  QR decomposition is an essential operation for solving linear equations and
obtaining least-squares solutions. In high-performance computing systems,
large-scale parallel QR decomposition often faces node faults. We address this
issue by proposing a fault-tolerant algorithm that incorporates `coded
computing' into the parallel Gram-Schmidt method, commonly used for QR
decomposition. Coded computing introduces error-correcting codes into
computational processes to enhance resilience against intermediate failures.
While traditional coding strategies cannot preserve the orthogonality of $Q$,
recent work has proven a post-orthogonalization condition that allows low-cost
restoration of the degraded orthogonality. In this paper, we construct a
checksum-generator matrix for multiple-node failures that satisfies the
post-orthogonalization condition and prove that our code satisfies the
maximum-distance separable (MDS) property with high probability. Furthermore,
we consider in-node checksum storage setting where checksums are stored in
original nodes. We obtain the minimal number of checksums required to be
resilient to any $f$ failures under the in-node checksum storage, and also
propose an in-node systematic MDS coding strategy that achieves the lower
bound. Extensive experiments validate our theories and showcase the negligible
overhead of our coded computing framework for fault-tolerant QR decomposition.
