Welcome to the Modular-qudit Kingdom.

A $$[[3,1,2]]_3$$ prime-qudit CSS code with 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. The codewords are \begin{align} \begin{split} | \overline{0} \rangle &= \frac{1}{\sqrt{3}} (| 000 \rangle + | 111 \rangle + | 222 \rangle) \\ | \overline{1} \rangle &= \frac{1}{\sqrt{3}} (| 012 \rangle + | 120 \rangle + | 201 \rangle) \\ | \overline{2} \rangle &= \frac{1}{\sqrt{3}} (| 021 \rangle + | 102 \rangle + | 210 \rangle)~. \end{split} \end{align} The elements in the superposition of each logical codeword are related to each other via cyclic permutations. Protection: Detects single qutrit errors and protects against a single-qutrit erasure. There does not exist a three-qubit code with analogous properties. Cousins: Approximate secret-sharing code. Cousin of: $$[[3,1,2]]_{\mathbb Z}$$ rotor code.
Modular-qudit code Also called a $$\mathbb{Z}_q$$-qudit code. Encodes $$K$$-dimensional Hilbert space into a $$q^n$$-dimensional ($$n$$-qudit) Hilbert space, with canonical qudit states $$|k\rangle$$ labeled by elements $$k$$ of the group $$\mathbb{Z}_q$$ of integers modulo $$q$$. Usually denoted as $$((n,K))_q$$ or $$((n,K,d))_q$$, whenever the code's distance $$d$$ is defined, and with $$q=p$$ when the dimension is prime. Protection: A convenient and often considered error set is the modular-qudit analogue of the Pauli string basis for qubit codes. For a single qudit, this set consists of products of powers of the qudit Pauli matrices $$X$$ and $$Z$$, which act on computational basis states $$|k\rangle$$ for $$k\in\mathbb{Z}_q$$ as \begin{align} X\left|k\right\rangle =\left|k+1\right\rangle \,\,\text{ and }\,\,Z\left|k\right\rangle =e^{i\frac{2\pi}{q}k}\left|k\right\rangle ~, \end{align} with addition performed modulo $$q$$. For multiple qudits, error set elements are tensor products of elements of the single-qudit error set. Parent of: Modular-qudit stabilizer code.
An $$((n,K,d))_q$$ modular-qudit code whose logical subspace is the joint eigenspace of commuting qudit Pauli operators forming the code's stabilizer group $$\mathsf{S}$$. Traditionally, the logical subspace is the joint $$+1$$ eigenspace, and the stabilizer group does not contain $$e^{i \phi} I$$ for any $$\phi \neq 0$$. The distance $$d$$ is the minimum weight of a qudit Pauli string that implements a nontrivial logical operation in the code. Protection: Detects errors on up to $$d-1$$ qudits, and corrects erasure errors on up to $$d-1$$ qudits. More generally, define the normalizer $$\mathsf{N(S)}$$ of $$\mathsf{S}$$ to be the set of all operators that commute with all $$S\in\mathsf{S}$$. A stabilizer code can correct a Pauli error set $${\mathcal{E}}$$ if and only if $$E^\dagger F \notin \mathsf{N(S)}\setminus \mathsf{S}$$ for all $$E,F \in {\mathcal{E}}$$.
Let $$C$$ be a quantum cyclic code on $$n$$ prime-dimensional qudits. $$C$$ is a Frobenius code if there exists a positive integer $$t$$ such that $$n$$ divides $$p^t +1$$. Protection: Protects against Pauli noise.
Stub. Cousins: Kitaev surface code.
An $$((n,K,d))_q$$ modular-qudit stabilizer code admitting a set of stabilizer generators that are either $$Z$$-type or $$X$$-type Pauli strings. The stabilizer generator matrix, taking values from $$\mathbb{Z}_q$$, is of the form \begin{align} H=\begin{pmatrix}0 & H_{Z}\\ H_{X} & 0 \end{pmatrix} \label{eq:parityq} \end{align} such that the rows of the two blocks must be orthogonal \begin{align} H_X H_Z^T=0~. \label{eq:commQ} \end{align} The above condition guarantees that the $$X$$-stabilizer generators, defined in the symplectic representation as rows of $$H_X$$, commute with the $$Z$$-stabilizer generators associated with $$H_Z$$. Parents: Modular-qudit stabilizer code.
A family of stabilizer codes whose generators are few-body $$X$$-type and $$Z$$-type Pauli strings associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface (with a qudit located at each edge of the tesselation). The code has $$n=E$$ many physical qudits, where $$E$$ is the number of edges of the tesselation, and $$k=2g$$ many logical qudits, where $$g$$ is the genus of the surface. Protection: When defined on an $$L\times L$$ square tiling of the torus, protects against $$L$$ errors. More generally, the code distance is the number of edges in the shortest non contractible cycle in the tesselation or dual tesselation [9]. Cousins: Kitaev surface code, String-net code.
Also called prime-qudit polynomial code (QPyC). Prime-qudit CSS code constructed using two Reed-Solomon codes. Parents: Modular-qudit CSS code. Cousin of: Galois-qudit RS code.
A family of CSS codes extending Hamming-based CSS codes to prime qudits of dimension $$p$$ by expressing the qubit code stabilizers in local-dimension-invariant (LDI) form [11]. Parents: Modular-qudit CSS code.

## References

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[11]
Arun J. Moorthy and Lane G. Gunderman, “Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise”. 2110.11510