  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.
