This paper presents a family of high-rate quantum Low-Density Parity-Check (LDPC) codes with limited long-range interactions, suitable for implementation in neutral atom quantum registers. These codes outperform surface codes in error correction when the two-qubit nearest neighbor gate error probability is below ~0.1%. The codes are designed for two-dimensional static neutral atom qubit architectures with open boundaries, where long-range connectivity is achieved via Rydberg-blockade interactions. The codes are native to these architectures and require multiple laser colors to enable transitions to different Rydberg states for different interatomic distances. The paper analyzes a family of high-rate quantum LDPC codes constructed via hypergraph product (HGP) method, referred to as La-cross codes. These codes have a structure reminiscent of a long-armed cross stitch pattern and are characterized by their high encoding rate, low stabilizer weight, and moderate non-locality. The codes are shown to have a larger code distance and lower logical error probability compared to surface codes for sufficiently small physical error probabilities. The paper also discusses the implementation of these codes on neutral atom quantum computers. The codes are implemented using Rydberg atoms, where long-range gates are achieved via Rydberg-blockade interactions. The paper presents a detailed error model for these implementations, considering both hardware-specific and hardware-agnostic noise scenarios. The results show that the La-cross codes can achieve lower logical error probabilities than surface codes for physical error probabilities as low as ~0.1%. The paper concludes that the La-cross codes are promising candidates for implementation on near-term neutral atom quantum computers due to their high encoding rate, low stabilizer weight, and moderate non-locality. The codes are also shown to be compatible with experimental platforms, such as neutral atom registers, and can be implemented with a reduced overhead compared to surface codes. The paper highlights the potential of these codes for achieving fault-tolerant quantum computing with low qubit overhead and high error suppression.This paper presents a family of high-rate quantum Low-Density Parity-Check (LDPC) codes with limited long-range interactions, suitable for implementation in neutral atom quantum registers. These codes outperform surface codes in error correction when the two-qubit nearest neighbor gate error probability is below ~0.1%. The codes are designed for two-dimensional static neutral atom qubit architectures with open boundaries, where long-range connectivity is achieved via Rydberg-blockade interactions. The codes are native to these architectures and require multiple laser colors to enable transitions to different Rydberg states for different interatomic distances. The paper analyzes a family of high-rate quantum LDPC codes constructed via hypergraph product (HGP) method, referred to as La-cross codes. These codes have a structure reminiscent of a long-armed cross stitch pattern and are characterized by their high encoding rate, low stabilizer weight, and moderate non-locality. The codes are shown to have a larger code distance and lower logical error probability compared to surface codes for sufficiently small physical error probabilities. The paper also discusses the implementation of these codes on neutral atom quantum computers. The codes are implemented using Rydberg atoms, where long-range gates are achieved via Rydberg-blockade interactions. The paper presents a detailed error model for these implementations, considering both hardware-specific and hardware-agnostic noise scenarios. The results show that the La-cross codes can achieve lower logical error probabilities than surface codes for physical error probabilities as low as ~0.1%. The paper concludes that the La-cross codes are promising candidates for implementation on near-term neutral atom quantum computers due to their high encoding rate, low stabilizer weight, and moderate non-locality. The codes are also shown to be compatible with experimental platforms, such as neutral atom registers, and can be implemented with a reduced overhead compared to surface codes. The paper highlights the potential of these codes for achieving fault-tolerant quantum computing with low qubit overhead and high error suppression.