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Matrix code Encodes $$K$$ states (codewords) in an $$m\times n$$-dimensional matrix of coordinates over a field (e.g., the Galois field $$GF(q)$$ or the complex numbers $$\mathbb{C}$$). Parents: Error-correcting code (ECC).
Also called a vector rank-metric code. A linear code over $$GF(q^N)$$ that corrects errors over rank metric instead of the traditional Hamming distance. Every element $$GF(q^N)$$ can be written as an $$N$$-dimensional vector with coefficients in $$GF(q)$$, and the rank of a set of elements is rank of the matrix formed by their coefficients. Protection: Set of vectors $$\{x_1, x_2, \ldots, x_M\}$$ determines a rank code with distance $$d=\min d(x_i, x_j)$$. The code with distance $$d$$ corrects all errors with rank of the error not greater than $$\lfloor (d-1)/2\rfloor$$. Parents: Linear $$q$$-ary code, Rank-metric code. Cousins: Maximum-rank distance (MRD) code.
Also called a Delsarte code. Each codeword is a matrix over $$GF(q)$$, with codewords forming a $$GF(q)$$-linear subspace, and with the metric being the rank of the difference of matrices. The distance $$d$$ is the minimum rank of all nonzero matrices in the code. Rank-metric codes on $$n\times m$$ matrices are denoted as $$[n\times m,k,d]_q$$. Protection: Protects against errors with rank $$\leq \lfloor \frac{d-1}2 \rfloor$$. Parents: Matrix code. Parent of: Gabidulin code, Maximum-rank distance (MRD) code.
Code designed for wireless transmission of information (via, e.g., radio waves) such that the sender can send multiple times from multiple locations. A spacetime code uses a modulation scheme to encode a message into signals that are sent at different times through different antennas, thereby utilizing both spatial and temporal (i.e., spacetime) degrees of freedom. Parents: Matrix code. Parent of: Spacetime block code (STBC).
Also called an optimal rank-distance code. An $$[n\times m,k,d]_q$$ rank-metric code whose parameters are such that the Singleton-like bound \begin{align} k \leq \max(n, m) (\min(n, m) - d + 1) \end{align} become an equality. Parents: Rank-metric code. Cousin of: Gabidulin code.
In a space-time block code, $$n$$ spatially separated channels transmit symbols in $$T$$ time slots. These symbols can be arranged in a $$T\times n$$ matrix where the columns correspond to the channels, and the rows correspond to the time slots. The codewords $$\{X\}$$ are $$T\times n$$ matrices such that the codeword difference matrices have rank $$n$$, and $$\min_{X\neq 0}\det(XX^*)$$ is maximized. Protection: Provides protection against errors due to thermal noise and destructive interference arising from traversing an environment with scattering, reflection, and/or refraction. Parents: Spacetime code (STC). Parent of: Orthogonal Spacetime Block Code (OSTBC).
The codewords are $$T\times n$$ matrices as defined for spacetime codes, with the additional condition that columns of the coding matrix are orthogonal. The parameter $$n$$ is the number of channels, and $$T$$ is the number of time slots. Protection: If the matrix $$C-C'$$, where $$C$$ and $$C'$$ are distinct codewords, has minimum rank $$b$$, the code has diversity order $$bn_R$$ (see Ref. [7], Sec. 28.2.1), where $$n_R$$ is the number of receivers. The maximum possible diversity order is $$nn_R$$. Parents: Spacetime block code (STBC). Parent of: Alamouti code.
The simplest OSTBC, with two time slots, two channels, and with unitary coding matrix \begin{align} \begin{pmatrix}c_{1} & c_{2}\\ -c_{2}^{\star} & c_{1}^{\star} \end{pmatrix}~, \end{align} where $$c_i$$ are complex numbers.

## References

[1]
E. M. Gabidulin, Theory of Codes with Maximum Rank Distance, Problemy Peredachi Informacii, Volume 21, Issue 1, 3–16 (1985)
[2]
R. M. Roth, “Maximum-rank array codes and their application to crisscross error correction”, IEEE Transactions on Information Theory 37, 328 (1991). DOI
[3]
P. Delsarte, “Bilinear forms over a finite field, with applications to coding theory”, Journal of Combinatorial Theory, Series A 25, 226 (1978). DOI
[4]
V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction”, IEEE Transactions on Information Theory 44, 744 (1998). DOI
[5]
E. M. Gabidulin, "Theory of Codes with Maximum Rank Distance", Problemy Peredachi Informacii, Volume 21, Issue 1, 3–16 (1985)
[6]
S. M. Alamouti, “A simple transmit diversity technique for wireless communications”, IEEE Journal on Selected Areas in Communications 16, 1451 (1998). DOI
[7]
W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021). DOI