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 $1.3_{-0.9}^{+3.0} \times 10^{-13}$ with a $[[2304,1156,\leq 14]]$ code and $2.9_{-1.5}^{+3.1} \times 10^{-11}$ with a $[[1152,580,\leq 12]]$ code. These results approach the teraquop regime, highlighting the promise of this code family for practical ultra-high-rate quantum error correction.

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