This paper introduces a novel deep-learning assisted video list decoding method for error-prone video transmission systems. Unlike traditional list decoding techniques, our proposed system uses a Transformer-based no-reference image quality assessment method to select the highest-scoring reconstructed video candidate after reception. Three new components are defined and used in the Transformer-assisted image quality evaluation metric: neighborhood-based patch fidelity aggregation, discriminant color texture transformation and ranking-constrained penalty loss function. We have also created our own database of non-uniformly distorted images, similar to those that might result from transmission errors, in a High Efficiency Video Coding (HEVC) context. In our specific testing context, our improved Transformer-assisted method has a decision accuracy of 100% for intra-coded image, while, for errors occurring in an inter image, it is 96%. Notably, in the few cases where a wrong choice is made, the selected candidate's quality remains similar to the intact frame. Code: https://github.com/Yujing0926/Robust-Video-List-Decoding-Using-a-Deep-Learning-Approach.