Two-error perfect quantum codes over non-prime-power local dimensions

Problem

Does a nontrivial pure perfect quantum code correcting arbitrary errors on two qudits exist for some local dimension \(q\) with at least two distinct prime factors? Let \(q\geq2\) and \(n\geq1\) be integers, and let \(\mathcal C\subseteq(\mathbb C^q)^{\otimes n}\) be a code of integer dimension \(K>1\) with orthogonal projector \(P\). No stabilizer structure is assumed, and \(K\) need not be a power of \(q\). On one site let \(X\lvert j\rangle=\lvert j+1 \bmod q\rangle\) and \(Z\lvert j\rangle=\omega^j\lvert j\rangle\) with \(\omega=e^{2\pi i/q}\), and use the phase-free Weyl basis \(\{X^aZ^b: 0\leq a,b\leq q-1\}\) of \(q^2\) operators, including the identity. Let \(\mathcal E_2\) be the set of \(n\)-fold tensor products of these operators with weight at most two, where weight counts non-identity sites. The code is a pure two-error-correcting code when

\begin{equation} PE^\dagger FP=\delta_{E,F}\,P, \qquad E,F\in\mathcal E_2, \tag{1} \end{equation}

where \(\delta_{E,F}\) is the Kronecker delta, so that distinct correctable errors map the code space to mutually orthogonal subspaces. Put

\begin{equation} V_2(n,q):=\lvert\mathcal E_2\rvert =1+n(q^2-1)+\binom n2(q^2-1)^2 . \tag{2} \end{equation}

Equation (1) implies \(K\,V_2(n,q)\leq q^n\), and the code is Hamming-perfect when

\begin{equation} K\,V_2(n,q)=q^n . \tag{3} \end{equation}

The question asks whether Eqs. (1) and (3) hold simultaneously for some \(n\), some integer \(K>1\), and some \(q\) that is not a prime power. A complete answer either exhibits such a code or proves that none exists for any non-prime-power \(q\).

Source

Cazorla García proposes, in Section 5.2 of the arXiv version, adapting his Diophantine method for classical perfect two-error-correcting codes over non-prime-power alphabets to perfect two-error-correcting quantum codes, noting that the classification of Li and Xing assumes prime-power local dimension [Caz24], [LX13]. The question is derived from this documented gap; neither source states it as a named problem or conjecture.

Progress

Every Weyl operator of weight between \(1\) and \(4\) is proportional to \(E^\dagger F\) for some distinct \(E,F\in\mathcal E_2\), so Eq. (1) forces distance at least five. Rains’s quantum Singleton bound \(K\leq q^{n-2d+2}\) holds for every integer alphabet size (Theorem 2 of the arXiv version) and gives \(K\leq q^{n-8}\). Combined with Eq. (3) and \(K\in\mathbb Z\), it yields the necessary conditions

\begin{equation} V_2(n,q)\geq q^8,\qquad V_2(n,q)\mid q^n,\qquad n\geq 9 . \tag{4} \end{equation}

These are direct deductions, not separate existence theorems [Rai99].
For prime-power \(q\), Li and Xing prove a quantum analogue of Lloyd’s theorem (Theorem 7) and show that the dimension of a pure perfect code is a power of \(q\) (Lemma 8). Theorem 9 of the arXiv version concludes that every nontrivial pure perfect quantum code has parameters \(\bigl(\bigl((q^{2l}-1)/(q^2-1),\,q^{n-2l},\,3\bigr)\bigr)_q\), so minimum distance three, whether or not it is a stabilizer code. Hence no pure perfect two-error-correcting code exists at prime-power local dimension. This classifies parameters, not codes up to equivalence, and the prime-power hypothesis enters the proof. The arXiv version omits the proof of Theorem 9 as long, stating that it follows the classical Tietäväinen–van Lint argument [LX13].
A hypothetical code yields a purely arithmetic consequence. With \(B_2(n,Q):=1+n(Q-1)+\binom n2(Q-1)^2\), the integers \(Q:=q^2\) and \(M:=Kq^n\) satisfy \(B_2(n,Q)=V_2(n,q)\) and hence, by Eq. (3),

\begin{equation} M\,B_2(n,Q)=Q^n , \tag{5} \end{equation}

the classical sphere-packing equation for a perfect two-error-correcting code of \(M\) words of length \(n\) over \(Q\) symbols. This deduction does not produce a classical code, so a classical nonexistence theorem transfers automatically only when its proof uses Eq. (5) alone. Cazorla García studies exactly this Diophantine equation in Sections 2 and 3 [Caz24].
Lemma 3.1 and Table 2 of the arXiv version of that paper list all solutions of Eq. (5) with \(n\geq5\) for every non-prime-power \(Q\geq6\) with either \(Q\leq200\) and \(Q\notin\{94,166\}\), or \(Q\leq600\) and all prime factors of \(Q\) in \(\{2,3,5,7,11\}\); solutions occur only for \(Q\in\{15,21,46\}\). For every non-prime-power \(q\leq24\) the square \(Q=q^2\) lies in this range and is not in that set, and \(n\geq9\) by Eq. (4). The transfer therefore excludes pure perfect two-error-correcting quantum codes for

\begin{equation} q\in\{6,10,12,14,15,18,20,21,22,24\}, \tag{6} \end{equation}

which are all non-prime-power integers up to \(24\). This list is a deduction from the cited lemma, not a quantum theorem stated in the source. It uses only the lemma, not the paper’s classical nonexistence theorem, whose proof also invokes Lloyd’s theorem [Caz24].
Bennett proves that no classical perfect two-error-correcting code exists over a non-prime-power alphabet whose largest prime factor is at most \(13\) (Theorem 4). For alphabet size \(2^\alpha p^\beta\) with \(p\) an odd prime and integers \(\alpha,\beta\geq1\), such a code requires \(\alpha>20\), \(\beta\leq2519\), \(p>10^{10}\), and \(p\equiv3\pmod 8\) (Theorem 2). Both proofs combine the sphere-packing condition with integer roots of the classical Lloyd polynomial, so they do not transfer to quantum codes through Eq. (5) alone, and the preprint does not treat quantum codes [Ben26]. A subsequent preprint characterizes classical perfect codes as exact minimizers of a quadratic discrepancy and records that existence of perfect two-error-correcting codes over non-prime-power alphabets remains open; the characterization is not an existence proof [Zab26].

Comment

No construction and no nonexistence proof covering all non-prime-power local dimensions was located. The exclusions above cover only the dimensions in Eq. (6), so the smallest non-prime-power dimension they leave undecided is \(q=26\). The prime-power classification, the arithmetic exclusions, and the classical results above do not classify the quantum codes considered here. Bennett’s classical arguments use Lloyd’s theorem at alphabet size \(Q\). The quantum analogue of Li and Xing, whose Lloyd polynomial coincides with the classical one at \(Q=q^2\), is proved under the prime-power hypothesis, so extending those exclusions to non-prime-power \(q\) requires that ingredient. Arithmetic feasibility of Eqs. (4) and (5) would not by itself construct a code. Literature checked through 15 September 2026.

References

[Rai99]: E. M. Rains, “Nonbinary quantum codes,” IEEE Transactions on Information Theory 45(6), 1827–1832 (1999).DOI arXiv
[LX13]: Z. Li and L. Xing, “Classification of \(q\)-ary perfect quantum codes,” IEEE Transactions on Information Theory 59(1), 631–634 (2013).DOI arXiv
[Caz24]: P.-J. Cazorla García, “Perfect codes over non-prime power alphabets: an approach based on Diophantine equations,” Mathematics 12(11), 1642 (2024).DOI arXiv
[Ben26]: M. A. Bennett, “Perfect \(2\)-codes over arbitrary alphabets,” arXiv preprint (2026).arXiv
[Zab26]: A. L. Zabokritskiy (Yohananov), “Perfect codes as exact minimizers of quadratic discrepancy in \(q\)-ary Hamming spaces,” arXiv preprint (2026).arXiv

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