Welcome to the Matrix Kingdom.

Matrix-based 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 $$\mathbb{C}$$).
Also called tensor code, Kroneckerian code, or product code. A matrix-based code constructed out of two linear binary or $$q$$-ary codes $$C_A,C_B$$ in an outer-product construction denoted as $$C_A \otimes C_B$$. Its dual is sometimes called the check-product code [5; Lemma 3.3]. Protection: For linear codes $$C_A=[n_A,k_A,d_A]$$ and $$C_B=[n_B,k_B,d_B]$$, the resulting tensor code is $$C_A \otimes C_B=[n_A n_B,k_A k_B,d_A d_B]$$. Tensor codes can be useful for protecting against burst errors .
Matrix computation code Encoding that provides an extra redundancy for distributed matrix computation algorithms such as matrix multiplication. Parallelized algorithms distribute a desired computation over many nodes, and a key performance bottleneck is due to some nodes completing their individual tasks much later than other nodes. Matrix computation codes provide a layer of redundancy such that the computation can be performed without having all nodes finish their piece of the computation. Protection: Allows computation to complete without waiting for stragglers, or nodes that either do not finish or finish their portion of the computation much later than all other nodes. Parents: Matrix-based 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-based 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-based code. Parent of: Spacetime block code (STBC).
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: Rank-metric code, Linear $$q$$-ary code. Cousins: Maximum-rank distance (MRD) code.
An LRPC code of rank $$d$$ is a rank-metric code that, when interpreted as a linear code over $$GF(q^m)$$, admits an $$(n-k)\times n$$ parity-check matrix whose entries span a subspace of $$GF(q^m)$$ that is at most $$d$$-dimensional. Parents: Rank-metric code. Cousins: Low-density parity-check (LDPC) code.
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. , 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


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