Quantum MDS Codes

• Quantum MDS codes are quantum error-correcting codes that meet the quantum Singleton bound, optimally balancing code length, logical qudits, and minimum distance.
• They are constructed using advanced techniques like Hermitian GRS codes and constacyclic methods, ensuring self-orthogonality and superior error correction.
• Recent developments include entanglement-assisted and convolutional variants, expanding parameter ranges and enhancing the robustness of quantum communication systems.

Quantum maximum-distance-separable (MDS) codes, or quantum MDS codes, are a distinguished class of quantum error-correcting codes that saturate the quantum Singleton bound, offering the best-possible tradeoff between code length, number of logical qudits, and minimum distance. The construction and classification of quantum MDS codes—especially those with large minimum distance or for a wide spectrum of parameters—constitutes a foundational topic in quantum coding theory, with profound connections to both classical algebraic coding and quantum information.

1. Fundamental Definitions and Quantum Singleton Bound

Quantum error-correcting codes encode $k$ logical qudits into $n$ physical qudits of dimension $q$, denoted $[[n,k,d]]_q$, where $d$ is the code's minimum distance. The quantum Singleton bound, proved for stabilizer codes, reads: $2d \leq n - k + 2$ Codes achieving equality are called quantum MDS codes. Quantum MDS codes are sought after for their optimality in correcting errors and, in many parameter regimes, their constructed families establish limits for quantum communication and storage (Wang et al., 2014, Liu et al., 2020, Sari et al., 2017, Wan et al., 2023).

2. The Hermitian Construction and Dual-Containing Codes

Most infinite families of quantum MDS codes are constructed via the Hermitian construction (sometimes called the Ashikhmin–Knill or Hermitian CSS method), a stabilizer code framework using classical codes over $\mathbb{F}_{q^2}$. Given an $[n,k_c,d_c]_{q^2}$ code $C$ such that $C^{\perp_H}\subseteq C$ (Hermitian self-orthogonal), the induced quantum code has parameters $[[n,2k_c-n,\geq d_c]]_q$. If $C$ is classical MDS, then the quantum code also attains the quantum Singleton bound, making it a quantum MDS code (Wang et al., 2014, Jin et al., 2013, 0812.4514).

The self-orthogonality can be realized via explicit conditions on the defining set of a constacyclic code, or in terms of systems of homogeneous equations for GRS codes. For constacyclic constructions, dual-containment is typically equivalent to a defining-set disjointness criterion $Z\cap(-q)Z=\emptyset$ modulo some group order, derived from the algebraic structure of the code and the specific inner product (Wang et al., 2014, Chen et al., 2014, Lu et al., 2018).

3. Major Construction Methodologies

A. Quantum MDS Codes from Generalized Reed–Solomon (GRS) Codes

The GRS framework underlies many quantum MDS families. Explicitly, for $q$ a prime power, the classical GRS code over $\mathbb{F}_{q^2}$ of length $n$, with distinct evaluation points $a_1,\ldots,a_n$ and multipliers $v_1,\ldots,v_n$, is Hermitian self-orthogonal if

$\sum_{i=1}^{n} v_i^{q+1} a_i^{qi+j} = 0$

for all $0\leq i,j < k$, where $k$ is the dimension (0812.4514, Wan et al., 2023, Fang et al., 2018). Meeting these equations often involves partitioning the evaluation points into cosets of subgroups or applying linear-algebraic criteria.

Classic GRS-based quantum MDS codes include:

• $[[n, n-2k, k+1]]_q$ for GRS codes of length $n$ and dimension $k$ (0812.4514), and
• Infinite families for various $n$ constructed by parameterizing the coset structure, including $n=q^2+1$, $n=(q^2+2)/3$ (for $3\,|\,q+1$), or $n$ dividing $q^2-1$ or $q^2+1$ (Jin et al., 2013, Wan et al., 2023, Wan, 2023).

B. Quantum MDS Codes from Constacyclic Codes

Constacyclic codes over $\mathbb{F}_{q^2}$ extend cyclic code constructions, using automorphisms defined by roots of unity of order dividing $q+1$ or $q-1$. For lengths $n=\lambda(q-1)$ using $q+1=\lambda r$ (with $r$ even or odd), families of quantum MDS codes were constructed with parameters $[[n, n-2d+2, d]]_q$ for suitable $d$ [$2\le d\le (q+1)/2+\lambda-1$ or $(q+1)/2+\lambda/2-1$], often exceeding the $q/2+1$ minimum distance barrier as soon as $\lambda>1$ (Wang et al., 2014).

Families leveraging cyclotomic coset analysis, BCH bounds, and algebraic number theory yield further codes for $n=(q^2-1)/r$ and $n=(q^2+1)/r$, breaking previous minimum-distance records for many parameter regimes (Sari et al., 2017, Chen et al., 2014, Lu et al., 2018).

C. Other Constructions and Parameter Ranges

Quantum MDS codes have been constructed for a broad spectrum of parameter sets—via extended versions of GRS codes, hybrid cyclic/constacyclic approaches, and by utilizing intersection properties of cosets in multiplicative subgroups (Fang et al., 2019, Wang et al., 2024). Codes with lengths $n\equiv 0, 1\, (\mathrm{mod}\, (q\pm1)/2)$ and minimum distances significantly exceeding $q/2+1$ have recently been reported (Wan, 2023).

A key structural insight from the theory is that, under the (classical) MDS conjecture, $q$-ary quantum MDS codes can exist only for $n\leq q^2+1$, with some families attaining the full MDS length spectrum for lengths up to $q^2+1$ or $q^2+2$ (the latter for $q=2^m$) (Grassl et al., 2015).

4. Explicit Parameters and Achievable Domains

A wide variety of parameter sets is realized within this framework. The table below summarizes some pivotal constructions (where $q$ is a prime power):

Construction Type	Length $n$	Dimension $k$	Minimum Distance $d$	Key Reference
GRS-based (extended)	$q^2+1$	$n-2d+2$	$d$, $2\leq d\leq q+1$	(Jin et al., 2013, Ball, 2019)
Constacyclic $r|q+1$, even $r$	$\lambda(q-1)$	$n-2d+2$	$2\leq d\leq (q+1)/2+\lambda-1$	(Wang et al., 2014)
Constacyclic $r|q+1$, odd $r$	$\lambda(q-1)$	$n-2d+2$	$2\leq d\leq (q+1)/2+\lambda/2-1$	(Wang et al., 2014)
GRS/Hybrid coset-based	arbitrary $n\equiv 0,1\,(\mathrm{mod}\,(q\pm1)/2)$	$n-2d$	$d=k+1>q/2+1$	(Wan, 2023)
Cyclic over $q^2$	$n=(q^2+1)/a$	$n-2d+2$	$2\leq d\leq$ (range linear in $q$)	(Lu et al., 2018)

Explicit examples are worked out for small $q$ throughout the literature (e.g., $q=5$ with $n=26$, $d=2,3,4,6$ (Jin et al., 2013)).

Important recent constructions (e.g., (Wang et al., 2024, Wan et al., 2023, Fang et al., 2018)) systematically expand the known quantum MDS lengths and distance domains, including many new $n > q+1$ cases with $d > q/2+1$.

5. Entanglement-Assisted and Convolutional Quantum MDS Codes

Generalizing the standard (stabilizer) formalism, entanglement-assisted quantum MDS (EAQMDS) codes allow quantum codes to be built from any classical code via the use of pre-shared entanglement between sender and receiver. The necessary condition of self-orthogonality is relaxed; instead, the entanglement consumption parameter $c$ is determined by the dimension of the code hull or the rank of the gram matrix of the parity-check matrix (Chen, 2022, Zheng et al., 2022, Jin et al., 2019).

The parameters are of the form $[[n, n-k-h, k+1, k-h]]_q$, with flexibility in entanglement consumption and code length, and EAQMDS codes exist for all $n\leq q^2+1$, covering the "second MDS range" beyond $q+1$ (Chen, 2022).

Quantum MDS convolutional codes, constructed from Hermitian self-orthogonal GRS codes, further extend the theory to the setting of streaming (memory-based) quantum error correction. Eighteen new infinite families are known, with free distances achieving the generalized convolutional quantum Singleton bound (Ding et al., 2015).

6. Significance, Classification, and Open Questions

Quantum MDS codes unify and far generalize known quantum code constructions. The GRS-based framework is particularly powerful: for every previously known stabilizer quantum MDS code, there exists a Hermitian self-orthogonal GRS code with the same parameters (0812.4514). The landscape is now understood to comprise codes derived from GRS, extended GRS, constacyclic, and coset-based constructions, as well as their hybrid and entanglement-assisted variants.

Key advances include:

• Extension of the minimum distance well beyond the $q/2+1$ barrier for many new lengths, especially as $q$ grows (Jin et al., 2013, Wang et al., 2014, Wan, 2023).
• Resolution of open existence problems for certain lengths, e.g., for $n=l^2+1$ and $d=l$ (Liu et al., 2020).
• Filling of previously unattainable parameter domains, both in terms of $n$ and $d$, via algebraic, combinatorial, and number-theoretic techniques (Fang et al., 2019, Fang et al., 2018, Wan et al., 2023).

Open problems include the search for families with $d > q+1$ (except a handful of sporadic codes (Ball, 2019)), the classification for all $n\leq q^2+1$, the complete characterization of hull dimensions for Hermitian self-orthogonal GRS codes, and the exploration of analogous constructions in asymmetric quantum and finite-rate entanglement-assisted settings.

7. Representative References

The following primary research articles underlie the contemporary theory of quantum MDS codes:

• "Quantum generalized Reed-Solomon codes: Unified framework for quantum MDS codes" (0812.4514)
• "A Construction of New Quantum MDS Codes" (Jin et al., 2013)
• "New quantum MDS codes derived from constacyclic codes" (Wang et al., 2014)
• "New quantum mds constacylıc codes" (Sari et al., 2017)
• "Some constructions of quantum MDS codes" (Ball, 2019)
• “Constructions of quantum MDS codes” (Liu et al., 2020)
• "Quantum MDS Codes with length $n\equiv 0,1($mod$\,\frac{q\pm1}{2})$" (Wan, 2023)
• "New Quantum MDS codes from Hermitian self-orthogonal generalized Reed-Solomon codes" (Wan et al., 2023)
• "Some New Constructions of Quantum MDS Codes" (Fang et al., 2018)
• "Quantum MDS Codes over Small Fields" (Grassl et al., 2015)
• "New MDS Entanglement-Assisted Quantum Codes from MDS Hermitian Self-Orthogonal Codes" (Chen, 2022)
• "New constructions of quantum MDS convolutional codes derived from generalized Reed-Solomon codes" (Ding et al., 2015)

These references comprehensively cover the principal methodologies, results, comparison with prior constructions, and open research directions in the field of quantum MDS codes.