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.