A new mathematical preprint reports a way to tell, from a code’s global discrepancy score, how badly its covering pattern fails. In every nontrivial perfect-code case covered by the analysis, the excess total quadratic ball discrepancy has an explicit positive lower bound proportional to the fixed-radius tiling defect. In the model, holes, overlaps and other multiplicity errors therefore leave a quantifiable trace in the score.
Here, a code is an N-element subset of a finite q-ary Hamming space: fixed-length strings built from an alphabet of q symbols. The analysis ranges over arbitrary such subsets, holding the alphabet size, length and cardinality fixed; it does not impose a minimum distance or an error-correction capability. A perfect code is the ideal case in which balls of the relevant radius tile the space with the required multiplicity.
The paper asks whether excess discrepancy controls the failure of perfect tiling, rather than measuring proximity to one particular code. Its use of stability refers to an explicit inequality linking the discrepancy gap with the ball-covering defect, while the stronger question of closeness to a chosen perfect code remains separate.
A score for imperfect tiling
To make the comparison, the proof places discrepancy and defect in common nonnegative Fourier coordinates and combines spectral identities with moment constraints. It then derives coercive bounds, or inequalities that link a nonzero covering defect to an excess score. The result is a proof-based comparison across code classes, not an empirical estimate.
The same framework produces a defect dictionary. It bounds holes and defective ambient points using the tiling defect, and connects normalized defect quantities with chi-square and total variation under uniform ball noise. The estimates also cover overlaps and translate into Rényi-divergence bounds, giving several ways to read the same mathematical shortfall.
The one-error case is the cleanest
The clearest result comes with one-error parameters. At admissible parameters, every N-word code obeys a lower bound with coefficient one. If no perfect code exists for those parameters, the gap above the benchmark is at least 4/N², and the coefficient-one bound is uniformly sharp.
A separate one-error certificate keeps the parameter dependence visible. It sandwiches the discrepancy excess between a lower coefficient of κ_{n,q}/q² and an upper coefficient of ρ⁽¹⁾_{n,q}, both applied to the normalized defect; the paper also proves κ_{n,q}≥q². These are certified comparison coefficients, not claims that the constants are optimal, and finding optimal parameter-specific values remains open.
Two errors bring arithmetic conditions
For two errors, the result for q≥4 and n≥5 requires the stated divisibility condition and two integral Lloyd-root conditions. It supplies a positive lower coefficient, written ϑ_{n,q}, and a no-perfect gap with numerator 4ϑ_{n,q} over N². Fixed-q asymptotic formulas describe the coefficient’s behavior, but the theorem does not assert that the arithmetic parameters or a perfect code actually exist.
The paper completes the two-error comparison across the alphabet sizes allowed by its hypotheses. For q=2, the arithmetic reduction leaves the binary length-five case and gives coefficient one; for q=3, it leaves the ternary length-eleven case and gives coefficient 274/9. For q≥4, the coefficient is ϑ_{n,q}.
An exact bounded search illustrates how restrictive those conditions are, without settling the question outside the search. Over 2≤q≤300 and 5≤n≤4000, it found both arithmetic conditions only at (2,5) and (3,11). Among 123 q≥4 root pairs in that range, none also met the divisibility condition. The bounded search does not rule out admissible parameters beyond those limits.
Finite certificates sharpen the picture
One special result covers every two-word binary code of length 2e+1. Such codes satisfy the repetition-code stability inequality, and coefficient one is sharp for every odd length. The certificate is specific to the two-word family rather than a general theorem for all code sizes.
For the ternary Golay parameters, the certificate uses coefficient 274/9 and benchmark 7 446 692/311; every perfect two-error-correcting code in that setting attains the benchmark. The binary Golay certificate uses coefficient 4136/25 and benchmark 3 277 860 456/2^23; every perfect three-error-correcting code there attains its benchmark.
What the result does not establish
The result has a deliberately narrower meaning than a code-reconstruction theorem. It controls the ball-multiplicity profile—the number of fixed-radius balls covering each ambient point—not the symmetric-difference distance between a near-minimizer and a particular perfect code. Whether near-minimal discrepancy forces that stronger kind of closeness is left open.
Other caveats are built into the arithmetic. In the q≥4 two-error branch, the formal benchmark may not be realizable by a code; the displayed fixed-alphabet asymptotics do not establish infinitely many admissible parameter pairs; and certified coefficients may be smaller than the true optimum. The paper also leaves open whether analogous certificates can be obtained for larger error parameters or other association schemes.
The document is an arXiv preprint, version 1, dated 26 August 2026. Its acknowledgment reports assistive use of OpenAI ChatGPT and OpenAI Codex, says the authors verified the manuscript, and points to an exact-arithmetic verifier archived on Zenodo, version 1.0.3. No funding statement is reported in the supplied text.
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
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