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No binary [[14,3,d≥5]] quantum stabilizer code exists

Brandon Li

Published Sep 22, 2026• 0 citations
Publisher

Abstract

We prove that no binary qubit stabilizer code with parameters [[14,3,d>=5]] exists, settling the remaining distance gap for stabilizer codes encoding three qubits into fourteen. The proof first applies a signed affine-shadow inequality and an integrality congruence to restrict low-weight stabilizer words. Exact split-shadow infeasibility certificates then exclude weight-two words and all configurations with three weight-four checks, leaving a unique weight-four check. That check yields a ten-coordinate binary-additive code of size 1,024, minimum distance at least five, and symplectic hull dimension four. We reconstruct all 37 relevant equivalence classes by exhaustive lengthening: 25 have the wrong hull dimension, and the other twelve violate a necessary 64-coset capacity bound. Together with the known [[14,3,4]] construction, this proves d_max(14,3)=4 for binary stabilizer codes.

DOI
10.5281/zenodo.22885774

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We prove that no binary qubit stabilizer code with parameters [[14,3,d>=5]] exists, settling the remaining distance gap for stabilizer codes encoding three qubits into fourteen. The proof first applies a signed affine-shadow inequality and an integrality congruence to restrict low-weight stabilizer words. Exact split-shadow infeasibility certificates then exclude weight-two words and all configurations with three weight-four checks, leaving a unique weight-four check. That check yields a ten-coordinate binary-additive code of size 1,024, minimum distance at least five, and symplectic hull dimension four. We reconstruct all 37 relevant equivalence classes by exhaustive lengthening: 25 have the wrong hull dimension, and the other twelve violate a necessary 64-coset capacity bound. Together with the known [[14,3,4]] construction, this proves d_max(14,3)=4 for binary stabilizer codes.
We prove, with computer assistance, that the optimal minimum distance of abinary stabilizer code on 141414 qubits encoding 333 logical qubits is 444.Equivalently, no binary quantum stabilizer code with parameters$\code{14,3,5}$ exists, including degenerate codes. In the binary symplecticmodel, this is equivalent to the nonexistence of an 111111-dimensional isotropicsubspace $C\subseteq\F_2^{28}$ satisfying$\min\{\wt(v):v\in C^{\perps}\setminus C\}\ge 5$.The argument separates the two cases determined by the parity of thestabilizer weights. MacWilliams identities and integrality eliminate theall-even case. In the odd case, the shadow enumerator of Rains is combined withtwo averages over the 135135135 self-dual extensions of the code: one for theirweight enumerators and one for their shadows. An incidence computation inan eight-dimensional orthogonal geometry of plus type then forces threelow-weight stabilizer profiles. Integrality and a collision count amongweight-three shadow words eliminate these profiles. All parameter-specificcalculations are finite and exact, and reproducible source code accompanies the paper.The nonexistence theorem is additionally formalized and machine-checked inLean~4 over mathlib, with an axiom audit reporting only standard foundationalprinciples.
Quantum low-density parity-check (qLDPC) codes can be implemented by measuring only low-weight checks, making them compatible with noisy quantum hardware and central to the quest to build noise-resilient quantum computers. A fundamental open question is how constraints on check weight limit the achievable parameters of qLDPC codes. Here, we study stabilizer and subsystem codes with constrained check weight, combining analytical arguments with numerical optimization to establish strong upper bounds on their parameters. We show that stabilizer codes with checks of weight at most three cannot have nontrivial distance. We also prove tight tradeoffs between rate and distance for broad families of CSS stabilizer and subsystem codes with checks of weight at most four and two, respectively. Notably, our bounds are applicable to general qLDPC codes, as they rely only on check-weight constraints without assuming geometric locality or special graph connectivity. In the finite-size regime, we derive numerical upper bounds using linear programming techniques and identify explicit code constructions that approach these limits, delineating the landscape of practically relevant qLDPC codes with tens or hundreds of physical qubits.
Low check weight is a crucial code property for fault-tolerant quantum computing, which underlies the strong interest in quantum low-density parity-check (qLDPC) codes. Here, we initiate the theory of weight-constrained stabilizer codes from various foundational perspectives including the complexity of computing code weight and the explicit boundary of feasible low-weight codes in both theoretical and practical settings. We first prove that computing the optimal generator weight of a stabilizer code is NP\mathsf{NP}NP-hard, motivating efficiently computable bounds. We derive analytical lower bounds on check weight in terms of code rate and distance, identifying the minimum weights needed for single-qubit error detection and correction, as well as the sharp distance and rate limits of weight-three error-detecting codes. Matching constructions show that these bounds are tight in several regimes. To establish refined finite-size constraints, we develop a linear programming framework based on quantum weight enumerators subject to generator-weight constraints, yielding exact optimal weights for all parameter combinations with n≤9n\le9n≤9. Finally, we show that the same framework can incorporate architecture-dependent constraints, using the 127-qubit IBM Eagle chip as a concrete example. Our study brings the weight as a crucial parameter into coding theory and provides guidance for code design and utility in practical scenarios.
In this research, we propose a novel construction of quantum stabilizer code based on a binary formalism. First, from any binary vector of even length, we generate the parity-check matrix of the quantum code from a set composed of elements from this vector and its relations by shifts via subtraction and addition. We prove that the proposed matrices satisfy the condition constraint for the construction of quantum codes. Finally, we consider some constraint vectors which give us quantum stabilizer codes with various dimensions and a large minimum distance with code length from six to twelve digits.
Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define XXX and ZZZ-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with XXX-, YYY-, and ZZZ-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the YYY-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ2^22 hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.
Abstract Calderbank, Rains, Shor, and Sloane (see [10]) showed that quantum stabilizer codes correspond to additive quaternary codes in binary projective spaces, which are self‐orthogonal with respect to the symplectic form. A geometric description is given in [8, 19]. In [8] the notion of a quantum cap is introduced. Quantum caps are equivalent to quantum stabilizer codes of minimum distance when the code is linear over . In this paper, we determine the valuesksuch that there exists a quantumk‐cap in , corresponding to pure linear quantum codes, proving, by exhaustive search, that no 11, 37, 39‐quantum caps exist. Moreover we give examples of quantum caps in not already known in the literature.