The quantum MDS conjecture

Quantum maximum-distance-separable (MDS) codes protect quantum information as well as the quantum Singleton bound allows. The conjecture says that for qudits of dimension $q$, such codes built from stabilizers are never longer than $q^2+1$, apart from one exception. This is the quantum counterpart of the classical MDS conjecture over a field with $q^2$ elements.

Problem

Let $q$ be a prime power. A stabilizer code $[[n,k,d]]_q$ is a $q^k$-dimensional subspace of $(\mathbb C^q)^{\otimes n}$: the joint $+1$ eigenspace of an abelian group of $n$-qudit Pauli operators that contains no nontrivial multiple of the identity. Its distance $d$ means it detects every error acting on fewer than $d$ qudits. Every quantum code obeys the quantum Singleton bound $k\le n-2d+2$, and a code is MDS when equality holds [5].

Conjecture. Every stabilizer MDS code $[[n,n-2(d-1),d]]_q$ with $d\ge3$ and $k\ge1$ has length $n\le q^2+1$, except when $q$ is even and $d=4$, where $n\le q^2+2$ [1, 3, 5].

A counterexample is a single stabilizer code above this limit. A proof must cover every prime power $q$, every $d\ge3$ and every $k\ge1$.

Background

Grassl, Beth and Rötteler raised the question of the maximal length of quantum MDS codes in 2004 [1]. In 2006 Ketkar, Klappenecker, Kumar and Sarvepalli argued that the classical MDS conjecture would imply the limit above, with $q^2+2$ allowed in sporadic cases [1]. The classical conjecture says a nontrivial linear MDS code of dimension $k$ over $\mathbb F_Q$ has length at most $Q+1$, except when $Q$ is even and $k\in\{3,Q-1\}$, where $Q+2$ is allowed [1]. Huber and Grassl stated the quantum version as a conjecture in 2020 [3]. A stabilizer code corresponds to an additive code over $\mathbb F_{q^2}$, closed under addition but not necessarily linear, so the implication actually needs the MDS conjecture for additive codes [4]. MDS codes with $k=0$ are absolutely maximally entangled states, a separate question [3].

What is known

• Every quantum MDS code with $d\ge3$, stabilizer or not, has $n\le q^2+d-2$ (Huber and Grassl, 2020; for stabilizer codes, Ketkar et al., 2006) [1, 3]. This proves the conjecture for $d=3$.
• For $q=2^m$ the codes $[[q^2+2,q^2-4,4]]_q$ of Grassl and Rötteler meet the bound $q^2+2$ at $d=4$, which is why the conjecture has an exception [2, 3].
• Huber and Grassl note that the conjecture is otherwise unresolved for $d>3$ [3]. The source record adds an argument through Bush's 1952 bound on orthogonal arrays that rules out $d=4$ for odd $q$ as well, so the open cases have $d\ge5$.
• Ball, Gamboa and Lavrauw (2023) classified the additive MDS codes over $\mathbb F_4$, $\mathbb F_8$ and $\mathbb F_9$. This proves the conjecture for $q=2$ and $q=3$. For $q=4$ a counterexample would have to come from an additive MDS code of length 18 over $\mathbb F_{16}$ that is not linear over $\mathbb F_4$ [5].
• Ball proved the classical MDS conjecture for linear codes over prime fields [3], but $q^2$ is never prime, and the conjecture for additive codes is open beyond small fields [4, 5].

What would count as progress

• Settling $q=4$: classify, or rule out, the additive MDS codes of length 18 over $\mathbb F_{16}$ that are not $\mathbb F_4$-linear [5]. A computer classification should be reproducible by an independent search.
• A counterexample for some $q$: an explicit stabilizer matrix of an MDS code longer than the limit. Its parameters can be checked by computer for small $q$.
• A proof of the MDS conjecture for additive codes over $\mathbb F_{q^2}$ for a new infinite family of $q$, which would prove the quantum conjecture for those $q$ [4].

References

1. A. Ketkar, A. Klappenecker, S. Kumar and P. K. Sarvepalli, "Nonbinary stabilizer codes over finite fields", IEEE Transactions on Information Theory 52, 4892–4914 (2006). doi:10.1109/TIT.2006.883612
2. M. Grassl and M. Rötteler, "Quantum MDS codes over small fields", 2015 IEEE International Symposium on Information Theory (ISIT), 1104–1108 (2015). doi:10.1109/ISIT.2015.7282626
3. F. Huber and M. Grassl, "Quantum codes of maximal distance and highly entangled subspaces", Quantum 4, 284 (2020). doi:10.22331/q-2020-06-18-284
4. S. Ball, A. Centelles and F. Huber, "Quantum error-correcting codes and their geometries", Annales de l'Institut Henri Poincaré D 10, 337–405 (2023). doi:10.4171/aihpd/160
5. S. Ball, G. Gamboa and M. Lavrauw, "On additive MDS codes over small fields", Advances in Mathematics of Communications 17, 828–844 (2023). doi:10.3934/amc.2021024

Source

Imported from QIQCOP Zoo op_2a6fc2b70a85f846, curated by Bikun Li, Qicheng Tang, Changhao Li, Chengkai Zhu, Minbo Gao, Zhong-Xia Shang, Bin Cheng and Naixu Guo. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.