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arXiv:2610.30088

The strengthened quantum Hamming bound of Yu, Lai, and Oh

Valery, Grishin, Aryeh Lev Zabokritskiy

quant-phSubmitted Oct 21, 2026• Announced Oct 22, 2026• 0 citations

v1 · Oct 21, 2026 v2 · Oct 22, 2026

Abstract

We prove a conjecture of Yu, Lai, and Oh: their rounded family of strengthened quantum Hamming bounds, which interpolates between Hamming-type bounds and the Singleton bound, holds for arbitrary exact quantum subspaces of dimension $K>1$. The result covers every integer local dimension $p\ge2$, every integer $t\ge1$, both distances $2t+1$ and $2t+2$, and every interpolation parameter $e\in\{0,\ldots,t\}$. Yu, Lai, and Oh proved these bounds for pure codes and for general codes of distance three or four; the case $e=0$ contains the ordinary quantum Hamming bound for general codes, recently proved by Zhang and Chen. More generally, for $p\ge3$, we prove the inequality $Kf(0)\le p^n\mathbb{E}f(X)$, where $X\sim\operatorname{Bin}(n,1-p^{-2})$, for every real polynomial $f$ of degree at most $D$, with $2\le D<d$, that is nonnegative on $\{0,\ldots,n\}$ and positive at zero; here $n$ and $d$ are the code length and distance.

Comments: 106 pages; computational certificates and reproduction code: https://doi.org/10.5281/zenodo.23535642

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We prove a conjecture of Yu, Lai, and Oh: their rounded family of strengthened quantum Hamming bounds, which interpolates between Hamming-type bounds and the Singleton bound, holds for arbitrary exact quantum subspaces of dimension

K>1

. The result covers every integer local dimension

p\ge2

, every integer

t\ge1

, both distances

2t+1

and

2t+2

, and every interpolation parameter

e\in\{0,\ldots,t\}

. Yu, Lai, and Oh proved these bounds for pure codes and for general codes of distance three or four; the case

e=0

contains the ordinary quantum Hamming bound for general codes, recently proved by Zhang and Chen. More generally, for

p\ge3

, we prove the inequality

Kf(0)\le p^n\mathbb{E}f(X)

, where

X\sim\operatorname{Bin}(n,1-p^{-2})

, for every real polynomial

f

of degree at most

D

, with

2\le D<d

, that is nonnegative on

\{0,\ldots,n\}

and positive at zero; here

n

and

d

are the code length and distance.

The quantum Hamming bound arises from a direct dimension estimate for exact quantum error-correcting codes. For nondegenerate codes, distinct correctable errors map the code space into orthogonal subspaces, so dimension counting directly gives the bound; degeneracy allows these subspaces to overlap and invalidates the argument. Although linear-programming, asymptotic, and structural approaches have advanced the problem from several directions, the general local-dimension case has remained open. Here we prove the quantum Hamming bound for every local dimension

q\ge3

. Building on the exact two-center Fourier reduction, we develop new overlap estimates for nonbinary Hamming spaces. For

q\ge4

, Jacobi estimates and monotonicity give a uniform gap; qutrits form the boundary case where the gap vanishes and are handled by a filtered correction. This completes the finite-length nonbinary case; together with the existing qubit result, it establishes the quantum Hamming bound for arbitrary local dimension.

We report two analytical bounds for quantum error-correcting codes that do not have preexisting classical counterparts. Firstly the quantum Hamming and Singleton bounds are combined into a single tighter bound, and then the combined bound is further strengthened via the well-known Lloyd's theorem in classical coding theory, which claims that perfect codes, codes attaining the Hamming bound, do not exist if the Lloyd's polynomial has some non-integer zeros. Our bound characterizes quantitatively the improvement over the Hamming bound via the non-integerness of the zeros of the Lloyd's polynomial. In the case of 1-error correcting codes our bound holds true for impure codes as well, which we conjecture to be always true, and for stabilizer codes there is a 1-logical-qudit improvement for an infinite family of lengths.

The quantum Hamming bound was originally put forward as an upper bound on the parameters of nondegenerate quantum codes, but over the past few decades much work has been done to show that many degenerate quantum codes must also obey this bound. In this paper, we show that there is a Hamming-like bound stricter than the quantum Hamming bound that applies to degenerate

t

-error-correcting stabilizer codes of length greater than some positive integer

N(t)

. We show that this bound holds for all single-error-correcting degenerate stabilizer codes, forcing all but a handful of optimal distance-3 stabilizer codes to be nondegenerate.

Hybrid codes simultaneously encode both quantum and classical information into physical qubits. We give several general results about hybrid codes, most notably that the quantum codes comprising a genuine hybrid code must be impure and that hybrid codes can always detect more errors than comparable quantum codes. We also introduce the weight enumerators for general hybrid codes, which we then use to derive linear programming bounds. Finally, inspired by the construction of some families of nonadditive codes, we construct several infinite families of genuine hybrid codes with minimum distance two and three.

Abstract — Optimal quantum stabilizer codes of distance 3 are explicitly constructed for all lengths except for the following four families of lengths 8fm − {1, 2} and fm+2 − {2, 3} with fm = 4 m −1 3 and m ≥ 2 being integer, for which our codes are of the best parameters known and are only one logical qubit less than the quantum Hamming bound. The optimality of our codes is ensured by saturating either the quantum Hamming bound or a stronger bound for three families of lengths 8fm + {1, 2} and fm+2 − 1 with m ≥ 1 derived from the linear programming bound. For the lengths less than 128 three previously unknown codes [[36, 29, 3]], [[37, 30, 3]] and [[81, 73, 3]] have been found. Index Terms — quantum error correction, 1-error correcting stabilizer codes, quantum Hamming bound, linear programming bound, optimal codes I.