C2 Codes: Quantum, Classical & Geometric Insights

• C2 codes are families of codes defined in quantum, classical, and geometric contexts, each with unique algebraic, combinatorial, or geometric structures.
• In the quantum setting, C2 codes use hypergraph product constructions from cyclic codes to achieve low logical error rates and constant-depth syndrome extraction, enhancing fault tolerance.
• In classical and geometric frameworks, quasi-cyclic codes of index 2 and complex spherical 2-codes provide precise duality conditions and optimal distance bounds, enabling efficient error correction and design of tight two-distance sets.

C2 codes refer to several distinct, highly structured families of codes that arise in classical and quantum information theory as well as combinatorial geometry. These include: (1) quantum hypergraph product codes constructed from the product of a cyclic code with itself; (2) quasi-cyclic codes of index 2 in classical coding theory; and (3) complex spherical two-distance sets—often called complex 2-codes—in geometric coding theory. Each context confers a precise algebraic, combinatorial, or geometric structure and offers unique optimality or performance properties relative to their domain.

1. Quantum Hypergraph Product Codes from Cyclic Codes: The “C2” Construction

C2 codes in the context of quantum error correction are quantum low-density parity-check (LDPC) codes formed as a hypergraph product (HGP) of a binary cyclic code with itself. Given a classical cyclic code $C$ of length $n$ and dimension $k$ with minimum distance $d_C$, its parity-check matrix $H$ is circulant, with each row being a cyclic shift of a generator vector of weight $w$. The hypergraph product quantum code is constructed via Calderbank-Shor-Steane (CSS) checks as: $$H_X = [H \otimes I_n \mid I_{n-k} \otimes H^{T}], \;\;\; H_Z = [I_n \otimes H \mid H^{T} \otimes I_{n-k}].$$ This yields code parameters:

• Blocklength: $N = n^2 + (n-k)^2$
• Number of logical qubits: $K = k^2$
• Minimum distance: $d = d_C$ (when $d_{C^{T}} \geq d_C$, as is often achievable in cyclic families).

Notably, both $X$- and $Z$-type stabilizer generators are regular of weight $2w$, enforcing uniform low-weight checks. C2 codes systematically outperform prior HGPs (optimized with progressive edge growth, simulated annealing, RL, etc.): for example, the code $[[450, 32, 8]]$ (from $[15,7,8]$ cyclic) achieves per-round logical error rate $\approx 4.5 \times 10^{-7}$ at $p=10^{-3}$ with $\approx 14.1$ overhead, while $[[882,50,10]]$ attains $2 \times 10^{-8}$ with overhead $\approx 17.6$, significantly lower than surface or bivariate bicycle codes.

C2 codes are particularly suited for trapped ion (QCCD) architectures, supporting a planar cyclic layout that enables regular, constant-depth syndrome extraction: syndrome rounds proceed via cyclically shifting ancillae and enacting CNOT/CZ gates with aligned data, requiring depth $4w+2$ per round, independent of $n$.

2. Classification and Structure of Quasi-Cyclic Codes of Index 2 (“QC Index 2” Codes)

In classical coding theory, a quasi-cyclic code of index 2 and length $2m$ over $\mathbb{F}_q$ is defined as a code invariant under shift by $m$, or equivalently as an $R$-submodule of $R^2$ where $R = \mathbb{F}_q[x]/(x^m-1)$. Every such code $C$ has a unique characterization: $$C = \langle (g_{11}(x), g_{12}(x)),\; (0, g_{22}(x)) \rangle_R,$$ where $g_{11}(x), g_{22}(x)$ divide $x^m-1$, and $\deg g_{12}(x) < \deg g_{22}(x)$, with $g_{11}g_{22}$ dividing $(x^m-1)g_{12}$. The dimension is $2m - \deg g_{11} - \deg g_{22}$. When $\gcd(q, m) = 1$, a one-generator criterion holds: $g_{11}g_{22} \equiv 0 \mod (x^m-1)$.

Duals with respect to the standard (Euclidean), symplectic, and Hermitian inner products are fully described. For instance, the Euclidean dual admits generators: $$C^{\perp_e} = \langle \left(\tfrac{x^m-1}{g_{11}}, 0\right),\, \left(-\tfrac{x^m-1}{g_{11}} g_{12}^{\circ}, \tfrac{x^m-1}{g_{22}} g_{22}^{\circ}\right) \rangle_R,$$ where $f^{\circ}(x) = x^m f(x^{-1})$. Self-orthogonality and dual-containing properties have explicit algebraic conditions, and analogous statements hold under symplectic and Hermitian forms.

Minimum distance is bounded below by

$$d(C) \ge \min\left\{ d(C_2),\; d(C_4),\; d(C_1) + d(C_3) \right\},$$

where $C_1 = \langle g_{11} \rangle$, $C_2 = \langle g_{22} \rangle$, $C_3 = \langle \gcd(g_{12}, g_{22}) \rangle$, $C_4 = \langle \gcd(g_{11}g_{12}, g_{22}) \rangle$. This lower bound can be tight, as in explicit constructions over $\mathbb{F}_2$ for small $m$ (e.g., the unique $[6,3,3]$ code with $g_{11}(x)=x^2+x+1$, $g_{22}(x)=x-1$, $g_{12}(x)=1$) (Abdukhalikov et al., 1 Apr 2025).

3. Complex Spherical 2-Codes (“C₂-codes”): Geometric Two-Distance Sets

In combinatorial geometry, a complex spherical 2-code is a finite subset $X$ of the unit sphere in $\mathbb{C}^d$ such that the angle set $$A(X) = \{\langle x, y \rangle : x, y \in X, x\ne y\}$$ has cardinality $2$, with the two values being nonreal (conjugate) complex numbers. The existence and tightness of such sets are governed by the structure of associated tournaments and certain matrix conditions:

• If $d$ is odd, the absolute bound is $|X| \leq 2d+1$, tightness if and only if the tournament on $X$ (given by “angle $\alpha$”) is doubly regular of order $2d+1$.
• If $d$ is even, $|X| \leq 2d$, with equality if and only if $I+A-A^{T}$ is a skew Hadamard matrix of order $2d$.

Explicit tight examples exist via skew-symmetric D-optimal designs (e.g., for $d=4$, Paley’s $8\times8$ skew Hadamard matrix yields a tight 2-code in $\mathbb{C}^4$).

The embedding dimension of the associated tournament into $\mathbb{C}^d$ is determined by the spectrum of its Seidel matrix $S = i(A - A^{T})$. Representation dimension minimization relies on eigenvalue multiplicities and the so-called main angle (overlap with the all-ones vector), establishing exact embedding parameters via analytic formulas based on spectral interlacing and rank-one perturbation theory (Nozaki et al., 2015).

4. Tables of Representative Parameters and Performance

For quantum C2 codes, codes identified via exhaustive search (for $n \leq 40$, $w \leq 5$) have the following representative parameters:

Code [[N,K,d]]	Classical C [n,k,$d_C$]	Weight $w$	Overhead $N/K$
[[450, 32, 8]]	[15, 7, 8]	3	$\approx$14.1
[[882, 98, 8]]	[21, 7, 8]	4	9.0
[[882, 50, 10]]	[21, 5, 10]	3	17.6

Corresponding logical error rates (per qubit per round, depolarizing noise $p=10^{-3}$):

Code	Logical Error Rate	Overhead	Distance $d$
[[450, 32, 8]]	$4.5 \times 10^{-7}$	14.1	8
[[882, 50, 10]]	$2 \times 10^{-8}$	17.6	10
Surface [[113,1,8]]	$3 \times 10^{-6}$	113	8

These C2 codes exhibit logical error rates up to three orders of magnitude better than previous LDPC and surface codes at significantly reduced physical overhead.

5. Trade-offs, Advantages, and Limitations

Key trade-offs for quantum C2 codes include:

• Pros: Constant-depth and fully parallel syndrome extraction; regular low-weight stabilizers; constant encoding rate in the asymptotic limit ($K/N = (k/n)^2$); minimum distance grows linearly with that of the constituent cyclic code.
• Cons: Block length $N$ scales quadratically with $n$, producing large code "patches" for high minimum distance; implementation of cyclic shifts may conflict with hardware connectivity, especially for large $n$ or generator weight $w$.

For quasi-cyclic codes of index 2, algebraic characterizations ensure tractable generator and dual structure, but the block size and generator constraints may limit certain rate-distance combinations. Tightness in complex spherical 2-codes is fully characterized and only realized for highly restricted combinatorial structures (doubly regular tournaments, skew Hadamard matrices), imposing severe constraints on dimensions and cardinalities.

6. Future Directions and Open Problems

Potential directions for advancing quantum C2 codes include exploring codes with increased generator weight $w$ to raise $d_C$, combining cyclic symmetry with graph-expansion methods such as progressive edge growth, and optimizing circuit schedules for simultaneous logical operations and ancilla shifts. Design of fault-tolerant logical gates (using lattice surgery or code-switching) and hardware-specific optimizations (for overlap between cyclic shifts and quantum gates) remain open for achieving further performance gains.

In classical quasi-cyclic codes of index 2, extension to higher index or composite-length settings, and exploitation of symplectic or Hermitian duality structures for cryptographic or post-quantum applications, remain active research streams.

The classification of tight and maximal complex spherical 2-codes is complete up to isomorphism; however, the construction of new families connected to previously unclassified combinatorial designs or the realization of such sets in physical quantum state deployments present plausible future investigations.