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 research, authored by Yongsheng Tang, Heqian Xu, Ting Yao, and Xiaoshan Kai, focuses on rings of the type F plus u times F, where F is the finite field with q raised to the power of 2m elements, q is an odd prime power, m is a positive integer, and u is an indeterminate satisfying the deceptively simple relation u squared equals one. Because u squares to one rather than to zero, the ring is not a chain ring; its ideals do not stack neatly in a single linear hierarchy. This seemingly technical distinction matters enormously. Chain rings have long been the workhorse of code construction over finite rings, but non-chain rings of this type offer a richer internal structure, and the new work shows how to harvest that richness for quantum coding purposes.

The central obstacle in building quantum codes from classical codes is that a quantum code cannot be assembled from just any classical code. The most productive construction routes pass through the so-called dual-containing condition: a classical code must contain its own dual, or more precisely its Hermitian dual, before it can be converted into a quantum stabilizer code. Verifying and engineering this condition directly over an unfamiliar ring is difficult. The Chinese team’s first key move is to define a class of Gray maps, functions that translate codewords over the ring R into codewords over the much better understood finite field with q to the 2m elements. Crucially, these maps are designed to preserve the Hermitian dual-containing property. If a linear code over the ring contains its Hermitian dual, then its Gray image is a linear code over the field that also contains its Hermitian dual. The property survives the journey across the map, and that survival is what makes the whole construction work.

Once the Gray maps are in place, the Hermitian construction takes over. This classical technique, rooted in the pioneering work of Calderbank, Rains, Shor, and Sloane in the late 1990s, converts a classical code that contains its Hermitian dual into a quantum code over a smaller alphabet. Applied to Hermitian dual-containing constacyclic codes over the ring R, the construction yields a new class of q raised to the m-ary quantum codes. Constacyclic codes are a natural generalization of cyclic codes: shifting a codeword cyclically multiplies it by a fixed constant lambda rather than leaving it unchanged. This extra flexibility, controlled by the unit u in the new setting, expands the family of available codes well beyond what cyclic codes alone can offer.

The second major contribution concerns primitive quantum BCH codes, an important family with strong distance properties. The authors take the Hermitian dual-containing u-constacyclic codes over R, apply the Gray maps to obtain their images over the field, and then extract the subfield subcodes of those images. A subfield subcode is obtained by restricting a code over a large field to symbols drawn from a smaller subfield, a process that typically improves the code’s minimum distance and produces parameters of genuine practical interest. Through this route, the paper determines a family of q-ary primitive quantum BCH codes, extending a line of research that stretches back to the influential work of Aly, Klappenecker, and Sarvepalli on quantum and classical BCH codes.

The third strand of the paper introduces a different type of map with a different destination. Instead of mapping Hermitian dual-containing codes over R to Hermitian dual-containing codes over the field, this second class of maps converts the Hermitian dual-containing property over the ring into the trace dual-containing property over the field. The trace dual-containing condition is the entry ticket for the Symplectic construction, an alternative route to quantum codes that produces codes over the smaller alphabet of size q raised to m. Using this second pipeline, the authors obtain yet another class of q raised to the m-ary quantum codes from the same pool of Hermitian dual-containing u-constacyclic codes over R. Two independent mechanisms, Hermitian and Symplectic, now feed off the same algebraic source, effectively doubling the harvest.

The technical machinery underlying these results is worth appreciating. A constacyclic code of length n over R can be represented as an ideal in a quotient ring of polynomials, and over rings of the form F plus uF the polynomial x raised to n minus lambda factors in a way that permits a complete description of all such codes through their generating polynomials. The Hermitian dual of such an ideal is again an ideal, described by a reciprocal polynomial relationship, and the dual-containing condition translates into divisibility constraints among the generators. The Gray maps then act coordinate-wise, expanding each ring symbol into a pair or block of field symbols, and the careful design of the maps ensures that the Hermitian inner product relations are maintained throughout. This interplay between ring-theoretic ideal structure, polynomial algebra, and linear maps over finite fields is the engine room of the entire paper.

What makes the contribution notable within the field is its place in a research trajectory that the same community has been steadily building. Tang, Zhu, Kai, and Ding produced early quantum codes from dual-containing cyclic codes over finite rings in 2016. Subsequent work by Tang and colleagues extended the approach to constacyclic codes over polynomial residue rings and to rings of the form F plus uF in characteristic two. Other groups, including Wang, Kai, Sun, and Zhu, explored Hermitian dual-containing constacyclic codes over rings of the form F plus vF with q squared elements. The new paper pushes the program into the case where the base field has q raised to 2m elements and the nilpotent-style indeterminate u squares to one rather than to zero, a combination that had not been systematically treated with both Hermitian and Symplectic constructions in parallel.

The practical significance of new code families lies in their parameters. A quantum code is characterized by its length, its dimension, and its minimum distance, the latter determining how many qubit errors it can correct. Codes with favorable combinations of these three numbers are scarce, and tables such as Markus Grassl’s codetables.de track the best known bounds. Every new construction that produces codes with competitive parameters enriches the toolbox available to theorists designing fault-tolerant protocols, and the authors report that the quantum codes emerging from their constructions include codes with good parameters, alongside families that are new additions to the known landscape of quantum error-correcting codes.

The work also carries conceptual weight for the mathematics of coding theory itself. Finite rings once sat at the periphery of coding research, viewed as curiosities compared to finite fields, but three decades of development have established them as a fertile source of classical codes with unexpected structure. The present study strengthens the bridge between ring-based classical coding and quantum stabilizer theory by demonstrating that dual-containing properties, the crucial currency of quantum constructions, can be transported across carefully chosen maps without loss. Each new bridge of this kind means that a larger body of classical algebraic knowledge can be repurposed for quantum applications, a pattern that has repeatedly accelerated progress in the field.

The paper also reflects the collaborative and well-supported state of Chinese research in quantum information mathematics. The work was supported by multiple grants from the National Natural Science Funds of China, together with funding from the Natural Science Foundation of Anhui Province and several provincial programs supporting research teams and young scientists. The authors acknowledge Doctor Sun Zhonghua for helpful suggestions that improved the presentation of the paper, and they declare no competing financial interests.

For a field racing toward practical quantum computers, incremental algebraic advances of this kind accumulate into real capability. Fault-tolerant quantum computation will demand families of error-correcting codes tailored to hardware constraints, and the mathematical repertoire from which such codes can be drawn determines how much design freedom engineers ultimately possess. By showing that finite non-chain rings of the form F plus uF, with u squared equal to one, can serve as reliable factories for quantum codes through both Hermitian and Symplectic constructions, Tang, Xu, Yao, and Kai have widened that repertoire in a rigorous and reusable way. The study appeared in Quantum Information Processing, volume 25, article number 305, after being received in November 2025 and accepted in August 2026, and it stands as a further demonstration that the deepest resources for protecting quantum information often lie in the oldest and most classical branches of algebra.