A new mathematical preprint says that when a code’s total quadratic discrepancy is close to its lowest possible benchmark, its correction-radius balls must cover the surrounding Hamming space more evenly. The paper calls this a quantitative form of tiling stability: excess discrepancy gives an explicit positive bound on the squared deviation of the covering multiplicity from one, for every nontrivial perfect-code parameter set covered by the theorem.
In plain terms, the result connects a global score with a local picture of coverage. A multiplicity of one means each ambient point is covered exactly once by a correction-radius ball. The theorem says that competitors with the same prescribed number of codewords cannot match the benchmark while allowing arbitrarily uneven coverage, within the finite settings and arithmetic conditions established in the manuscript.
A bound over every code of the required size
The analysis quantifies over every code of a prescribed cardinality. The competitors are arbitrary codes of that size; they are not required to share a perfect code’s minimum distance or to be linear. The comparison is made either with a same-cardinality perfect code when one exists or with a formal algebraic benchmark in conditional two-error cases.
The authors reach the result with spectral expansions, moment identities and a pointwise minorant, then compare the resulting discrepancy with a representation of the tiling defect based on the Lloyd polynomial. Those tools turn the difference between the benchmark and a competitor into a certified lower bound.
The strength of the certificate depends on the error-correction setting. For one-error parameters, the comparison can use a uniform coefficient of one, and the paper describes that coefficient as uniformly sharp. Parameter-specific certificates can be much larger. In the two-error branch, a positive coefficient is available under the stated arithmetic conditions even before an actual perfect code is known to exist.
Stronger certificates in selected cases
The preprint also gives more detailed results for selected parameter families. For alphabet sizes of at least four, it reports a closed form for the two-error coefficient together with asymptotic behavior as the code length grows while the alphabet is held fixed. At the ternary Golay parameters, the reported minimum of the relevant ratio is 274/9.
Other listed examples include a coefficient of one for binary repetition codes with error radius at least three, and a coefficient of 4136/25 for the binary Golay code. These are parameter-specific certificates.
The large-alphabet result comes with an important qualification. For alphabet sizes of at least four, the manuscript says that no nontrivial perfect two-error code is currently known, while some non-prime-power cases remain open under the paper’s restrictions. The formal benchmark therefore supports a conditional stability statement, not a general existence result.
What the measure captures
The authors interpret the multiplicity defect as an error in smoothing by uniform noise over Hamming balls. From the same stability inequalities, they derive bounds involving holes, overlaps, the fraction of uncovered ambient points, total variation and order-two Rényi divergence. In this reading, a small defect means the code’s ball-noise output is closer to uniform in the quantified sense.
The conclusion is narrower than saying that a low-discrepancy code is close to a particular perfect code. The theorem controls the covering multiplicities, not the symmetric difference between a competitor and a chosen code.
The result is conditional on the finite Hamming-space and arithmetic settings covered by the theorem. A separate open question is whether a small multiplicity defect forces geometric proximity to some perfect code.
A preprint with an exact-arithmetic check
The supplied manuscript is an arXiv version 2 preprint dated 27 September 2026. It reports no funding source. Its acknowledgment discloses assistive use of OpenAI ChatGPT and OpenAI Codex.
The acknowledgment also reports an exact-arithmetic verifier archived on Zenodo, version 1.0.3, along with quantitative finite-certificate appendices intended to reproduce the specified calculations.
Paper data and sources
Original title: Quantitative tiling stability from quadratic discrepancy in Hamming spaces
Authors: Valery, Grishin, Aryeh Lev Zabokritskiy
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text