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The PIDOTS neutral particle transport code utilizes a red/black implementation of the Parallel Gauss-Seidel algorithm to solve the SN approximation of the neutron transport equation on 3D Cartesian meshes. PIDOTS is designed for execution on massively parallel platforms and is capable of using the full resources of modern, leadership class high performance computers. Initial testing revealed that some configurations of PIDOTS's Integral Transport Matrix Method solver demonstrated unexpectedly poor parallel scaling. Work at Idaho and Los Alamos National Laboratories then revealed that this inefficiency was a result of the accumulation of high-cost latency events in the complex blocking communication networks employed during each PIDOTS iteration. That work explored the possibility of minimizing those inefficiencies while maintaining a blocking communications model. While significant speedups were obtained, it was shown that fully mitigating the problem on general-purpose platforms was highly unlikely for a blocking code. This work continues that analysis by implementing a deeply interleaved non-blocking communication model into PIDOTS. This new model benefits from the optimization work performed on the blocking model while also providing significant opportunities to overlap the remaining un-mitigated communication costs with computation. Additionally, our new approach is easily transferable to other similarly spatially decomposed codes. The resulting algorithm was tested on LANL's Trinity system at up to 32,768 processors and was found at that processor count to effectively hide 100% of MPI communication cost – equivalently 20% of the red/black phase time. It is expected that the implemented interleaving algorithm can fully support far higher processor counts and completely hide communication costs up ~50% of total iteration time. [ABSTRACT FROM AUTHOR]