Here is a list of codes related to quantum MDS codes or entanglement-assisted MDS codes.
| Code | Description |
|---|---|
| Asymmetric quantum code (AQC) | Quantum systems can be roughly characterized by two types of noise, a bit-flip noise that maps canonical basis states into each other, and a phase-flip noise that induces relative phases between superpositions of such basis states. A code cannot protect against both types of noise arbitrarily well, and there is a tradeoff between the two types of protection. An AQC is one that performs much better against one type of noise than the other type. Such codes typically have tunable distances against each noise type and include CSS codes, GKP codes, and QSCs. |
| Constacyclic code | A block code \(C\) of length \(n\) over an alphabet \(R\) is \(\alpha\)-constacyclic (or \(\alpha\)-twisted) if, for each string \(c_1 c_2 \cdots c_n\in C\), the string \(\alpha c_n, c_1, \cdots, c_{n-1} \in C\) [1; Def. 3.2.7]. A \(-1\)-constacyclic code is called negacyclic. |
| Cyclic linear \(q\)-ary code | A \(q\)-ary code of length \(n\) is cyclic if, for each codeword \(c_1 c_2 \cdots c_n\), the cyclically shifted string \(c_n c_1 \cdots c_{n-1}\) is also a codeword. A cyclic code is called primitive when \(n=q^r-1\) for some \(r\geq 2\). A shortened cyclic code is obtained from a cyclic code by taking only codewords with the first \(j\) zero entries, and deleting those zeroes. |
| EA MDS code | EA Galois-qudit code whose parameters make the EAQECC Singleton bound [2; Thm. 6] become an equality. |
| EA qubit stabilizer code | A code constructed using a variation of the stabilizer formalism designed to utilize pre-shared entanglement between sender and receiver. A code is typically denoted as \([[n,k;e]]\) or \([[n,k,d;e]]\), where \(d\) is the distance of the EA code and \(e\) is the number of required pre-shared maximally entangled Bell states (ebits). While other entangled states can be used, there is always a choice of generators such that Bell states suffice while still using the fewest ebits. |
| Galois-qudit GRS code | A true \(q\)-Galois-qudit stabilizer code constructed from GRS codes via either the Hermitian construction [3–5] or the Galois-qudit CSS construction [6,7]. |
| Galois-qudit RS code | A Galois-qudit CSS code family (with \(q>n\)) constructed using two RS codes over \(\mathbb{F}_q\). |
| Galois-qudit quantum RM code | True Galois-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes via the Galois-qudit Hermitian construction, the Galois-qudit CSS construction, or directly from their parity-check matrices [8][9; Sec. 4.2]. |
| Generalized RS (GRS) code | An \([n,k,n-k+1]_q\) linear code that is a modification of the RS code where codeword polynomials are multiplied by additional prefactors [10; Def. 15.3.19]. |
| Good QLDPC code | Also called asymptotically good QLDPC codes. A family of QLDPC codes \([[n_i,k_i,d_i]]\) whose asymptotic rate \(\lim_{i\to\infty} k_i/n_i\) and asymptotic distance \(\lim_{i\to\infty} d_i/n_i\) are both positive. |
| Hermitian Galois-qudit code | An \([[n,k,d]]_q\) true Galois-qudit stabilizer code constructed from a Hermitian self-orthogonal linear code over \(\mathbb{F}_{q^2}\) using the one-to-one correspondence between the Galois-qudit Pauli matrices and elements of the Galois field \(\mathbb{F}_{q^2}\). |
| Maximum distance separable (MDS) code | A \(q\)-ary linear code whose parameters satisfy the Singleton bound with equality. |
| Perfect-tensor code | Block quantum code encoding one subsystem into an odd number \(n\) subsystems whose encoding isometry is a perfect tensor. This code stems from an AME\((n,q)\) AME state, or equivalently, a \(((n+1,1,\lfloor (n+1)/2 \rfloor + 1))\) code. |
| Quantum data-syndrome (QDS) code | Stabilizer code designed to correct both data qubit errors and syndrome measurement errors simultaneously due to extra redundancy in its stabilizer generators. |
| Quantum maximum-distance-separable (MDS) code | A type of block quantum code whose parameters satisfy the quantum Singleton bound with equality. |
| Quantum quadratic-residue (QR) code | Galois-qudit \([[n,1]]_q\) pure self-dual Galois-qudit CSS code constructed from a dual-containing QR code via the Galois-qudit CSS construction. For \(q\) not divisible by \(n\), its distance satisfies \(d^2-d+1 \geq n\) when \(n \equiv 3\) modulo 4 [11; Thm. 40] and \(d \geq \sqrt{n}\) when \(n\equiv 1\) modulo 4 [11; Thm. 41]. |
| Singleton-bound approaching AQECC | A member of an approximate quantum code family of rate \(R\) that can tolerate adversarial errors nearly saturating the quantum Singleton bound of \((1-R)/2\). The formulation of such codes relies on a notion of quantum list decoding [12,13]. |
| Skew-cyclic CSS code | A member of a family of Galois-qudit CSS codes constructed from skew-cyclic classical codes over rings [14; Thm. 5.8]. See related study [15] that uses cyclic codes over rings. |
| Subsystem Galois-qudit stabilizer code | Galois-qudit generalization of a subsystem qubit stabilizer code. Can be obtained by taking a Galois-qudit stabilizer code and assigning some of its logical qudits to be gauge qudits. |
| \([[2m,2m-2,2]]\) error-detecting code | Self-complementary and self-dual CSS code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. The code is constructed via the CSS construction from an SPC code and a repetition code [16; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [17]. |
| \([[3,1,2]]_3\) Three-qutrit code | A \([[3,1,2]]_3\) prime-qudit CSS code that is the smallest qutrit stabilizer code to detect a single-qutrit error. It has stabilizer generators \(ZZZ\) and \(XXX\). The code defines a quantum secret-sharing scheme and serves as a minimal model for the AdS/CFT holographic duality. It is also the smallest non-trivial instance of a quantum maximum distance separable code (QMDS), saturating the quantum Singleton bound. |
| \([[3,1,2]]_4\) Galois-qudit code | Three-Galois-qudit CSS code over \(\mathbb{F}_4=\{0,1,\omega,\omega^2\}\) that encodes one logical Galois qudit and detects a single-qudit error. |
| \([[4,2,2]]\) Four-qubit code | A four-qubit hyperbolic self-dual CSS stabilizer code that is the smallest two-logical-qubit stabilizer code to detect a single-qubit error. It is unique for its parameters [18; Thm. 8]. |
| \([[5,1,3]]\) Five-qubit perfect code | Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error. |
| \([[5,1,3]]_4\) Galois-qudit CSS code | Five-Galois-qudit CSS code over \(\mathbb{F}_4=\{0,1,\omega,\omega^2\}\) that encodes one logical Galois qudit and corrects a single-qudit error. |
| \([[6,2,3]]_{q}\) code | Six-qudit MDS error-correcting code defined for Galois-qudit dimension \(q=3\) [19], \(q=2^2\) [20], and \(q \geq 5\) [19][11; Exam. 33]. This code cannot exist for qubits (\(q=2\)). |
| \([[6,4,2]]\) error-detecting code | Self-complementary six-qubit code with rate \(2/3\) that is unique for its parameters, up to equivalence [17; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [21]. |
| \([[7,3,3]]_{q}\) code | Seven-qudit MDS error-detecting code defined for Galois-qudit dimension \(q=3\) [19] and \(q \geq 7\) [19][11; Exam. 33]. This code cannot exist for qubits (\(q=2\)). |
| \([[9,1,5]]_3\) quantum Glynn code | Nine-qutrit pure Hermitian code that is the smallest qutrit stabilizer code to correct two-qutrit errors. |
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