Structured Quasi-Gray labelling for Reed-Muller Grassmannian Constellations

The high-SNR capacity of a block-faded non-coherent channel can be achieved by a multi-dimensional Grassmannian modulation. This paper proposes an analytical quasi-Gray labelling for Reed-Muller Grassmannian constellations. The proposed labelling method enables to reduce the block error rate (BLER) of a system with error correction code, by minimizing the average Hamming distance between labels of neighboring modulation symbols. In particular, we first define an inherent generation label from which the Reed-Muller Grassmannian constellation is constructed, and show it fulfills a homogeneity property. Based on this, a bijective linear mapping between a quasi-Gray labelling and the former labelling is proposed, where the bijective linear mapping matrix is obtained by a low-complexity algorithm. Numerical results show that the proposed quasi-Gray labelling method can achieve 40% reduction of the average Hamming distance between neighboring modulation symbols compared to the generation label or random labelling. Finally, link-level simulation results further demonstrate that the proposed quasi-Gray labelling can effectively reduce the BLER.