Quantum error correction is widely believed to be essential for large-scale quantum computation, but the required qubit overhead remains a central challenge. Quantum low-density parity-check codes can substantially reduce this overhead through high-rate encodings, yet finite-size instances with practical logical error rates often achieve encoding rates only around or below 1/10. Here, building on a recent ultra-high-rate construction by Kasai, we identify new structural conditions on the underlying affine permutation matrices that make encoding rates exceeding 1/2 compatible with efficient implementation on reconfigurable neutral atom arrays. These conditions define a co-designed family of ultra-high-rate quantum codes that supports efficient syndrome extraction and atom rearrangement under realistic parallel control constraints. Using a hierarchical decoder with high accuracy and good throughput, we study the performance under a circuit-level noise model with p=0.1%, achieving per-logical-per-round error rates of 1.3 (+3.0/−0.9) × 10^−13 with a [[2304,1156,≤14]] code and 2.9 (+3.1/−1.5) × 10^−11 with a [[1152,580,≤12]] code. These results approach the teraquop regime, highlighting the promise of this code family for practical ultra-high-rate quantum error correction.