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.

翻译：我们针对高速率应用引入了公平密度奇偶校验（FDPC）码。具体而言，我们从维度为 $2\sqrt{n} \times n$ 的基础奇偶校验矩阵 $H_b$ 出发，其中 $n$ 为码块长度，且 $H_b$ 每行和每列中"1"的数量分别等于 $\sqrt{n}$ 和 $2$。我们提出了一种确定性组合方法用于选取基础矩阵 $H_b$，假设 $n=4t^2$（其中整数 $t \geq 2$）。通过获取 $H_b$ 的置换版本（例如，对其列进行随机置换）并将它们堆叠在一起，我们进一步扩展该方法，从而得到维度满足 $k \geq n-2s\sqrt{n}+s$（其中 $s \geq 2$）的码，称为 $s$ 阶 FDPC 码。我们提出了明确表征和限定新码重量的方法，并利用这些方法推导出其在最大似然（ML）译码下错误概率的联合型近似上界。对于二进制擦除信道（BEC），我们证明在广泛的信道参数范围内，FDPC 码的近似 ML 界紧密跟随随机编码上界（RCU）。此外，值得注意的是，在低复杂度最小和译码器下，FDPC 码在二进制输入加性高斯白噪声（B-AWGN）信道上的传输性能比 5G-LDPC 码提高了约 0.5dB（当 $n=1024$ 且码率 $=0.878$ 时）。进一步地，我们提出了一种新译码器，该译码器结合了加权最小和消息传递（MP）译码算法与一种新的渐进列表（PL）译码组件，称为 MP-PL 译码器，以进一步提升 FDPC 码的性能。本文为高速率场景下适用于超高吞吐量（高频/光通信）应用的新型码构造与译码算法研究开辟了新途径。