The central objects of the study live on an unusual mathematical landscape. Rather than working over a single finite field F_q, the authors work over the direct product R_q = F_q × (F_q + vF_q), where the symbol v satisfies the idempotency relation v² = v. The second component, F_q + vF_q, is a small ring of characteristic p (where q = p^s) containing a nilpotent-free but non-field element; elements of this ring have the form a + bv with a and b in F_q, and multiplication follows from v² = v. Because a vector space over this ring decomposes neatly into a direct sum of two copies of F_q, codes over R_q behave like “mixed” codes that blend two field-based code components of different sizes into one structure. Codes of this kind generalize a lineage of constructions studied over the past two decades, from Z_2Z_4-additive cyclic codes through Z_2Z_2[u]-cyclic and constacyclic codes, and they are attractive to coding theorists precisely because a single code over R_q can yield several different codes over F_q simultaneously.

The “constacyclic” property is the structural heart of the paper. A linear code of length n over R_q is constacyclic if shifting every coordinate cyclically and multiplying by a fixed unit constant λ maps the code back to itself; in the present work the unit is λ = (1, 1−2v) in the product ring. When λ = 1 such codes are cyclic, and when λ = −1 they are negacyclic, so constacyclic codes encompass both classical cases. The researchers first construct two Gray maps, functions that translate length-n codewords over R_q into length-3n codewords over the plain field F_q. These maps are distance-preserving in an appropriate sense, which means that parameters such as the Hamming distance of the resulting field code can be controlled through the structure of the original code over the ring. Gray maps of this type are the standard bridge from ring-based coding theory to the finite-field codes that ultimately specify quantum code parameters, and having two distinct maps gives the construction extra flexibility in how the two ring components are unpacked into field symbols.

With the Gray maps in place, the paper delivers a full algebraic characterization of all (1, 1−2v)-constacyclic codes of length n over R_q and, crucially, of their dual codes. Because the length-n shift over the product ring splits naturally according to the two factors F_q and F_q + vF_q, every constacyclic code decomposes into a pair of constacyclic codes over the field component and the ring component respectively. Each component is generated by a single polynomial factor of x^n − λ modulo the ambient ring polynomial, so the entire code family is parametrized by a small set of divisor polynomials. The duals satisfy a corresponding factorization: the dual of a constacyclic code with unit λ is constacyclic with reciprocal unit λ^(−1), and the generating polynomials of the dual are reciprocal to the original ones. This clean polynomial description is what makes the family tractable for the quantum constructions that follow.

A distinctive feature of the study is its detailed treatment of Euclidean hulls and Euclidean sums. The Euclidean hull of a code C is the intersection C ∩ C^⊥, where C^⊥ denotes the dual under the standard Euclidean inner product; the hull measures how much of a code is self-orthogonal. Hulls have become a hot topic in recent coding theory because the dimension of the hull governs how many entanglement-assisted resources a quantum code derived from C would require, and because hull-variability problems connect to algebraic-geometry questions about finite fields. Liu and Liu determine, for every (1, 1−2v)-constacyclic code, the precise structure of its hull and of the Euclidean sum C + C^⊥, again expressed through the factorization of generating polynomials. This means a researcher can now read off the self-orthogonality properties of any code in the family directly from its polynomial description, without performing brute-force inner-product computations on generator matrices.

The quantum payoff arrives through two classical-to-quantum conversion recipes. The first is Steane’s construction, the 1996 enlargement method that builds a quantum stabilizer code from a pair of nested classical codes in which one code contains the dual of the other — the ancestor of the celebrated Calderbank–Shor–Steane (CSS) scheme, which itself grew out of Peter Shor’s pioneering 1995 nine-qubit code. The second is “quantum construction X,” a propagation technique in the spirit of Construction X from classical coding theory, which enlarges a code by combining it with auxiliary codes to push its minimum distance upward while keeping the dimension favorable. Applied to the Euclidean sums and hulls of the (1, 1−2v)-constacyclic codes — paired with auxiliary linear codes of the same length over R_q — these two methods yield families of q-ary QEC codes whose parameters [[n, k, d]] encode the number of physical qubits protected, the number of logical qubits carried, and the number of errors that can be corrected.

To demonstrate that the theory is not merely formal, the authors construct concrete examples of new QEC codes arising from the Euclidean sums and hulls of their constacyclic codes. The stated purpose is to enrich the variety of available quantum error-correcting codes, a goal that matters because tables of best-known quantum code parameters still contain many gaps. Every new [[n, k, d]] code with parameters competitive against existing entries is a potential asset for quantum communication protocols, since larger minimum distances translate directly into lower logical error rates for a fixed physical overhead. The mixed-ring setting is particularly effective at generating codes whose parameters would be awkward to reach through straightforward field-based constructions, because the two ring components contribute code components of differing field sizes that merge into richer composite structures after the Gray map is applied.

The broader context of this line of research stretches back to the foundations of quantum error correction. Shor’s 1995 scheme demonstrated that quantum information, despite its extreme fragility under decoherence, could be redundantly encoded; Steane and Calderbank, Rains, Shor and Sloane then established the stabilizer formalism and the CRSS framework for nonbinary stabilizer codes over finite fields, later generalized by Ashikhmin and Knill. Since then, a large research community has mined families of classical codes — BCH codes, cyclic codes, negacyclic codes, skew constacyclic codes, and codes over an expanding zoo of finite rings including F_q + uF_q, F_q + vF_q + v²F_q, and various non-chain rings — for quantum constructions. Recent contributions in Quantum Information Processing and related journals have extracted quantum maximum-distance-separable codes, entanglement-assisted codes, and quantum synchronizable codes from such families. The present work extends this program to the product ring F_q × (F_q + vF_q) with a constacyclic unit that is neither 1 nor −1, filling a previously open case.

Why do mathematicians persist in exploring ever-more-exotic rings for quantum codes? The answer lies in a trade-off between algebraic convenience and parameter richness. Rings with idempotent or nilpotent elements allow codes to be assembled from several field-level components at once, so that a single well-chosen constacyclic code over the ring can produce multiple distinct q-ary quantum codes with different lengths and distances after Gray mapping. Moreover, the constacyclic property preserves the cyclic symmetry that makes encoding and decoding circuits efficient — a property that matters practically, since a code that cannot be encoded and decoded with manageable circuit depth offers little benefit to a quantum computer designer regardless of its theoretical parameters. The complete duality theory developed by Liu and Liu ensures that the self-orthogonality conditions required by Steane’s construction can be verified at the polynomial level, streamlining the search for good quantum codes dramatically compared with matrix-level approaches.

The authors acknowledge support from the Research Funds of Hubei Province (Grant No. Q20164505) and the talent project of Hubei Polytechnic University (Grant No. 16xjzo8R). Both authors contributed equally to the work, which was received by the journal on 7 May 2025, accepted on 14 July 2026, and classified under the mathematics subject classifications 94B15 and 94B65, covering linear codes over rings and quantum coding theory respectively. The paper reports that no datasets were generated or analyzed beyond the theoretical constructions themselves.

For the quantum computing community, the study arrives at a moment when the demand for good error-correcting codes is intensifying. As hardware platforms scale toward hundreds and thousands of physical qubits, the question of which classical code families feed the best quantum stabilizer constructions has become an active frontier of applied mathematics. The complete structural theory of (1, 1−2v)-constacyclic codes over F_q × (F_q + vF_q) — their Gray images, duals, hulls, and sums — hands researchers a new, fully mapped territory in which to search for quantum codes with improved parameters, and the concrete examples included in the paper provide immediate entry points into databases of best-known quantum codes. Whether the next generation of fault-tolerant quantum machines will use codes born from mixed product rings remains an open question, but the algebraic inventory from which such codes may be drawn has just grown measurably larger.

The work, published in the journal Quantum Information Processing by Ruihu Li and Yang Liu of Air Force Engineering University in Xi’an, Yuezhen Ren of Xi’an Polytechnic University, and Chaofeng Guan of Zhengzhou University, establishes a systematic correspondence between symplectic subspaces—geometric structures living inside the symplectic group over finite fields—and quaternary additive codes, the algebraic objects that encode the parameters of entanglement-assisted stabilizer codes. This correspondence is more than an aesthetic curiosity. It provides a constructive bridge that allows the authors to translate long-standing open problems about optimal EAQECCs into geometric questions about subspaces, where powerful counting and classification techniques become available. In doing so, the team has resolved several open problems concerning optimal entanglement-assisted codes and entanglement-assisted quantum maximum distance separable codes, a class of codes that achieve the best possible trade-off between length, dimension, and error-correcting capability.

To appreciate why this matters, it helps to trace the lineage of the field. Quantum error correction was born in the mid-1990s, when Peter Shor demonstrated in 1995 that a quantum state could be protected by spreading it across nine physical qubits, and Andrew Steane independently showed shortly afterward that error-correcting ideas from classical coding theory could be imported into the quantum setting. The modern framework of stabilizer codes, crystallized in Daniel Gottesman’s 1997 doctoral thesis, recast the problem in the language of the Pauli group: quantum states are protected by measuring operators that form an abelian subgroup, the stabilizer, whose eigenvalues reveal the syndrome of an error without revealing the encoded information. In 1998, Calderbank, Rains, Shor, and Sloane made a decisive connection to classical coding theory by showing that stabilizer codes over qubits correspond to self-orthogonal additive codes over the finite field GF(4). This correspondence turned the search for good quantum codes into a problem about classical codes satisfying a self-orthogonality constraint, and it has driven the field ever since.

But the self-orthogonality requirement is a straitjacket. Requiring the stabilizer to be an abelian group means the associated classical code must be self-orthogonal with respect to a suitable inner product, and many excellent classical codes fail that test. Entanglement assistance, introduced by Todd Brun, Igor Devetak, and Min-Hsiu Hsieh in a landmark 2006 paper in Science, removes this restriction. The central idea is elegant: if the stabilizer generators fail to commute, the anticommuting parts can be absorbed by sharing pre-existing entangled pairs—ebits—between the sender and receiver. With a supply of entanglement, essentially any classical quaternary code, self-orthogonal or not, can be converted into a quantum code. An EAQECC with parameters [[n, k, d; c]] encodes k logical qubits into n physical qubits, corrects errors of weight up to the floor of (d−1)/2, and consumes c shared ebits in the process. The entanglement acts as a catalytic resource, purchased in advance and consumed to buy superior error-correcting performance.

The cost of that resource makes the parameter c a central object of study. Researchers including Mark Wilde and Todd Brun derived optimal entanglement formulas that determine the minimum number of ebits a code needs, while Chi-Kwong Lai and collaborators developed duality theories and linear-programming bounds tailored to the entanglement-assisted setting. A rich landscape of bounds—the quantum Singleton bound, the quantum Hamming bound, and entanglement-assisted variants—governs what parameter triples are achievable, and codes that meet these bounds with equality are prized as optimal. Entanglement-assisted quantum maximum distance separable codes, or EAQMDS codes, are the quantum analogues of the celebrated classical MDS codes: they saturate the Singleton bound, meaning their minimum distance is as large as algebraically possible for their length and dimension. Constructing such codes explicitly, for many different lengths and over many field sizes, is one of the most active pursuits in quantum coding theory, and one where the new work makes its most striking contribution.

The key technical insight of the new paper lies in how it characterizes EA stabilizer codes. In the additive-code picture, an entanglement-assisted code is built from a pair of codes C and D over GF(4), or equivalently from a single additive code together with information about its symplectic dual. The number of ebits required, the dimension of the encoded space, and the minimum distance all translate into combinatorial properties of these codes. What Li and colleagues show is that these properties can be understood through the geometry of the symplectic group Sp(2m, q), the group of linear transformations preserving a symplectic form on a 2m-dimensional vector space over a finite field. Subspaces of this vector space come in families classified by their dimension and by how they intersect their own symplectic orthogonal complements—whether they are totally isotropic, nonisotropic, or something in between. The authors establish precise relations between such symplectic subspaces and the quaternary additive codes arising in EAQECC constructions, allowing parameters of EA stabilizer codes to be read off directly from geometric data.

This geometric dictionary pays off immediately. Counting arguments over families of symplectic subspaces—an approach pioneered by Zhexian Wan in his monograph on the geometry of classical groups over finite fields, and long a staple of finite-geometry-based coding constructions—let the researchers establish the existence of EAQECCs with parameter sets that had eluded previous constructions, and in several cases to prove optimality where only bounds existed before. The work also builds on the authors’ own earlier discoveries: in 2023, Guan, Li, Liu, and Ma showed in IEEE Transactions on Information Theory that certain quaternary additive codes genuinely outperform their linear counterparts, a finding that highlighted how relaxing linearity enlarges the space of good codes. The additive setting is technically delicate—quaternary additive codes need not be linear over GF(4), which complicates classical duality theory—but the symplectic geometric framework embraces them naturally, since symplectic orthogonality is defined at the level of vector spaces over the base field GF(2).

Beyond resolving existence and optimality questions, the authors point to a second, practical payoff: the design of encoding and decoding quantum circuits for EA stabilizer codes. The stabilizer formalism translates directly into Clifford-group circuits, with each stabilizer generator corresponding to a measured operator implemented by controlled-Pauli gates, and the symplectic representation makes these circuits explicit: Clifford operations act as symplectic transformations on the binary representation of Pauli operators. By grounding EAQECC parameters in symplectic subspace structure, the new framework provides a cleaner route from a code’s abstract definition to the concrete circuitry a hardware engineer would deploy, potentially easing the path from mathematical construction to working fault-tolerant logic.

The broader context makes the contribution timely. As quantum processors scale from hundreds toward thousands of qubits, the overhead of error correction dominates resource estimates for useful quantum computation. Entanglement-assisted schemes occupy an interesting position in this economy: they demand a communication and entanglement-distribution infrastructure, since ebits must be established between communicating parties before coding begins, but in exchange they relax the algebraic constraints on the underlying classical codes, often yielding shorter codes or larger minimum distances for the same block length. In settings where entanglement distribution is feasible—such as future quantum networks and quantum repeater chains, where entanglement is precisely the commodity being distributed—EAQECCs could be the natural encoding layer. Knowing exactly which parameter sets are attainable, and which are optimal, tells network architects what protection they can buy for a given budget of qubits and ebits.

The paper also contributes to a longer-running conversation about fundamental bounds. Recent years have seen intense scrutiny of the quantum Singleton bound and its entanglement-assisted generalizations, including entropic proofs by Markus Grassl, Florian Huber, and Andreas Winter, and results showing that entanglement can allow codes to beat the ordinary Singleton bound. Precise constructions that saturate or approach these bounds, anchored in firm geometric ground as the new work proposes, sharpen our understanding of the ultimate limits of quantum communication. With the field’s standard reference tables, maintained by Grassl at codetables.de, still containing many open entries for EAQECC parameters, techniques that systematically generate new codes and prove their optimality are exactly what the community needs to fill in the map.

The research was supported by the National Natural Science Foundation of China under Grant No. U21A20428 and by the Natural Science Foundation of Shaanxi Province. It arrives as part of a visible surge of mathematical sophistication in quantum coding theory, where finite geometry, combinatorial design theory, and algebraic curves over finite fields all supply construction tools. What distinguishes the present contribution is the systematic nature of the link it forges: rather than producing isolated parameter sets, it offers a framework in which entire families of entanglement-assisted codes can be analyzed, compared, and optimized through the lens of symplectic geometry. As the demand for efficient, hardware-compatible error correction intensifies, such unifying mathematical structures may prove to be among the most valuable assets the field has.