## 1 Introduction
Recently, a Kronecker product least squares (KPLS) receiver is proposed in [23] to estimate the symbol and channel matrices. More recently, it is shown in [24] that a KPLS receiver can be extended to all the tensor-based systems. Although the KPLS receiver is a non-iterative and low-complexity solution, it needs the related core tensor unfolding to be right-invertible, which is a relatively harsh condition in signal design. Inspired by [21] and [22], we considered a simple tensor space-time coding scheme for multiple-antenna systems, along with an efficient receiver. The allocation factor and the space-time code factor in the TST coding scheme in [22] are independent, while the allocation factor in our coding scheme is also a three-dimensional space-time code factor. Thanks to the special structure of the proposed coding scheme, the received signal can be constructed as a Tucker-2 model [25; 26], which has uniqueness property under some suitable conditions. Then, a robust semi-blind receiver based on optimized LM algorithm is presented for joint channel and symbol estimation. Uniqueness and identifiability issues for the constructed Tucker-2 model are also discussed in this paper. Compared with existing receivers, the proposed receiver has a better estimation performance. Moreover, the proposed semi-blind receiver can be extended to the multi-user massive MIMO system. For the low-rank channel,the proposed receiver still has good performance for joint symbol and channel estimation even in the shorter length of code and information symbol, and larger number of data streams. The organization of this paper is as follows. Section 2 presents a brief overview of the Tucker model. In Section 3, the system model is presented and the associated tensor signal model is formulated. Section 4 briefly reviews the receiver with the ALS algorithm and describes the proposed semi-blind receiver based on the optimized LM algorithm. Section 5 extends the proposed semi-blind receiver to multi-user massive MIMO systems for joint symbol and channel estimation. In Section 6, some simulation results are shown to demonstrate the performance of our semi-blind receiver. Conclusions are drawn in Section 7. _Notation_: Scalars, vectors, matrices, and tensors are denoted by lower-case letters \((a,b,\cdots)\), boldface lower-case letters \((\mathbf{a},\mathbf{b},\cdots)\), boldface capitals \((\mathbf{A},\mathbf{B},\cdots)\), and underlined boldface capitals \((\mathbf{\underline{A}},\mathbf{\underline{B}},\cdots)\), respectively. \(\mathbf{A}^{T}\), \(\mathbf{A}^{H}\), \(\mathbf{A}^{-1}\), and \(\mathbf{A}^{+}\) represent transpose, conjugate transpose, inverse, and Moore-Penrose pseudo-inverse of the matrix \(\mathbf{A}\), respectively. \(\left\|\mathbf{A}\right\|_{F}\) denotes the Frobenius norm of \(\mathbf{A}\). \(\mathbf{I}_{M}\) denotes the \(M\times M\) identity matrix. The operator \(vec\left(\cdot\right)\) stacks the columns of its matrix argument to a vector, while \(unvec\left(\cdot\right)\) represents the inverse vectorization operation. The Kronecker matrix product is denoted by \(\otimes\). The term \(D_{i}\left(\mathbf{A}\right)\) corresponds the diagonal matrix out of the \(i\)-th row of \(\mathbf{A}\).