}+s$, for some $s \geq 2$, referred to as order-$s$ FDPC codes. We propose methods to explicitly characterize and bound the weight distribution of the new codes and utilize them to derive union-type approximate upper bounds on their error probability under Maximum Likelihood (ML) decoding. For the binary erasure channel (BEC), we demonstrate that the approximate ML bound of FDPC codes closely follows the random coding upper bound (RCU) for a wide range of channel parameters. Also, remarkably, FDPC codes, under the low-complexity min-sum decoder, improve upon 5G-LDPC codes for transmission over the binary-input additive white Gaussian noise (B-AWGN) channel by almost 0.5dB (for $n=1024$, and rate $=0.878$). Furthermore, we propose a new decoder as a combination of weighted min-sum message-passing (MP) decoding algorithm together with a new progressive list (PL) decoding component, referred to as the MP-PL decoder, to further boost the performance of FDPC codes. This paper opens new avenues for a fresh investigation of new code constructions and decoding algorithms in high-rate regimes suitable for ultra-high throughput (high-frequency/optical) applications.