Complete Sampling of the $uv$ Plane with Realistic Radio Arrays: Introducing the RULES Algorithm, with Application to 21 cm Foreground Wedge Removal

We introduce the Radio-array $uv$ Layout Engineering Strategy (RULES), an algorithm for designing radio arrays that achieve complete coverage of the $uv$ plane, defined as, at minimum, regular sampling at half the observing wavelength ($\lambda$) along the $u$ and $v$ axes within a specified range of baseline lengths. Using RULES, we generate $uv$-complete layouts that cover the range $10\lambda\leq|(u,v)|\leq 100\lambda$ with fewer than 1000 antennas of diameter $5\lambda$, comparable to current and planned arrays. We demonstrate the effectiveness of such arrays for mitigating contamination from bright astrophysical foregrounds in 21 cm Epoch of Reionization observations,particularly in the region of Fourier space known as the foreground wedge,by simulating visibilities of foreground-like sky models over the 130-150 MHz band and processing them through an image-based power spectrum estimator. We find that with complete $uv$ coverage, the wedge power is suppressed by sixteen orders of magnitude compared to an array with a compact hexagonal layout (used as a reference for a sparse $uv$ coverage). In contrast, we show that an array with the same number of antennas but in a random configuration only suppresses the wedge by three orders of magnitude, despite sampling more distinct $uv$ points over the same range. We address real-world challenges and find that our results are sensitive to small antenna position errors and missing baselines, while still performing equally or significantly better than random arrays in any case. We propose ways to mitigate those challenges such as a minimum redundancy requirement or tighter $uv$ packing density.