  Tensor-based modulation (TBM) has been proposed in the context of unsourced
random access for massive uplink communication. In this modulation,
transmitters encode data as rank-1 tensors, with factors from a discrete vector
constellation. This construction allows to split the multi-user receiver into a
user separation step based on a low-rank tensor decomposition, and independent
single-user demappers. In this paper, we analyze the limits of the tensor
decomposition using Cram\'er-Rao bounds, providing bounds on the accuracy of
the estimated factors. These bounds are shown by simulation to be tight at high
SNR. We introduce an approximate perturbation model for the output of the
tensor decomposition, which facilitates the computation of the log-likelihood
ratios (LLR) of the transmitted bits, and provides an approximate achievable
bound for the finite-length error probability. Combining TBM with classical
forward error correction coding schemes such as polar codes, we use the
approximate LLR to derive soft-decision decoder showing a gain over
hard-decision decoders at low SNR.
