  This paper presents a finite-rate deep-learning (DL)-based channel state
information (CSI) feedback method for massive multiple-input multiple-output
(MIMO) systems. The presented method provides a finite-bit representation of
the latent vector based on a vector-quantized variational autoencoder (VQ-VAE)
framework while reducing its computational complexity based on shape-gain
vector quantization. In this method, the magnitude of the latent vector is
quantized using a non-uniform scalar codebook with a proper transformation
function, while the direction of the latent vector is quantized using a
trainable Grassmannian codebook. A multi-rate codebook design strategy is also
developed by introducing a codeword selection rule for a nested codebook along
with the design of a loss function. Simulation results demonstrate that the
proposed method reduces the computational complexity associated with VQ-VAE
while improving CSI reconstruction performance under a given feedback overhead.
